Models and Simulation Techniques from Stochastic Geometry J I

Home , Ising model, Poisson point process

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JJ II Models and simulation techniques from stochastic geometry J I

Page 1 of 163 Wilfrid S. Kendall Department of Statistics, University of Warwick Go Back A course of 5 × 75 minute lectures at the Madison Probability Intern Program, June-July 2003 Full Screen

Quit Introduction Home Page Stochastic geometry is the study of random patterns, whether of points, line seg- ments, or objects. Title Page Related reading: Stoyan et al. (1995); Contents Barndorff-Nielsen et al. (1999);

JJ II Møller and Waagepetersen (2003) and the recent expository essays edited by Møller (2003);

J I The Stochastic Geometry and Statistical Applications section of Advances in Applied Probability.

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J I 1 A rapid tour of point processes 7

Page 5 of 163 2 Simulation for point processes 45

3 Perfect simulation 75 Go Back 4 Deformable templates and vascular trees 103

Full Screen 5 Multiresolution Ising models 123

Close A Notes on the proofs in Chapter 5 145

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Contents Chapter 1

JJ II A rapid tour of point processes J I

Page 7 of 163 Contents Go Back 1.1 The Poisson point process ...... 8 1.2 Boolean models ...... 23 Full Screen 1.3 General deﬁnitions and concepts ...... 27 1.4 Markov point processes ...... 33 Close 1.5 Further issues ...... 40

Quit 1.1. The Poisson point process Home Page The simplest example of a random point pattern is the Poisson point process, which models “completely random” distribution of points in the plane R2, or 3-space R3, Title Page or other spaces. It is a basic building block in stochastic geometry and elsewhere, with many symmetries which typically reduce calculations to computations of area

Contents or volume.

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Figure 1.1: Which of these point clouds exhibits interaction? Quit Home Page Deﬁnition 1.1 Let X be a random point process in a window W ⊆ Rn deﬁned as a collection of random variables X(A), where A runs through Title Page the Borel subsets of W ⊆ Rn. The point process X is Poisson of intensity λ > 0 if the following are true: Contents 1. X(A) is a Poisson random variable of mean λ Leb(A);

2. X(A1),..., X(An) are independent for disjoint A1,..., An. JJ II Examples: J I stars in the night sky; Page 9 of 163

Go Back patterns used to deﬁne collections of random discs;

Full Screen ... or space-time structures;

Close Brownian excursions indexed by local time(!). Quit • In dimension 1 we can construct X using a Markov chain (“number of points Home Page in [0, t]”) and simulate it using Exponential random variables for inter-point spacings. Higher dimensions are trickier! See Theorem 1.5 below.

n Title Page • We can replace λ Leb (and W ⊆ R ) by a nonnegative σ-ﬁnite measure µ on a measurable space X , in which case X is an inhomogeneous Poisson point process with intensity measure µ. We should also require µ to be Contents diffuse (P [µ({x}) = 0] = 1 for every point {x}) in order to avoid the point process faux pas of placing than one point onto a single location. JJ II

Exercise 1.2 Show that any measurable map will carry one (perhaps J I inhomogeneous) Poisson point process into another, so long as – the image of the intensity measure is a diffuse measure; Page 10 of 163 – and we note that the image Poisson point process is degenerate if the image measure fails to be σ-ﬁnite. Go Back

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Exercise 1.3 Show that in particular we can superpose two indepen- Close dent Poisson point processes to obtain a third, since the sum of independent Poisson random variables is itself Poisson.

Quit • It is a theorem that the second, independent scattering, requirement of Deﬁ- Home Page nition 1.1 is redundant! In fact even the ﬁrst requirement is over-elaborate:

Title Page Theorem 1.4 Suppose a random point process X satisﬁes

Contents P [X(A) = 0] = exp (−µ(A)) for all measurable A. Then X is an (inhomogeneous) Poisson point process of intensity mea- JJ II sure µ.

J I – This can be proved directly. I ﬁnd it more satisfying to view the re- Page 11 of 163 sult as a consequence of Choquet capacity theory, since this approach extends to cover a theory of random closed sets (Kendall 1974). Com- pare inclusion-exclusion arguments and “correlation functions” used Go Back in the lattice case in statistical mechanics (Minlos 1999). – We don’t have to check every measurable set:1 it is enough to consider Full Screen sets A running through the ﬁnite unions of rectangles! – But we cannot be too constrained in our choice of various A (the family Close of convex A is not enough).

1 Quit Certainly we need check only the BOUNDED measurable sets. • An alternative approach2 lends itself to simulation methods for Poisson point Home Page processes in bounded windows: condition on the total number of points.

Title Page Theorem 1.5 Suppose the window W ⊆ Rn has ﬁnite area. The following constructs X as a Poisson point process of ﬁnite intensity λ: Contents – X(W ) is Poisson of mean λ Leb(W ); – Conditional on X(W ) = n, the random variables X(A) are JJ II constructed as point-counts of a pattern of n independent points U1,..., Un uniformly distributed over W . So J I X(A) = # {i : Ui ∈ A} .

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Exercise 1.6 Generalize Theorem 1.5 to (inhomogeneous) Poisson Full Screen point processes and to the case of W of inﬁnite area/volume.

2The construction described in Theorem 1.5 is the same as the one producing the link via condi- Quit tioning between Poisson and multinomial contingency table models. Home Page Exercise 1.7 Independent thinning. Suppose each point ξ of X, produced as in Theorem 1.5, is deleted independently of all other points Title Page with probability 1 − p(ξ) (so p(ξ) is the retention probability). Show that the resulting thinned random point pattern is an inhomogeneous Poisson point process with intensity measure p(u) Leb( du). Contents

JJ II Exercise 1.8 Formulate the generalization of the above exercise J I which uses position-dependent thinning on inhomogeneous Poisson point processes.

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Quit • The process produced by conditioning on X(W ) = n has a claim to being Home Page even more primitive than the Poisson point process.

Title Page Deﬁnition 1.9 A Binomial point process with n points deﬁned on a window W is produced by scattering n points independently and uni- Contents formly over W .

JJ II However you should notice

J I – point counts in disjoint subsets are now (perhaps weakly) negatively correlated: the more in one region, the fewer can fall in another;

Page 14 of 163 – the total number n of points is fixed, so we cannot extend the notion of a (homogeneous) Binomial point process to windows of infinite measure; Go Back – if you like that sort of thing, the Binomial point process is the simplest representative of a class of point processes (with finite total number of Full Screen points) analogous to canonical ensembles in statistical physics. The Poisson point process relates similarly to grand canonical ensembles Close (grand, because points are indistinguishable and we randomize over the total number of particles). Of course these are trivial cases because there is no interaction between points — but more of that later! Quit Here’s a worry.3 It is easy to prove that the construction of Theorem 1.5 produces Home Page point count random variables which satisfy Definition 1.1 and therefore delivers a Poisson point process. But what about the other way round? Can we travel from Poisson point counts to a random point pattern? Or is there some subtle concealed Title Page random behaviour in Theorem 1.5 going beyond the labelling of points, which cannot be extracted from a point count specification? Contents The answer is that the construction and the point count definition are equivalent.

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Close 3Perhaps you think this is more properly a neurosis to be dealt with by personal counselling?! Consider that a neurotic concern here is what allows us to introduce measure-theoretic foundations Quit which are most valuable in further theory. Home Page Theorem 1.10 Any Poisson point process given by Deﬁnition 1.1 induces a unique probability measure on the space of locally ﬁnite point patterns, Title Page when this is furnished with the σ-algebra generated by point counts.

Contents Here is a sketch of the proof. Take W = [0, 1)2. For each n = 1, 2, . . . , make a dyadic dissection of [0, 1)2 into JJ II 4n different sub-rectangles [k2−n, (k + 1)2−n) × [`2−n, (` + 1)2−n). For a given level n, visualize the pattern Pn obtained by painting sub-rectangles black if they J I are occupied by points, white otherwise. The point counts for each of these 4n sub-rectangles are independent Poisson of mean 4−nλ: by subadditivity of proba-

Page 16 of 163 bility:

n −n −n P [one or more point counts > 1] ≤ 4 (1 − (1 + 4 λ) exp −4 λ Go Back = 4n(4−nλ)2θ exp −4−nλθ

Full Screen for some θ ∈ (0, 1) (use the mean value theorem for x 7→ 1 − (1 + x) exp(−x)), which is summable in n. So by Borel-Cantelli we deduce that for all large enough n the black sub-rectangles in pattern Pn contain just one Poisson point each. Or- Close dering these points randomly yields the construction in Theorem 1.5.

Quit Simulation techniques for Poisson point processes are rather direct and straight- Home Page forward: here are two possibilities. 1. We can use Theorem 1.5 to reduce the problem to the algorithm Title Page • draw N = n from a Poisson random variable distribution; • draw n independent values from a Uniform distribution over the re- Contents quired window.

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def poisson process (region ← W, intensity ← alpha): J I pattern ← [] for i ∈ range (poisson (alpha)): Page 17 of 163 pattern.append (uniform (region ← W )) return pattern

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Quit 2. Quine and Watson (1984) describe a generalization of the 1-dimensional Home Page “Markov chain” method mentioned above. Suppose we wish to simulate a Poisson point process of intensity λ in ball(o,R). We describe the two- dimensional case; the generalization to higher dimensions is straightfor- Title Page ward. 4

Contents Using polar coordinates and power transform of radius, produce a measure-preserving map

2 JJ II (0,R /2] × (0, 2π] → ball(o,R) √ (1.1) (s, θ) 7→ ( 2s, θ) J I √ where ( 2s, θ) is expressed in polar coordinates, and [0,R2/2] × [0, 2π] and the ball(o,R) are endowed with the measure λ Leb. Be- Page 18 of 163 ing measure-preserving, the map preserves point processes. But the following Exercise 1.11 shows a Poisson(λ) point process on Go Back the product space [0,R2/2] × [0, 2π] can be thought of as a 1- dimensional Poisson(2λπ) point process of “half-squared radii” on [0,R2/2] (which may be constructed using a sequence of independent Full Screen Exponential(2λπ) random variables), each point of which is associated with an “angle” which is uniformly distributed on [0, 2π]. Close

4The Quine-Watson method is particularly suited to the construction of Poisson point processes Quit which are to be used to build geometric tessellations . . . Home Page Exercise 1.11 Suppose X is a Poisson point process of intensity λ on the rectangle [a, b] × [0, 1]. Show that it may be realized as a Title Page one-dimensional Poisson point process on [a, b] of intensity λ, en- riched by attaching to each one-dimensional point an independent Uniform([0, 1]) random variable to provide the second coordinate. Contents

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J I Simply suppose the construction is given by Theorem 1.5, show the ﬁrst coordinate of each point provides the required one-dimensional Poisson point Page 19 of 163 process, note the second coordinate is uniformly distributed as required!

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Close Exercise 1.12 Why is it irrelevant to this proof that the map image omits the centre o of ball(o,R)?

Quit Home Page Exercise 1.13 Check the details of modifying the Quine-Watson construction for higher dimensions, to show how to build a d-dimensional Title Page Poisson point process using independent sequences of Exponential random variables and random variables uniformly distributed on Sd−1. Contents

JJ II Exercise 1.14 Consider the following method of parametrizing lines: J I measure s the perpendicular distance from the origin and θ the angle made by the perpendicular to a reference line. How should you mod- ify the Quine-Watson construction to produce a Poisson line process Page 20 of 163 which is invariant under isometries of the plane? What about invariant Poisson hyperplane processes? Go Back

Full Screen What do large cells look like? In the early 1940s D.G. Kendall con- jectured a large cell should approximate a disk. We now know this is true, and more besides! (Miles 1995; Kovalenko 1997; Kovalenko Close 1999; Hug et al. 2003)

Quit Given a Poisson point process, we may construct a locally ﬁnite point pattern Home Page using the methods of Theorem 1.10, and enumerate the points of the pattern using some arbitrary rule. So we may speak of summing h(ξ) over the pattern points ξ; by Fubini the result will not depend on the arbitrary ordering so long as h is Title Page nonnegative.

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Theorem 1.15 If X is a Poisson point process of intensity λ and if h(x, φ) JJ II is a nonnegative measurable function of a point x and a locally ﬁnite point pattern φ then J I X Z E h(ξ, X \{ξ}) = E [h(u, X)] λ Leb( du) , (1.2) d Page 21 of 163 ξ∈X R

and indeed by direct generalization Go Back X E h(ξ1, . . . , ξn,X \{ξ1, ξ2, . . . , ξn}) = Full Screen disjoint ξ1,...,ξn∈X (1.3) Z Z n Close ··· E [h(u1, . . . , un,X)] λ Leb( du1),..., Leb( dun) .

d d R ···R

Quit Home Page Exercise 1.16 Use Theorem 1.5 to prove (1.2) for h(u, X) which is positive only when u lies in a bounded window W and which depends on X only on Title Page the intersection of the pattern X with the window W . Hence deduce the full Theorem 1.15.

Contents This delivers a variation on the independent scattering property of the Poisson point process. JJ II

Corollary 1.17 If g is a nonnegative measurable function of a locally ﬁnite J I point pattern φ then

Page 22 of 163 E [g(X \{u})|u ∈ X] = E [g(X)]

Go Back hP i Use Theorem 1.15 to calculate E ξ∈X I [ξ ∈ ball(u, ε)]g(X \{ξ}) . Interpret R [g(X \{u})|u ∈ X] λ Leb( u) Full Screen the expectation as ball(u,ε) E d We say the Palm distribution for a Poisson point process is the Poisson point pro- Close cess itself with the conditioning point added in!

Quit 1.2. Boolean models Home Page

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Contents In stochastic geometry the counterpart to the Poisson point process is the Boolean model; the set-union of (possibly random) geometric ﬁgures or grains located JJ II one at each point or germ of an underlying Poisson point process. Much of the amenability of the Poisson point process is inherited by the Boolean model, though nevertheless its analysis can lead to very substantial calculations and theory. J I

Page 23 of 163 Boolean models have the advantage of being simple enough that one can carry explicit computation. They have a long application history, for example being Go Back used by Robbins (1944,1945) to study bombing problems (how heavily to bomb a beach in order to ensure a high probability of landmine-clearance . . . ). Full Screen

Close Bombing problems and coverage: refer to Hall (1988, §3.5), Molchanov (1996a); also note application of Exercise 1.14. Quit Home Page Definition 1.18 A Boolean model Ξ is defined in terms of a germ process which is a Poisson point process X of intensity λ, and a grain distribution Title Page which defines a random set5. It is a random set which is formed by the union [ (U(ξ) + ξ) , Contents ξ∈X

where the U(ξ) are independent realizations of the random set, one for each JJ II germ or point ξ of the germ process. (Notice: the grains should also be independent of the germs!) J I The classic computation for a Boolean model Ξ is to determine the probability Page 24 of 163 P [o ∈ Ξ] that the resulting random set covers a speciﬁed point (here, the origin o). This reduces to a matter of geometry: if o is covered then it must be covered by at least one grain, so just think about where the corresponding germ might lie. Go Back If U is non-random then the germ will have to lie in the reﬂection Uˆ of U in the origin, so restrict attention to germs lying in Uˆ. If U is random then generalize Full Screen h i this: thin each germ ξ of the germ process using the probability P ξ ∈ Uˆ that

5 Close The random set is often not random at all (a disk of ﬁxed radius, say) or is random in a simple parametrized manner (a disk of random radius, or a random convex polygon deﬁned as the intersection of a sequence of random half-spaces); so we will follow Robbins (1944,1945) in treating this Quit informally, rather than developing the later and profound theory of random sets (Matheron 1975)! the corresponding grain covers o. The result is an inhomogeneous Poisson point h i Home Page process of intensity measure P u ∈ Uˆ Leb( du): calculate the probability of this process containing no points at all to produce the following: Title Page

Theorem 1.19 Suppose Ξ is a Boolean model with germ intensity λ and Contents typical random grain U. Then the area / volume fraction is

P [o ∈ Ξ] = 1 − exp (−λ E [Leb(U)]) . (1.4) JJ II

J I A similar argument produces the generalization

Page 25 of 163 Theorem 1.20 Suppose Ξ is a Boolean model with germ intensity λ and typical random grain U. Let K be a compact region. Then the probability Go Back of Ξ hitting K is h i P [Ξ ∩ K 6= ∅] = 1 − exp −λ E Leb(Uˆ ⊕ K) , (1.5) Full Screen

where Uˆ ⊕ K is a Minkowski sum: Close K1 ⊕ K2 = {x1 + x2 : x1 ∈ K1, x2 ∈ K2} . (1.6)

Quit Remember, from the discussion prior to Theorem 1.15, that we may view the Home Page germs as a sequence of points (using some arbitrary method of ordering); so simulation of Ξ need only depend on generating a suitable sequence of independent random sets U1, U2,.... Title Page Harder analytical questions include: • how to estimate the probability of complete coverage? Contents • in a high-intensity Boolean model, what is the approximate statistical shape of a vacant region? JJ II • as intensity increases, how to describe the transition to inﬁnite connected J I regions of coverage? Useful further reading for these questions: Hall (1988), Meester and Roy (1996), Page 26 of 163 Molchanov (1996b).

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Quit 1.3. General deﬁnitions and concepts Home Page 1.3.1. General Point Processes

Title Page Other approaches to point processes include: • random measures (nonnegative integer-valued, charge singletons with 0, 1); Contents • measures on exponential measure spaces;

JJ II reflected in notation (a point process X can have points belonging to it (ξ ∈ X), can be used for point counts (X(E) when E is a subset of the ground space, and can be used to provide a reference measure so as to define other point processes J I via densities). Carter and Prenter (1972) establish essential equivalences. We can define general point processes as probability measures on the space of lo- Page 27 of 163 cally finite point patterns, following Theorem 1.10. This allows us immediately to define stationary point processes (the distribution respects translation symmetries, at least locally) and isotropic point processes (the same, but for rotations). The Go Back distribution of a general point process is characterized by its void probabilities as in Theorem 1.4. To each point process X (whether stationary, isotropic, or not) Full Screen there is an intensity measure given by Λ(E) = E [X(E)]. (For stationary point processes this has to be a multiple of Lebesgue measure!)

Quit We can also deﬁne the Palm distribution for X: Home Page

Deﬁnition 1.21 E [g(X)|u ∈ X] is required to solve Title Page   X f(ξ)g(X \{ξ}) = Contents E   ξ∈X∩ball(x,ε) Z f(u)E [g(X)|u ∈ X] Λ( du) . (1.7) JJ II ball(x,ε)

J I A somewhat dual notion is the conditional intensity: informally, if x is a location and φ is a locally ﬁnite point pattern not including the location x then the Page 28 of 163 conditional intensity λ(x; φ) is the inﬁnitesimal probability of a point pattern X containing x if X \{x} = φ:

Go Back [x ∈ X|X \{x} = φ] λ(x; φ) = P . (1.8) P [X \{x} = φ] Full Screen We shall not deﬁne this in rigorous generality, as its deﬁnition will be immediate

Close for the Markov point processes to which we will turn below.

Quit These considerations provide a general context for ﬁguring out constructions which Home Page move away from the “no interactions” property of Poisson point processes. 1. Poisson cluster processes (can be viewed as Boolean models in which the Title Page grains are simply ﬁnite random clusters of points). Neyman-Scott processes arise from clusters of independent points.

Contents 2. Randomize intensity measure: “doubly stochastic” or Cox process.

JJ II Exercise 1.22 If the number of points in a Neyman-Scott cluster is Poisson, then the Neyman-Scott point process is also a Cox process. J I

Page 29 of 163 Line processes present an interesting issue here! Davidson conjec- tured that a second-order stationary line process with no parallel lines is Cox. Kallenberg (1977) counterexample: Go Back Stoyan et al. (1995, Figure 8.3) provides an illustration.

Full Screen 3D variants with applications in Parkhouse and Kelly (1998).

Close 3. Or apply rejection sampling to a Poisson point process, rejection probability depending on statistics computed from the point pattern. Compare Gibbs measure formalism of statistical mechanics. Quit What sort of statistics might we use? Home Page • estimation of intensity λ by λˆ(X) = X(W )/ Leb(W ) (using sufﬁcient statistic X(W )); Title Page • measuring how close the point pattern is to a speciﬁc location using the spherical contact distribution function

Contents Hs(r) = P [dist(o,X) ≤ r] , (1.9)

estimated via a Boolean model Ξr based on balls U of ﬁxed radius r by JJ II [ Leb(Ξr ∩ W )/ Leb(W ) = Leb (U + ξ) ∩ W / Leb(W ); ξ∈X J I (1.10) • measuring how close pattern points come to each other using the nearest Page 30 of 163 neighbour distance function D(r) = P [dist(o,X) ≤ r|o ∈ X] . (1.11) Go Back Interpret conditioning on null event “o ∈ X” using Palm distributions;

Full Screen • The related reduced second moment measure 1 K(r) = [#{ξ ∈ X : dist(o, ξ) ≤ r} − 1|o ∈ X] , (1.12) λE Close estimating λK(r) by the number sr(X) of (ordered) neighbour pairs (ξ1, ξ2) in X for which dist(ξ1, ξ2) ≤ r (use Slivynak-Mecke Theorem 1.15). Quit We ignore the important issue of edge effects! Home Page

Exercise 1.23 Use Theorem 1.15 to prove that for a homogeneous Poisson Title Page point process D(r) = Hs(r) and K(r) = Leb(ball(o, r)).

Contents Here is a prototype of use of these statistics in rejection algorithms:

JJ II Exercise 1.24 Consider the following rejection simulation algorithm, not- ing that λˆ = X(W )/ Leb(W ) to connect with previous discussion: J I def reject ():

Page 31 of 163 X ← poisson process (region ← W, intensity ← lamda) while uniform (0, 1) > theta ∗ ∗(number (X)): X ← poisson process (region ← W, intensity ← lamda) Go Back return X

Show that the resulting point process is Poisson of intensity λ + θ. Full Screen

Close Use Theorem 1.5 to reduce this to a question about rejection sampling of Poisson random variables.

Quit 1.3.2. Marked Point Processes Home Page We can think of Boolean models as derived from marked Poisson point processes; to each germ point assign a random mark encoding the geometric data (such as d Title Page disk radius) of the grain. In effect the Poisson point process now lives on R × M where M is the mark space.

Contents • Poisson cluster processes as mentioned above: the grains are ﬁnite random clusters;

JJ II • segment and disk processes; • line and (hard) rod processes; J I • ﬁbre and surface processes. Compare Exercise 1.11; the intensity measure will factorize as µ×ν so long as the Page 32 of 163 marks are independent, where µ is the intensity of the basic Poisson point process and the measure ν is the distribution of the mark. Go Back But beware! it can be meaningful to consider improper mark distributions. Examples include:

Full Screen Decomposition of Brownian motion into excursions;

Close Marked planar Poisson point process for which the marks are disks of random radius R, density proportional to 1/r if r ≤ threshold. Quit 1.4. Markov point processes Home Page Consider interacting point processes generating by rejection sampling (Exercise 1.24 is simple example!). Title Page For the sake of simplicity, we suppose here that all our processes are two-dimensional. The general procedure is to use a statistic t(X) in the algorithm

Contents def reject (): X ← poisson process (region ← W, intensity ← lamda) JJ II while uniform (0, 1) > exp (theta ∗ t (X)): X ← poisson process (region ← W, intensity ← lamda) X J I return We can now use standard arguments from rejection simulation. The resulting point Page 33 of 163 process distribution has a density

constantθ × exp(θt(X)) (1.13) Go Back with respect to the Poisson point process X of intensity λ, for normalizing constant Full Screen constantθ = E [exp(θt(X))|X is Poisson(λ)] .

Quit If θt(X) > 0 is possible then the above algorithm runs into trouble (how do you Home Page reject a point pattern with probability greater than 1?!) However the probability density at Equation (1.13) can still make sense. Importance sampling could of course implement this. We can do better if the following holds: Title Page

Contents Deﬁnition 1.25 A point pattern density f(X) satisﬁes Ruelle stability if f(X) ≤ A exp(B#(X)) (1.14) JJ II for constants A > 0, B; where #(X) is the number of points in X.

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Page 34 of 163 Exercise 1.26 6Show that if the density exp(θt(X)) is Ruelle stable with constants as in 1.25 then the rejection algorithm above will produce a Go Back point pattern with the correct density if applied to a Poisson point process of density λeB instead of λ, and rejection probability is changed to exp(θt(X) − B#(X))/A instead of exp(θt(X)). Full Screen

Use t(X) = sr(X), for γ = exp(θ) ∈ (0, 1), the number of ordered pairs sepa- Close rated by at least r, to obtain the celebrated Strauss point process (Strauss 1975).

6 Quit Thanks for the comment, Tom! The statistic t(X) = Leb(Ξr ∩ W ), for γ = exp(−θ) results in a process known Home Page as the area-interaction point process (Baddeley and van Lieshout 1995), also known in statistical physics as the Widom-Rowlinson interpenetrating spheres model (Widom and Rowlinson 1970) in the case θ > 1. Title Page The (Papangelou) conditional intensity is easily deﬁned for this kind of weighted Poisson point process, which (borrowing statistical mechanics terminology) is Contents commonly known as a Gibbs point process: if the density of the Gibbs process with respect to a unit rate Poisson point process is given by f(X) then the conditional intensity at x may be deﬁned as JJ II f(φ ∪ {x} λ(x; φ) = . (1.15) J I f(φ)

Page 35 of 163 Exercise 1.27 Compute the conditional intensity for the Poisson point pro- Go Back cess of intensity λ (use its density with respect to a unit-intensity Poisson point process; see Exercise 1.24).

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Close Exercise 1.28 Compute the conditional intensity for the Strauss point process. Quit Home Page Exercise 1.29 Compute the conditional intensity for the area-interaction point process. Title Page All these conditional intensities λ(x; φ) share a striking property: they do not Contents depend on the point pattern φ outside of some neighborhood of x. This is an appealing property for a point process intended to model interactions: it is captured in Ripley and Kelly’s (1977) deﬁnition of a Markov point process: JJ II

J I Deﬁnition 1.30 A Gibbs point process is a Markov point process with respect to a neighbourhood relation ∼ if its density is everywhere positive7, and its conditional intensity λ(x; φ) depends only on x and {y ∈ φ : y ∼ x}. Page 36 of 163

The celebrated Hammersley-Clifford theorem tells us that the probability density Go Back for a Markov point process can always be factorized as a product of interactions, functions of ∼-cliques of points of the process. Ripley and Kelly (1977) presents Full Screen a proof for the point process case, while Baddeley and Møller (1989) provides a signiﬁcant extension. A recent treatment of Markov point processes in book form is given by Van Lieshout (2000). Close

7The positivity requirement ensures that the ratio expression for λ(x; φ) in Equation (1.15) is well- Quit defined! It can be relaxed to a hereditary property. However care must be taken to ensure they are well-defined: the danger of simply Home Page writing down a statistic t(X) and defining the density by Equation (1.13) is that the total integral E [exp(θt(X)|X is Poisson(λ)] Title Page may diverge so that the normalizing constant is not well-defined. (This is the case for the Strauss point process when γ > 1, as shown by Kelly and Ripley 1976. This Contents particular point process cannot therefore be used to model clustering — a major motivation for considering the more amenable area-interaction point process.) JJ II s (X) Exercise 1.31 Show that E γ r diverges for γ > 1 if the interaction J I radius r exceeds the maximum separation of potential points in the window W .

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Go Back Exercise 1.32 Bound exp(−θ Leb(Ξr ∩ W )) above in terms of the number of points in W (show the interaction satisﬁes Ruelle-stability), and hence Full Screen deduce that the area-interaction point process is deﬁned for all real θ.

Quit Can we bias using perimeter instead of area of Ξr in area-interaction, or even Home Page “holiness” (Euler characteristic: number-of-components minus number-of-holes)? Like area, these characteristics are local (lead to formal densities whose conditional intensities satisfy the local requirement of Theorem 1.30). The Hadwiger Title Page characterization theorem (Hadwiger 1957) more-or-less says that linear combina- tions of these three measures exhaust all local geometric possibilities! Can the Contents result be normalized?

JJ II Exercise 1.33 Use the argument of Exercise 1.32 to show that exp(θ perimeter(Ξr ∩ W )) can be normalized for all θ.

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Page 38 of 163 Exercise 1.34 Argue similarly to show that exp(θ euler(Ξr ∩ W )) can be normalized for all θ > 0. Go Back

There can be no more components than disks/grains of the Boolean model Ξr. Full Screen

Quit Home Page What about exp(θ euler(Ξr ∩ W )) for θ < 0? Key remark is Theorem 4.3 of Kendall et al. (1999): a union of

Title Page N closed disks in the plane (of whatever radii, not nec- essarily the same for each disk) can have at most 2N − 5 holes. Contents

JJ II Separate, trickier argument: disks can be replaced by J I polygons satisfying a “uniform wedge” condition.

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Full Screen There are counterexamples . . . .

Quit 1.5. Further issues Home Page Before moving on to develop machinery for simulating realizations of Markov point processes, we make brief mention of a couple of recent pieces of work. Title Page 1.5.1. Poisson point processes and set-indexed martingales Contents There are striking applications of set-indexed martingales to distributional results for the areas of adaptively random regions. Recall the notion of stopping time T JJ II for a ﬁltration {Ft}. Here is a variant on the standard deﬁnition: we require

{ω : ∆(ω) t} ∈ F where ∆(ω) = {ω : t T (ω)} . (1.16) J I t Makes sense for partially-ordered family of closed sets (using set-inclusion); hence Page 40 of 163 martingales, and Optional Sampling Theorem (Kurtz 1980). Zuyev (1999) used this to give an elegant proof of results of Møller and Zuyev (1996):

Go Back Theorem 1.35 Suppose ∆ is a compact stopping set for X a Poisson point Full Screen process of intensity α, such that

P [X(∆) = n] > 0 and does not depend on α . Close Then Leb(∆) given X(∆) is Gamma(n, α).

Quit Examples: minimal centred ball containing n points (easy: compare Exercise Home Page 1.11); regions constructed using Delaunay or Voronoi tessellations. Proof: use likelihood ratio for α (furnishing a martingale), hence

Title Page h z Leb(∆) i E e | X(∆) = n, α = α =

z Leb(∆) n −n −(α−α0) Leb(∆) Contents P e I [X(∆) = n]α α0 e | α = α0 . P [X(∆) = n | α = α0] JJ II

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(a) Delaunay circumdisk (b) Voronoi ﬂower Close Cowan et al. (2003) use this technique to consider the extent to which these exam-

Quit ples lead to decompositions as in the minimal centred ball. 1.5.2. Stationary random countable dense sets Home Page We conclude this lecture with a cautionary example! Without straying very from the theory of random point processes one can end up in the bad part of the Measure Title Page Theory town. Suppose for whatever reason one wishes to consider random point sets X which are countable but are also dense (set of directions of a line process?). Contents We need to be precise about what we can observe, so that we genuinely treat the different points of X on equal terms. So suppose we can observe any event in the hit-or-miss σ-algebra generated by [X ∩ A] 6= ∅, as A runs through the Borel sets. JJ II Suppose that X is stationary. (We thus exclude the situations considered by Aldous and Barlow (1981), where J I account is taken of how various points are constructed — in effect the dense point process is obtained by projection from a conventional locally ﬁnite point process.)

Page 42 of 163 There are two ways to deﬁne such sets:

(a) measurable subset of Ω × R; Go Back

(b) Borel-set-valued random variable, hit-or-miss measurable. Full Screen Equivalent for case of closed sets (Himmelberg 1975, Thm. 3.5). But (warning!) Close the set of countable sets is not hit-or-miss measurable, though constructive and weak countability are equivalent (section theorem).

Quit What can we say about stationary random countable dense sets? Home Page Theorem 1.36 (Kendall 2000, Thm. 4.3, specialized to Euclidean case.) If E is any hit-or-miss measurable event then either P [E] = 0 or P [E] = 1. Title Page Whether 0 or 1 depends on E but not on X!

Contents

Exercise 1.37 Compare these two stationary random countable dense sets: JJ II (a) For Y a unit-intensity Poisson point process on R×(0, ∞), the countable set obtained by projection onto the ﬁrst coordinate. J I (b) The set Q + Z, where Q is the set of rational numbers, and Z is a [0, 1] Page 43 of 163 Uniform( ) random variable. They are very different, but appear indistinguishable . . . . Go Back

This bad behaviour should not be surprising: for example Vitali’s non-measurable Full Screen set depends on there being no measurable random variable taking values in Q + Z. (Contrast our use of enumeration for locally ﬁnite point sets in Theorem 1.15.)

Close Careless thought betrays intuition!

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Contents Chapter 2

JJ II Simulation for point processes J I

Page 45 of 163 Contents Go Back 2.1 General theory ...... 48 2.2 Measure-theoretic approach ...... 51 Full Screen 2.3 MCMC for Markov point processes ...... 54 2.4 Uniqueness and convergence ...... 60 Close 2.5 The MCMC zoo ...... 61 2.6 Speed of convergence ...... 66

Quit Computations with Poisson process, Boolean model, cluster processes can be very Home Page direct, and simulation is usually direct too. But what about Markov point processes? Title Page • Simple-minded rejection procedures (§1.4) are usually infeasible.

Contents • What about accept-reject procedures sequentially in small sub-windows, thus constructing a point-pattern-valued Markov chain whose invariant dis- JJ II tribution is the desired point process?

• Or proceed to limit: add/delete single points? J I Markov chain Monte Carlo (MCMC) Gibbs’ sampler, Metropolis-Hastings! Page 46 of 163 Poisson point process as invariant distribution of spatial immigration-death

Go Back process after Preston (1977), also Ripley (1977, 1979). To deal with more general Markov point processes it is convenient ﬁrst to work

Full Screen through the general theory of detailed balance equations for Markov chains on general state-spaces. We will show how to use this to derive simulation algorithms for our example point processes above, and discuss the more general Metropolis- Close Hastings (MH) formulation after Metropolis et al. (1953) and Hastings (1970), which allows us to view MCMC as a sequence of proposals of random moves

Quit which may or may not be rejected. Simple example of detailed balance Home Page Underlying the Poisson point process example: the process X of total number of points alters as follows Title Page • birth in time interval [0, ∆t) at rate α∆t;

Contents • death in time interval [0, ∆t) at rate X∆t. Then we can solve JJ II P [X(t) = n − 1 and birth in [t, t + ∆t)] ≈ π(n − 1) × α∆t = π(n) × n∆t ≈ P [X(t) = n and death in [t, t + ∆t)] J I by α αne−α Page 47 of 163 π(n) = π(n − 1) = ... = . n n!

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Quit 2.1. General theory Home Page • Ergodic theory for Markov chains: invariant distribution X Title Page π(y) = π(x)p(x, y) x Contents or (continuous time) X π(x)q(x, y) = 0 . x JJ II • Detailed balance; J I – reversibility π(x)p(x, y) = π(y)p(y, x); – dynamic reversibility π(x)p(x, y) = π(˜y)p(˜y, x˜); Page 48 of 163

Go Back Exercise 2.1 Show detailed balance (whether reversibility or dynamical reversibility!) implies invariant distribution. Full Screen

Quit P • improper distributions: does x π(x) converge? Home Page R 1 Pn • estimate θ(y) dπ(y) using limn→∞ n m=1 θ(Ym);

Title Page Deﬁnition 2.2 Geometric ergodicity: there is R(x) > 0 with Contents (n) n kπ − p (x, ·)ktv ≤ R(x)ρ

JJ II for all n, all starting points x.

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Page 49 of 163 Deﬁnition 2.3 Uniform ergodicity: there are ρ ∈ (0, 1), R > 0 not depending on the starting point x such that

Go Back (n) n kπ − p (x, ·)ktv ≤ Rρ

Full Screen for all n, uniformly for all starting points x;

Quit 2.1.1. History Home Page • Metropolis et al. (1953) • Hastings (1970) Title Page • Simulation tests Contents • Geman and Geman (1984) • Gelfand and Smith (1990) JJ II • Geyer and Møller (1994), Green (1995)

J I • ... Deﬁnition 2.4 Generic Metropolis algorithm: Page 50 of 163 • Propose: move x → y with probability r(x, y);

Go Back • Accept: move with probability α(x, y). Required: balance π(x)r(x, y)α(x, y) = π(y)r(y, x)α(y, x). Full Screen Deﬁnition 2.5 Hastings ratio: set α(x, y) = min{Λ(x, y), 1} where

Close π(y)r(y, x) Λ(x, y) = . π(x)r(x, y)

Quit 2.2. Measure-theoretic approach Home Page We follow Tierney (1998) closely in presenting a treatment of the Hastings ratio for general discrete-time Markov chains, since in stochastic geometry one frequently Title Page deals with chains on quite complicated state spaces. Here is the key detailed balance calculation in general form: compare the “ﬂuxes”

Contents π( dx)p(x, dy) and π( dy)p(y, dx). Both are absolutely continuous with respect to their sum: set Λ(y, x) π( dx)p(x, dy) JJ II = Λ(y, x) + 1 π( dx)p(x, dy) + π( dy)p(y, dx)

J I defining a possibly infinite Λ(y, x) up to nullsets of π( dx)p(x, dy)+π( dy)p(y, dx). Now Λ(y, x) is positive off a null-set N1 of π( dx)p(x, dy) and finite off a null-set

Page 51 of 163 N2 of π( dy)p(y, dx). Moreover N1 and N2 can be chosen as transposes of each other under the symmetry (x, y) ↔ (y, x), because they are deﬁned in terms of whether Λ(y, x)/(Λ(y, x) + 1) is 0 or 1, and so we may arrange1 for Go Back Λ(y, x) Λ(x, y) + = 1 . Λ(y, x) + 1 Λ(x, y) + 1 Full Screen

1 π( x)p(x, y) + π( y)p(y, x) Quit working to null-sets of d d d d throughout! Applying simple algebra, we deduce Λ(y, x)Λ(x, y) = 1 off N1 ∪ N2. We know Home Page 0 < Λ(y, x), Λ(x, y) < ∞ off N1 ∪ N2 , so Title Page Λ(y, x) Λ(y, x) Λ(x, y) min {1, Λ(x, y)} = min {1, Λ(x, y)} × Λ(y, x) + 1 Λ(y, x) + 1 Λ(x, y) Contents Λ(x, y) = min {Λ(y, x), Λ(y, x)Λ(x, y)} (Λ(y, x) + 1)Λ(x, y) JJ II Λ(x, y) = min {Λ(y, x), 1} (2.1) Λ(x, y) + 1 J I and therefore

Page 52 of 163 min {1, Λ(x, y)} π( dx)p(x, dy) = min {1, Λ(y, x)} π( dy)p(y, dx) . The kernel min{1, Λ(x, y)}p(x, dy) may have a deﬁcit (integral over y less than Go Back one): compensate by adding an atom Z 1 − min{1, Λ(x, u)}p(x, du) δx( dy) Full Screen to obtain a reversible kernel (actually a Metropolis-Hastings kernel!) Close Z 1 − min{1, Λ(x, u)}p(x, du) δx( dy) + min{1, Λ(x, y)}p(x, dy)

Quit In case π( dx)p(x, dy) and π( dy)p(y, dx) are absolutely continuous with respect Home Page to each other, π( dy)p(y, dx) Λ(x, y) = (2.2) π( dx)p(x, dy) Title Page is exactly the Hastings ratio. Otherwise it arises via the Metropolis map L1- projection arguments of Billera and Diaconis (2001), measurable case. Contents We note in passing that the Hastings ratio (as opposed to other possible accept/reject rules) yields the largest spectral gap and minimizes the asymptotic variance of ad- JJ II ditive functionals (Peskun 1973; Tierney 1998).

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Quit 2.3. MCMC for Markov point processes Home Page We now apply the above to simulation of a point process in a bounded window, whose distribution2 has density f(ξ) with respect to a Poisson point process mea- Title Page sure π1( dξ) of unit intensity (Geyer and Møller 1994; Green 1995). We need to design a point-pattern-valued Markov chain whose limiting distribution will be Contents f(ξ)π1( dξ), and we will constrain the Markov chain to evolve solely by adding and deleting points. Let λ(x; ξ) = f(ξ ∪ {x})/f(ξ) be the Papangelou conditional intensity for the JJ II target point process, as deﬁned in Equation (1.15). Suppose the point process satisﬁes some condition such as local stability, namely J I λ(x; ξ) ≤ γ (2.3)

Page 54 of 163 for some positive constant γ. We can weaken this: we need require only that Z Go Back λ(u; ξ) du < ∞ . W

Full Screen However the stronger condition is useful later, in CFTP, and implies Ruelle stability. Close 2Here we use the measurable space of locally ﬁnite point patterns, introduced in Theorem 1.10. Notice that we are dealing with a bounded window, so in fact the total number of points in the pattern Quit will always be ﬁnite. 2.3.1. Discrete time case Home Page Consider a Markov chain X which evolves as follows. Fix αn so that α0 = 0 and Z Title Page αn + αn+1 λ(x; ξ) du ≤ 1 . (2.4) W

Contents • suppose the current value of X is X = ξ with #(ξ) = n. Choose from:

– Death: with probability αn−1, choose a point y of ξ at random and JJ II delete it from ξ. – Birth: generate a new point x with sub-probability density J I αnλ(x; ξ)/(n + 1)

Page 55 of 163 and add it to ξ (in case of probability deficit, do nothing!); Arrange matters so that birth and death are exclusive (feasible by (2.4) above). Go Back It is tedious to write down the transition kernel for X: however the message of Λ(x, y) (as given in (2.2)) is that equality of probability fluxes suffices. Point

Full Screen pattern configurations are tricky (patterns are not ordered); however an equivalent task is to arrange for equality of birth and death fluxes when the target distribution is given by rejection sampling based on the construction of Theorem 1.5, recalling Close that f is a density with respect to a unit Poisson point process. Set Leb(W ) = A. We consider the exchange between ξ˜ and ξ where ξ˜ = ξ ∪ {x}, ξ = ξ˜ \{x}. ˜ Quit Suppose #(ξ) = n, so #(ξ) = n + 1. Death: the flux from ξ˜ to ξ is Home Page e−A dξ ... dξ dx α e−A dξ ... dξ dx f(ξ˜) 1 n ×(n + 1)!× n = α f(ξ˜) 1 n . (n + 1)! n + 1 n n + 1 Title Page

The factor (n + 1)! in the numerator of the left-hand side allows for the Contents (n+1)! ways of permuting ξ1, . . . , ξn, x to represent the same conﬁguration ˜ ξ. The death probability αn/(n + 1) allows for the requirement that the point chosen to die should be x and not one of ξ1, . . . , ξn. JJ II Birth: the ﬂux from ξ to ξ˜ is J I e−A dξ ... dξ α f(ξ) 1 n × n! × n λ(x; ξ) dx Page 56 of 163 n! n + 1 e−A dξ ... dξ = α f(ξ)λ(x; ξ) 1 n dx . (2.5) Go Back n n + 1

The factor n! in the numerator of the left-hand side allows for the n! ways Full Screen of permuting ξ1, . . . , ξn to represent the same conﬁguration ξ.

Close Using the formula (1.15) for the Papangelou conditional intensity, we deduce equality between these two ﬂuxes. It follows that X has the desired target dis-

Quit tribution as an invariant distribution. Home Page Exercise 2.6 Consider the special case λ(x; ξ) = β, choosing say

Title Page αn = 1/(1 + β Leb(W ))

for all n > 0. Write down the simulation algorithm. Check that the result- Contents ing kernel is reversible with respect to πβ, the distribution measure of the Poisson point process of rate β. JJ II

There is a very easy approach: recognize that there can be no spatial interaction, so J I it sufﬁces to set up detailed balance equations for the Markov chain which counts the total number of points . . . .

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Quit 2.3.2. Continuous Time case Home Page In the treatment above it is tempting to make αn as large as possible, to maximize the rate of change per time-step. However we can also consider the limiting case Title Page of a continuous-time model and reformulate the simulation in event-based terms, thus producing a spatial birth-and-death process. For small δ, set αn = (n + 1)δ for n to maintain Equation (2.4). (For larger n use smaller αn . . . .) Then Contents • suppose the current value of X is X = ξ. While #(ξ) is not too large, choose from: JJ II – Death: with probability #(ξ)δ, choose a point y of ξ at random and delete it from ξ. J I – Birth: generate a new point x with sub-probability density λ(x; ξ)δ, and add it to ξ (in case of probability deﬁcit, do nothing!); Page 58 of 163 Taking the limit as δ → 0, and reformulating as a continuous time Markov chain: Go Back • suppose the current value of X is X = ξ: • wait an Exponential(#(ξ) + R λ(x; ξ) dx) random time then Full Screen W R – Death: with probability #(ξ)/(#(ξ)+ W λ(x; ξ) dx), choose a point y ξ ξ Close of at random and delete it from ; – Birth: otherwise generate a new point x with probability density proportional to λ(x; ξ), and add it to ξ. Quit Home Page Exercise 2.7 Verify that the above algorithm produces an invariant distribution with Papangelou conditional intensity λ(x; ξ). Title Page

Contents Exercise 2.8 Determine a spatial birth-and-death process with invariant distribution which is a Strauss point process. JJ II

J I Example of Strauss process target.

Page 59 of 163 Exercise 2.9 Determine a spatial birth-and-death process with invariant distribution which is an area-interaction point process. Go Back

Note that the technique underlying these exercises suggests the following: a local Full Screen stability bound λ(x; ξ) ≤ M implies that the point process can be expressed as some complicated and dependent thinning of a Poisson point process of intensity

Close M. We shall substantiate this in Chapter 3, when we come to discuss perfect simulation for point processes.

Quit 2.4. Uniqueness and convergence Home Page We have been careful to describe the target distributions as invariant distributions rather than equilibrium distributions, because we have not yet introduced useful Title Page methods to determine whether our chains actually converge to their invariant distributions! As we have already hinted in Chapter 1, this can be a problem; indeed

Contents one can write down a Markov chain which purports to have the attractive case Strauss point process as invariant distribution, even though this point process is not well-defined! JJ II The full story here involves discussions of φ-recurrence and Harris recurrence, which we reluctantly omit for reasons of time. Fortunately there is a relatively simple technique which often works to ensure uniqueness of and convergence to J I an invariant distribution, once existence is established. Suppose we can ensure that the chain X visits the empty-pattern state infinitely often, and with finite mean Page 60 of 163 recurrence time (the empty-pattern state ∅ is an ergodic atom). Then the coupling approach to renewal theory (Lindvall 1982; Lindvall 2002) can be applied to show Go Back that the distribution of Xn converges in total variation to its invariant distribution, which therefore must be unique. This pleasantly low-tech approach foreshadows the discussion of small sets later Full Screen in this chapter, and also anticipates important techniques in perfect simulation.

Quit 2.5. The MCMC zoo Home Page The attraction of MCMC is that it allows us to produce simulation algorithms for all sorts of point processes and geometric processes. It is helpful however to have Title Page in mind a general categorization: a kind of “zoo” of different MCMC algorithms.

• Here are some important special cases of the MH sampler classiﬁed by type Contents of proposal distribution. Let the target distribution be π (could be Bayesian posterior . . . ). JJ II – Independence sampler: Choose q(x, y) = q(y) (independent of current location x) which you hope is “well-matched” to π; J I – Random walk sampler: Choose q(x, y) = q(y − x) to be a random walk increment based on current location x. Tails of target distribution Page 61 of 163 π must not be too heavy and (eg) density contours not too sharply curved (Roberts and Tweedie 1996b);

Go Back – Langevin algorithms (Also called, “Metropolis-adjusted Langevin”): Choose q(x, y) = q(y − σ2g(x)/2) where g(x) is formed using x and the “logarithmic gradient” of the density of π( dx) at x (possibly Full Screen adjusted) (σ2 is the jump variance). Behaves badly if logarithmic gradient tends to zero at inﬁnity; a condition requiring something of the Close order “tails must not be too light” (Roberts and Tweedie 1996a).

Quit • Slice sampler: task is to draw from density f(x). Here is a one-dimensional Home Page example! (But note, this is only interesting because it can be made to work in many dimensions . . . ) Suppose f unimodal. Deﬁne g0(y), g1(y) implicitly by the requirement that [g0(y), g1(y)] is the superlevel set {x : f(x) ≥ y}. Title Page

def mcmc slice move (x): Contents y ← uniform (0, f (x)) return uniform (g0 (y), g1 (y)) JJ II

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Go Back Figure 2.1: Slice sampler

Full Screen Roberts and Rosenthal (1999, Theorem 12) show rapid convergence (order of 530 iterations!) under a speciﬁc variation of log-concavity. Relates to

Close notion of “auxiliary variables”.

Quit • Simulated tempering: Home Page – Embed target distribution in a family of distributions indexed by some “temperature” parameter. For example, embed density f(x) (deﬁned Title Page over bounded region for x) in family of normalized

(f(x))1/τ Contents ; R (f(u))1/τ du

JJ II – Choose weights β0, β1, . . . , βk for selected temperatures τ0 = 1 < τ1 < . . . τk; – Now do Metropolis-Hastings for target density J I

1/τ0 βo(f(x)) = β0f(x) at level 0 Page 63 of 163 Z 1/τ1 1/τ1 β1(f(x)) / (f(u)) du at level 1 ... Go Back Z 1/τk 1/τk βk(f(x)) / (f(u)) du at level k

Full Screen where moves either propose changes in x, or propose changes in level i; (NB: choice of βi?) Close – Finally, sample chain only when it is at level 0.

Quit • Markov-deterministic hybrid: Need to sample exp(−E(q))/Z. Introduce Home Page “momenta” p and aim to draw from 1 π(p, q) = exp −E(q) − |p|2 /Z0 . Title Page 2

Fact: for Hamiltonian H(p, q) = E(q) + 1 |p|2, the dynamical system Contents 2 dq = (∂H/∂p) dt = p JJ II dp = −(∂H/∂q) dt = −(∂E/∂q)

has π(p, q) as invariant distribution (Liouville’s theorem). So alternate be- J I tween (a) evolving using dynamics and (b) drawing from Gaussian marginal for p. Discretization means (a) is not exact; can ﬁx this by use of MH rejec- Page 64 of 163 tion techniques. Cf : dynamic reversibility. Go Back

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Quit • Dynamic importance weighting (Liu, Liang, and Wong 2001): augment Home Page Markov chain X by adding an importance parameter W , replace search R for equilibrium (E [θ(Xn)] → θ(x)π( dx)) by requirement

Title Page Z lim [Wnθ(Xn,Wn)] ∝ θ(x)π( dx) . n→∞ E

Contents • Evolutionary Monte Carlo Liang and Wong (2000, 2001): use ideas of genetic algorithms (multiple samples, “exchanging” sub-components) and JJ II tempering. • and many more ideas besides . . . . J I

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Quit 2.6. Speed of convergence Home Page There is of course an issue as to how fast is the convergence to equilibrium. This can be analytically challenging (Diaconis 1988)! Title Page 2.6.1. Convergence and symmetry: card shufﬂing Contents Example: transposition random walk X on permutation group of n cards. Equi- librium occurs at strong uniform times T (Aldous and Diaconis 1987): JJ II Deﬁnition 2.10 The random stopping time T is a strong uniform time for the Markov chain X (whose equilibrium distribution is π) if

J I P [T = k, Xk = s] = πs × P [T = k] . Broder: notion of checked cards. Transpose by choosing 2 cards at random; LH, Page 66 of 163 RH. Check card pointed to by LH if either LH, RH are the same unchecked card; Go Back or LH is unchecked, RH is checked. Inductive claim: given number of checked cards, positions in pack of checked Full Screen cards, list of values of cards; the map determining which checked card has which value is uniformly random. Set Tm to be the time till m cards checked. The n−1 Close P independent sum T = m=0 Tm is a strong uniform time by induction. How big is T ? Quit Now Tm+1−Tm is Geometrically distributed with success probability (n−m)(m+ 2 Home Page 1)/n . So mean of T is

" n−1 # n−1 X X n2 1 1 Title Page T = + ≈ 2n log n . E m n + 1 n − m m + 1 m=0 m=0

Contents Pn−1 MOREOVER: T = m=0 Tm is a discrete version of time to complete infection for simple epidemic (without recovery) in n individuals, starting with 1 infective, JJ II individuals contacting at rate n2. A classic calculation tells us T/(2n log n) has a limiting distribution. J I 1 NOTE: group representation theory: correct asymptotic is 2 n log n. Page 67 of 163 The MCMC approach to substitution decoding also produces a Markov chain moving by transpositions on a permutation group of around n = 60 symbols. However Go Back it takes much longer to reach equilibrium than order of 60 log(60) ≈ 250 transpositions. Conditioning on data can have a drastic effect on convergence rates! Full Screen

Quit 2.6.2. Convergence and small sets Home Page

The most accessible theory (as expounded for example in Meyn and Tweedie Title Page 1993) uses the notion of small sets for Markov chains. In effect, small set theory allows us to argue as if (suitably regular) Markov chains are actually discrete,

Contents even if deﬁned on very general state-spaces. In fact this discreteness can even be formalized as the existence of a latent discrete structure.

JJ II Deﬁnition 2.11 C is a small set (of order k) if there is a positive integer k, J I positive c and a probability measure µ such that for all x ∈ X and E ⊆ X (k) p (x, E) ≥ c × I [x ∈ C] µ(E) . Page 68 of 163

Go Back Markov chains can be viewed as regenerating at small sets. The notion is useful for recurrence conditions of Lyapunov type: compare low-tech “empty pattern” Full Screen method.

Close Small sets play a crucial roleˆ in notions of periodicity, recurrence, positive recurrence, geometric ergodicity, making link to discrete state-space theory (Meyn and Tweedie 1993). Quit Foster-Lyapunov recurrence on small sets (Mykland, Tierney, and Yu 1995; Home Page Roberts and Tweedie 1996a; Roberts and Tweedie 1996b; Roberts and Rosenthal 2000; Roberts and Rosenthal 1999; Roberts and Polson 1994) Title Page

Contents Theorem 2.12 (Meyn and Tweedie 1993) Positive recurrence holds if one can ﬁnd a small set C, a constant β > 0, and a non-negative function V JJ II bounded on C such that for all n > 0

E [V (Xn+1)|Xn] ≤ V (Xn) − 1 + β I [Xn ∈ C] . (2.6) J I

Page 69 of 163 Proof: Let N be the random time at which X first (re-)visits C. It suffices3 to show E [N|X0] < V (X0) + constant < ∞ (then use small-set regeneration). Go Back By iteration of (2.6), we deduce E [V (Xn)|X0] < ∞ for all n. If X0 6∈ C then (2.6) tells us n 7→ V (Xn∧N ) + n ∧ N defines a nonnegative Full Screen supermartingale (I X(n∧N) ∈ C = 0 if n < N) . Consequently

Close E [N|X0] ≤ E [V (XN ) + N|X0] ≤ V (X0) .

Contents where the last step uses Inequality (2.6) applied when I [Xn−1 ∈ C] = 1.

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Quit This idea can be extended to give bounds on convergence rates. Home Page

Theorem 2.13 (Meyn and Tweedie 1993) Geometric ergodicity holds if one Title Page can ﬁnd a small set C, positive constants λ < 1, β, and a function V ≥ 1 bounded on C such that

Contents E [V (Xn+1)|Xn] ≤ λV (Xn) + β I [Xn ∈ C] . (2.7)

JJ II Proof: Deﬁne N as in Theorem 2.12. J I Iterating (2.7), we deduce E [V (Xn)|X0] < ∞ and more speciﬁcally n∧N n 7→ V (Xn∧N )/λ Page 71 of 163 is a nonnegative supermartingale. Consequently

Go Back N E V (XN )/λ |X0 ≤ V (X0) .

Full Screen Using the facts that V ≥ 1, λ ∈ (0, 1) and Markov’s inequality we deduce n P [N > n|X0] ≤ λ V (X0) , Close which delivers the required geometric ergodicity.

Quit Bounds may not be tight, but there are many small sets for most Markov chains Home Page (not just the trivial singleton sets, either!). In fact small sets can be used to show:

Title Page Theorem 2.14 (Kendall and Montana 2002) If the Markov chain has a measurable transition density p(x, y) then the two-step density p(2)(x, y) can be expressed (non-uniquely) as a non-negative countable sum Contents

(2) X p (x, y) = fi(x)gi(y) . JJ II Hence in such cases small sets of order 2 abound!

J I This relates to the classical theory of small sets and would not surprise its origina- tors; see for example Nummelin 1984. Page 72 of 163

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Quit Proof: Home Page Key Lemma, variation of Egoroff’s Theorem: 2 Let p(x, y) be an integrable function on [0, 1] . Then we can find subsets Aε ⊂ [0, 1], increasing as ε decreases, such that Title Page 1 (a) for any fixed Aε the “L -valued function” px is uniformly continuous on 0 0 Aε: for any η > 0 we can find δ > 0 such that |x − x | < δ and x, x ∈ Aε Contents implies Z 1 |px(z) − px0 (z)| dz < η ; JJ II 0

(b) every point x in Aε is of full relative density: as u, v → 0 so J I Leb([x − u, x + v] ∩ A ) ε → 1 . Page 73 of 163 u + v

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Moreover one can discover latent discrete Markov chains representing all Full Screen such Markov chains at period 2.

Quit However one cannot get below order 2: Home Page

Theorem 2.15 (Kendall and Montana 2002) There are Markov chains Title Page which possess no small sets of order 1.

Contents Proof: Key Theorem: There exist Borel measurable subsets E ⊂ [0, 1]2 of positive area which are JJ II rectangle-free, so that if A × B ⊆ E then Leb(A × B) = 0. (Stirling’s formula, discrete approximations, Borel-Cantelli.) J I

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Close Figure 2.2: No small sets!

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Contents Chapter 3

JJ II Perfect simulation J I

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Go Back 3.1 CFTP ...... 77 3.2 Fill’s (1998) method ...... 92 Full Screen 3.3 Sundry other matters ...... 94

Quit Establishing bounds on convergence to equilibrium is important if we are to get Home Page a sense of how to do MCMC. It would be very useful if one could ﬁnesse the whole issue by modifying the MCMC algorithm so as to produce an exact draw from equilibrium, and this is precisely the idea of perfect simulation. A graphic Title Page illustration of this in the context of stochastic geometry (speciﬁcally, the dead leaves model) is to be found at Contents http://www.warwick.ac.uk/statsdept/staff/WSK/dead.html.

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Quit 3.1. CFTP Home Page There are two main approaches to perfect simulation. One is Coupling from the Past (CFTP), introduced by Propp and Wilson (1996). Many variations have Title Page arisen: we will describe the original Classic CFTP, and two variants Small-set CFTP and Dominated CFTP. Part of the fascination of this area is the interplay

Contents between theory and actual implementation: we will present illustrative simulations.

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Quit 3.1.1. Classic CFTP Home Page Consider reﬂecting random walk run From The Past (Propp and Wilson 1996): • no ﬁxed start: run all starts simultaneously (Coupling From The Past); Title Page • extend past till all starts coalesce by t = 0; Contents • 3-line proof: perfect draw from equilibrium.

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Page 78 of 163 Figure 3.1: Classic CFTP.

Go Back Implementation: - “re-use” randomness; Full Screen - sample at t = 0 not at coalescence;

Close - control coupling efﬁciently using monotonicity, boundedness; - uniformly ergodic. Quit Here is the 3-line proof, with a couple of extra lines to help explain! Home Page

Theorem 3.1 If coalescence is almost sure then CFTP delivers a sample Title Page from the equilibrium distribution of the Markov chain X corresponding to the random input-output maps F(−u,v].

Contents Proof: For each time-range [−n, ∞) use input-output maps F(−n,t] to deﬁne JJ II −n Xt = F(−n,t](0) for − n ≤ t . J I We have assumed ﬁnite coalescence time −T for F . Then

Page 79 of 163 −n −T X0 = X0 whenever − n ≤ −T ; (3.1) −n 0 L(X0 ) = L(Xn) (3.2) Go Back If X converges to an equilibrium π then

Full Screen −T −n 0 dist(L(X ), π) = lim dist(L(X ), π) = lim dist(L(Xn), π) = 0 0 n 0 n (3.3) Close (dist is total variation) hence giving the required result.

Quit Here is a simple version of perfect simulation applied to the Ising model, Home Page following Propp and Wilson (1996) (but they do something much cleverer!). Here the probability of a given ±1-valued conﬁguration S is proportional to Title Page XX exp β SxSy x∼y Contents and our recipe is, pick a site x, compute conditional probability of Sx = +1 given rest of conﬁguration S, JJ II P exp(β y:y∼x (-1) × Sy) X ρ = = exp(−2β S ) , exp(β P (1) × S ) y J I y:y∼x y y:y∼x

and set S = +1 with probability ρ, else set S = −1. Page 80 of 163 x x Boundary conditions are “free” in the simulation: for small β we can make a perfect draw from Ising model over inﬁnite lattice, by coupling in space as well as Go Back time (Kendall 1997b; Haggstr¨ om¨ and Steif 2000). We can do better for Ising: Huber (2003) shows how to convert Swendsen-Wang methods to CFTP. Full Screen

Quit 3.1.2. A little history Home Page Kolmogorov (1936): 1 Consider an inhomogeneous (ﬁnite state) Markov transition kernel Pij(s, t). Can it be represented as the kernel of a Markov chain begun in Title Page the indeﬁnite past? Yes! Apply diagonalization arguments to select a convergent subsequence from the Contents probability systems arising from progressively earlier and earlier starts:

[X(0) = k] = Q (0) JJ II P k P [X(−1) = j, X(0) = k] = Qj(−1)Pjk(−1, 0) [X(−2) = i, X(−1) = j, X(0) = k] = Q (−2)P (−2, −1)P (−1, 0) J I P i ij jk ...

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1 Quit Thanks to Thorisson (2000) for this reference . . . . 3.1.3. Dead Leaves Home Page Nearly the simplest possible example of perfect simulation.

Title Page • Dead leaves falling in the forests at Fontainebleau; • Equilibrium distribution models complex images; Contents • Computationally amenable even in classic MCMC framework;

JJ II • But can be sampled perfectly with no extra effort! (And uniformly ergodic)

How can this be done? see Kendall and Thonnes¨ (1999) J I [HINT: try to think non-anthropomorphically . . . .] Occlusion CFTP. Related work: Jeulin (1997), Lee, Mumford, and Huang (2001). Page 82 of 163

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Close Figure 3.2: Falling leaves of perfection.

Quit 3.1.4. Small-set CFTP Home Page Does CFTP depend on monotonicity? No: we can use small sets. Consider a simple Markov chain on [0, 1] with triangular kernel. Title Page

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J I Figure 3.3: Small-set CFTP.

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• Note “ﬁxed” small triangle; Go Back • this can couple Markov chain steps; Full Screen • basis of small-set CFTP (Murdoch and Green 1998).

Close Extends to more general cases (“partitioned multi-gamma sampler”).

Quit Uniformly ergodic. Home Page Exercise 3.2 Write out the details of small-set CFTP. Deduce how to represent the output of simple small-set CFTP (uses just one small set) in terms of Title Page a Geometric random variable and draws from a single kernel.

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Exercise 3.3 One small set is usually not enough - coalescence takes much JJ II too long! Work out how to use several small sets to do partitioned small-set CFTP (“partitioned multi-gamma sampler”) (Murdoch and Green 1998). J I

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Quit 3.1.5. Point patterns Home Page Recall Widom-Rowlinson model (Widom and Rowlinson 1970) from physics, “area-interaction” model in statistics (Baddeley and van Lieshout 1995). Title Page The point pattern X has density (with respect to Poisson process): exp (− log(γ) × (area of union of discs centred on X)) Contents

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Figure 3.4: Perfect area-interaction. Go Back

• Represent as two-type pattern (red crosses, blue disks); Full Screen • inspect structure to detect monotonicity;

Close • attractive case γ > 1: CFTP uses “bounded” state-space (Haggstr¨ om¨ et al. 1999) in sense of being uniformly ergodic.

Quit All cases can be dealt with using Dominated CFTP (Kendall 1998). 3.1.6. Dominated CFTP Home Page

Title Page Foss and Tweedie (Foss and Tweedie 1998) show classic CFTP is essentially equivalent to uniform ergodicity (rate of convergence to equilibrium does not depend on start-point). For classic CFTP needs vertical coalescence: every possible Contents start from time −T leads to same result at time 0.

Can we lift this uniform ergodicity requirement? JJ II

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Go Back Figure 3.5: Horizontal CFTP. Full Screen

Close CFTP almost works with horizontal coalescence: all sufﬁciently early starts from a speciﬁc location * lead to same result at time 0. But how to identify when this

Quit has happened? Recall Lindley’s equation for the GI/G/1 queue identity Home Page

Wn+1 = max{0,Wn + Sn − Xn+1} = max{0,Wn + ηn} Title Page and its development

Contents Wn+1 = max{0, ηn, ηn + ηn−1, . . . , ηn + ηn−1 + ... + η1}

leading by time-reversal to JJ II Wn+1 = max{0, η1, η1 + η2, . . . , η1 + η2 + ... + ηn}

J I and thus the steady-state expression

Page 87 of 163 W∞ = max{0, η1, η1 + η2,...} .

Loynes (1962) discovered a coupling application to queues with general in- Go Back puts, which pre-ﬁgures CFTP.

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Quit Dominated CFTP idea: Generate target chains using a dominating process Home Page for which equilibrium is known. Domination allows us to identify horizontal coalescence by checking starts from maxima given by the dominating Title Page process. Thus we can apply CFTP to Markov chains which are merely geometrically er- Contents godic (Kendall 1998; Kendall and Møller 2000; Kendall and Thonnes¨ 1999; Cai and Kendall 2002) or worse (geometric ergodicity 6= Dominated CFTP!).

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Full Screen (a) Perfect Strauss point process. (b) Perfect conditioned Boolean model.

Close Figure 3.6: Two examples of non-uniformly ergodic CFTP.

Quit Description2 of dominated CFTP (Kendall 1998; Kendall and Møller 2000) for Home Page birth-death algorithm for point process satisfying the local stability condition λ(x; ξ) ≤ γ . Title Page Original Algorithm: points x die at unit death rate, are born in pattern ξ at total R R birth rate W λ(u; ξ) du, density λ(x; ξ)/ W λ(u; ξ) du, so local birth rate λ(x; ξ). Contents Perfect modification: underlying supply of randomness is birth-and-death process with unit death rate, uniform birth rate γ per unit area. Mark points independently with (example) Uniform(0, 1) marks. Define lower and upper processes L JJ II and U as exhibiting minimal and maximal birth rates available in range of configurations between L and U: J I minimal birth rate at x = min{λ(x; ξ): L ⊆ ξ ⊆ U} ,

Page 89 of 163 maximal birth rate at x = max{λ(x; ξ): L ⊆ ξ ⊆ U} .

Use marks to do CFTP till current iteration delivers L(0) = U(0), then return Go Back the agreed conﬁguration! Example uses Poisson point patterns as marks for area- interaction, following Kendall (1997a).

Full Screen “It’s a thinning procedure, Jim, but not as we know it.” Dr McCoy, Startrek, stardate unknown. Close

2 Quit Recall remark about dependent thinning! All this suggests a general formulation: Home Page Let X be a Markov chain on X which exhibits equilibrium behaviour. Embed the state-space X in a partially ordered space (Y, ) so that X is at bottom of Y, in the sense that for any y ∈ Y, x ∈ X , Title Page y x implies y = x .

Contents We may then use Theorem 3.1 to show:

JJ II Theorem 3.4 Deﬁne a Markov chain Y on Y such that Y evolves as X after it hits X ; let Y (−u, t) be the value at t a version of Y begun at time −u, J I (a) of ﬁxed initial distribution L(Y (−T, −T )) = L(Y (0, 0)), and

Page 90 of 163 (b) obeying funnelling: if −v ≤ −u ≤ t then Y (−v, t) Y (−u, t).

Suppose coalescence occurs: P [Y (−T, 0) ∈ X ] → 1 as T → ∞. Then Go Back lim Y (−T, 0) can be used for a CFTP draw from the equilibrium of X.

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Quit So how far can we go? It looks nearly possible to deﬁne the dominating process Home Page using the “Liapunov function” V of Theorem 2.13. There is an interesting link – Rosenthal (2002) draws it even closer – nevertheless:

Title Page Existence of Liapunov function doesn’t ensure dominated CFTP! The relevant Equation (2.7) is: Contents E [V (Xn+1)|Xn] ≤ λV (Xn) + β I [Xn ∈ C] . JJ II However one can cook up perverse examples in which this supermartingale inequality is satisﬁed, but there is bad failure of the stochastic dominance required V (X) :-( J I to use for dominated CFTP ... .

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Quit 3.2. Fill’s (1998) method Home Page The alternative to CFTP is Fill’s algorithm, which at ﬁrst sight appears quite different (though we here describe the beautiful work of Fill et al. 2000, which shows Title Page they are profoundly linked). The underlying notion is that of a strong uniform time T , which we have already encountered.

Contents P [T = t, X(t) = s] = π(s) × P [T = t]

JJ II Fill’s method is best explained using “blocks” to model input-output maps representing a Markov chain. J I

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Figure 3.7: The “block” model for input-output maps. Go Back First recall that CFTP can be viewed in a curiously redundant fashion as follows: Full Screen • Draw from equilibrium X(−T ) and run forwards;

Close • continue to increase T until X(0) is coalesced; • return X(0). Quit Key observation: By construction, X(−T ) is independent of X(0) and T so . . . Home Page • Condition on a convenient X(0); • Run X backwards to a ﬁxed time −T ; Title Page • Draw blocks conditioned on the X transitions;

Contents • If coalescence then return X(−T ) else repeat.

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J I Figure 3.8: Conditioning the blocks on the reversed chain.

Page 93 of 163 See Fill (1998), Thonnes¨ (1999), Fill et al. (2000).

Go Back Exercise 3.5 Can you ﬁgure out how to do a dominated version of Fill’s method?

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Close Exercise 3.6 Produce a perfect version of the slice sampler! (Mira, Møller, and Roberts 2001)

Quit 3.3. Sundry other matters Home Page

Title Page Layered Multishift: Wilson (2000b) and further work by Corcoran and others. Contents Basic idea: how to draw simultaneously from Uniform(x, x + 1) for all x ∈ R, and to try to couple the draws? Answer: draw a uniformly random unit span integer lattice,.... 3 JJ II

J I Exercise 3.7 Figure out how to replace “Uniform(x, x + 1)” by any other location-shifted family of continuous densities. (Hint: mixtures of uniforms . . . .) Page 94 of 163

Montana used these ideas to do perfect simulation of nonlinear AR(1) in his Go Back PhD thesis.

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3 2 Quit Uniformly random lattices in dimension play a part in line process theory. Read-once randomness: Wilson (2000a). The clue is in: Home Page

Title Page Exercise 3.8 Figure out how to do CFTP, using the “input-output” blocks picture of Figure 3.7, but under the constraint that you aren’t allowed to save any blocks! Contents

JJ II Perfect simulation for Peirls’ contours (Ferrari et al. 2002). The Ising model can be reformulated in an important way by looking only at the contours J I (lines separating ±1 values). In fact these form a “non-interacting hard-core gas”, thus permitting (at least in theory) Ferrari et al. (2002) to apply their

Page 95 of 163 variant of perfect simulation (Backwards-Forwards Algorithm)! Compare also Propp and Wilson (1996).

Go Back Perfect simulated tempering (Møller and Nicholls 1999; Brooks et al. 2002).

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Quit 3.3.1. Efﬁciency Home Page Burdzy and Kendall (2000): Coupling of couplings:. . .

JJ II Suppose |pt(x1, y) − pt(x2, y)| ≈ c exp(−µ2t) while P [τ > t|X(0) = (x1, x2)] ≈ c exp(−µt) ∗ J I Let X be a coupled copy of X but begun at (x2, x1):

Full Screen ≤ P [σ > t|τ > t, X(0) = (x1, x2)] (≈ c exp(−µ0t)) Close 0 Thus µ2 ≥ µ + µ.

Quit See also Kumar and Ramesh (2001). 3.3.2. Example: perfect epidemic Home Page Context • many models: SIR, SEIR, Reed-Frost Title Page • data: knowledge about removals Contents • draw inferences about other aspects – infection times JJ II – infectivity parameter – removal rate J I – ...

Page 97 of 163 Perfect MCMC for Reed-Frost model

Go Back • based on explicit formula for size of epidemic • (O’Neill 2001; Clancy and O’Neill 2002) Full Screen

Quit Question Home Page Can we apply perfect simulation to SIR epidemic conditioned on removals? • IDEA Title Page – SIR → compartment model Contents – conditioning on removals is reminiscent of coverage conditioning for Boolean model

JJ II • consider SIR-epidemic-valued process, considered as event-driven simulation, driven by birth-and-event processes producing infection and removal events . . . J I • then use reversibility, perpetuation, crossover to derive perfect simulation Page 98 of 163 algorithm for conditioned case.

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Quit • PRO: conditioning on removals appears to make structure “monotonic” Home Page • CON: Perpetuation and nonlinear infectivity rate combine to make funnelling difﬁcult Title Page • So: attach independent streams of infection proposals to each level k = I + R to “localize” perpetuation Contents • Speed-up by cycling through:

JJ II – replace all unobserved marked removal proposals (thresholds) – try to alter all observed removal marks (perpetuate . . . ) J I – replace all infection proposals (perpetuate . . . ) (Reminiscent of H-vL-M method for perfect simulation of attractive area- Page 99 of 163 interaction point process)

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Quit Consistent theoretical framework Home Page View Poisson processes in time as 0 : 1-valued processes over “infinitesimal pixels”: Title Page infinitesimal infinitesimal X = 1 = λ × measure P pixel pixel Contents • replace thresholds: work sequentially through “infinitesimal pixels”, reject replacement if it violates conditioning JJ II • replace removal marks: can incorporate into above step • replace infection proposals: work sequentially through “infinitesimal pix- J I els” first at level 1, then level 2, then level 3 . . . (TV scan); reject replacement if it violates conditioning. Page 100 of 163 Theorem 3.9 Resulting construction is monotonic (so enables CFTP)

Go Back Proof: Establish monotonicity for Full Screen • fastest path of infection (what is observed) • slowest feasible path of infection (what is used for perpetuation) Close Quit Implementation: Home Page It works! • speed-up? Title Page • scaling? Contents • reﬁne? • generalize? JJ II Applications J I • “omnithermal” variant to produce estimate of likelihood surface

Page 101 of 163 • generalization using crossover to produce full Bayesian analysis

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Contents Chapter 4

JJ II Deformable templates and J I vascular trees Page 103 of 163

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Full Screen 4.1 MCMC and statistical image analysis ...... 104 4.2 Deformable templates ...... 106 Close 4.3 Vascular systems ...... 107

Quit 4.1. MCMC and statistical image analysis Home Page How to use MCMC on images? Huge literature, beginning with Geman and Ge- man (1984). Title Page Simple example: reconstruct white/black ±1 image: observation Xij = ±1, “true image” Sij = ±1, X|S independent Contents exp (φS x ) [X = x |S ] = 2 ij ij P ij ij ij cosh(φ) JJ II while prior for S is Ising model, probability mass function proportional to   J I X exp β SijSi0j0  (i,j)∼(i0,j0) Page 104 of 163 Posterior for S|X proportional to Go Back   X X exp β SijSi0j0 + φ XijSij Full Screen (i,j)∼(i0,j0) ij

(Ising model with inhomogeneous external magnetic ﬁeld φX), and we can sample Close from this using MCMC algorithms. Easy CFTP for this example Quit “Low-level image analysis”: contrast “high-level image analysis”. 2 Home Page For example, pixel array {(i, j)} representing small squares in a window [0, 1] , and replace S by a Boolean model (Baddeley and van Lieshout 1993).

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Figure 4.1: Scheme for high-level image analysis. Go Back van Lieshout and Baddeley (2001), Loizeaux and McKeague (2001) provide per- Full Screen fect examples.

Quit 4.2. Deformable templates Home Page Grenander and Miller (1994): to answer “how can knowledge in complex systems be represented? use “deformable templates” (image algebras). Title Page • conﬁguration space;

Contents • conﬁgurations (prior probability measure from interactions) → image; • groups of similarities allow deformations; JJ II • template is representation within similarity equivalence class;

J I • “jump-diffusion”: analogue of Metropolis-Hastings. Besag makes link in discussion of Grenander and Miller (1994), suggesting discrete-time Langevin- Metropolis-Hastings moves. Page 106 of 163 Expect to need long MCMC runs; Grenander and Miller (1994) use SIMD tech- Go Back nology.

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Quit 4.3. Vascular systems Home Page We illustrate using an application to vascular structure in medical images (Bhalerao et al. 2001, Thonnes¨ et al. 2002, 2003). Figures: Title Page Figure 4.2(a) cerebral vascular system (application: preplanning of surgical procedures). Contents Figure 4.2(b) retinal vascular system (application: diabetic retinopathy).

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Full Screen (a) AVI movie of cerebral vascular (b) Retina vascular system: a system. 4002 pixel image. Close Figure 4.2: Examples of vascular structures.

Quit Plan: use random forests of planar (or spatial!) trees as conﬁgurations, since the Home Page underlying vascular system is nearly (but not exactly!) tree-like. Note: our tree edges are graph vertices in the Grenander-Miller formalism. Vary tree number, topologies and geometries using Metropolis-Hastings moves. Title Page Hence sample from posterior so as to partly reconstruct the vascular system. Issues: Contents • ﬁguring out effective reduction of the very large datasets which can be in- volved, how to link the model to the data, JJ II • careful formulation of the prior stochastic geometry model, • J I choice of the moves so as not to slow down the MCMC algorithm, • and use of multiscale information. Page 108 of 163

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Quit 4.3.1. Data Home Page For computational feasibility, use multiresolution Fourier Transform (Wilson, Cal- way, and Pearson 1992) based on dyadic recursive partitioning using quadtree. Title Page (Compare wavelets.)

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Go Back Figure 4.3: Multiresolution summary (planar case!).

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Quit Partitioning: based on mean-squared error criterion and the extent to which sum- Home Page marizing feature is elongated: alternative approach uses Akaike’s information criterion (Li and Bhalerao 2003). Figure 4.4(a) shows a typical feature, which in the planar case is a weighted bivariate Gaussian kernel, while Figure 4.4(b) shows Title Page the result of summarizing a 400 × 400 pixel retinal image using 1000 weighted bivariate Gaussian kernels. Contents

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(a) A typical feature (b) 1000 kernels for a retina. Close for the MFT.

Figure 4.4: Decomposition into features using quadtree. Quit This procedure can be viewed as a spatial variant of a treed model (Chipman et al. Home Page 2002), a variation on CART (Breiman et al. 1984) in which different models are ﬁtted to divisions of the data into disjoint subsets, with further divisions being made recursively until satisfactory models are obtained. Title Page Thus subsequent Bayesian inference can be viewed as conditional on the treed model inference. Contents

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Full Screen Figure 4.5: Final result of Multiresolution Fourier Transform Close

Quit 4.3.2. Likelihood Home Page Link summarized data (mixture of weighted bivariate Gaussian kernels k(x, y, e) to conﬁguration, assuming additive Gaussian noise. Title Page X X X f(x, y) = k(x, y, e) + noise → f˜(x, y) = k(x, y, e) . Contents τ∈F e∈τ e∈E

Split likelihood by rectangles Ri. JJ II N ˜ X ˜ J I `(F|f) = `i(F|f) + constant i=1 1 X h X X i2 Page 112 of 163 ` (F|f˜) = − f˜(x, y) − k(x, y, e) i 2σ2 (x,y)∈Ri τ∈F e∈τ 1 ZZ h X X i2 Go Back ≈ − f˜(x, y) − k(x, y, e) h (x, y) dx dy 2σ2 i Ri τ∈F e∈τ

Full Screen 2 Extend from Ri to all R , enabling closed form computation.

Close ZZ 2 ˜ 1 h ˜ X X i `i(F|f) ≈ − 2 f(x, y) − k(x, y, e) gi(x, y) dx dy (4.1) 2σ 2 R τ∈F e∈τ Quit 4.3.3. Prior Home Page Prior using AR(1) processes for location, width, extending along forests of trees τ produced by subcritical binary Galton-Watson processes. Title Page lengthe = α × lengthe0 + Gaussian innovation (4.2)

Contents for autoregressive parameter α ∈ (0, 1).

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(a) sub-critical binary (b) AR(1) process on (c) each edge repre- Page 113 of 163 branching process. edges. sented by a weighted Gaussian kernel.

Go Back Figure 4.6: Components of the prior.

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Quit 4.3.4. MCMC implementation Home Page We have now formulated likelihood and model. It remains to decide on a suitable MCMC process. Title Page

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Full Screen Figure 4.7: Reﬁned scheme for high-level image analysis.

Quit Changes are local (hence slow). Compute MH accept-reject probabilities using Home Page AR(1) location distributions and Y Y P [τ] = P [ node has n children ] (4.3) Title Page n=0,1,2 v∈τ:v has n children

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Close Figure 4.8: Reversible jump MCMC proposals.

Quit Computations for topological moves. Suppose family-size distribution is Home Page (1 − p)2 , 2p(1 − p) , p2

Title Page Hastings ratio for adding a speciﬁed leaf ` ∈ ∂ext(τ):

P [τ ∪ {`}] /#(∂int(τ ∪ {`}) ∪ ∂ext(τ ∪ {`})) Contents H(τ; `) = . P [τ ∪ {`}] /#(∂int(τ) ∪ ∂ext(τ))

Use induction on τ: P [τ ∪ {`}] = P [τ] p(1 − p),... JJ II 1 + #(V (τ)) + #(∂ (τ)) H(τ; `) = p(1 − p) ext J I 2 + #(V (τ)) + #(∂ext(τ))+1 where +1 is added to the denominator if ` is added to a vertex which already has Page 116 of 163 one daughter. It is clear that this Hastings ratio is always smaller than p(1 − p), and is approxi- Go Back mately equal to p(1 − p) for large trees τ. Consequently removals of leaves will always be accepted, and additions will usually be accepted, but there is a computational slowdown as the tree size increases. Full Screen

Quit Geometric ergodicity is impossible: consider second eigenvalue λ2 of the Markov Home Page kernel by Rayleigh-Ritz:

P P 2 x y |φ(x) − φ(y)| p(x, y)π(y) Title Page 1 − λ2 = inf P P 2 φ6=0 x y |φ(x) − φ(y)| π(x)π(y) P P c p(x, y)π(y) ≤ 2 x∈S y∈S Contents π(S)

whenever π(S) < 1/2. Taking S to be the set of all trees of depth n or more, and JJ II using the fact that such trees must have at least n sites, it follows 2 1 J I 1 − λ ≤ → 0 . 2 p(1 − p) n

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Quit Experiments: Home Page

MCMC for prior model Title Page First we implement only the moves relating to a single tree, and only to that tree’s topology. We change the tree Contents just one leaf at a time. Convergence is clearly very slow! moreover the animation clearly indicates the problem is JJ II to do with incipient criticality phenomena.

J I MCMC for prior model with more global changes Moves dealing with sub-trees will speed things up Page 118 of 163 greatly! We won’t use these as such – however we will be working with a whole forest of such trees, and using Go Back more “global” moves of join and divide.

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Close Conditioning on data will also help! Here is a crude par- ody of conditioning (pixels are black/white depending on whether or not they intersect the planar tree). Quit 4.3.5. Results Home Page Figure 4.9 presents snapshots (over about 50k moves) of a ﬁrst try, using accept/reject probabilities for a Metropolis-Hastings algorithm based on the above Title Page prior and likelihood.

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Full Screen Figure 4.9: AVI movie of ﬁxed-width MCMC.

Quit We improve the algorithm by adding Home Page (1) a further AR(1) process to model the width of blood vessels;

Title Page (2) a Langevin component to the Metropolis-Hasting proposal distribution, bi- asing the proposal according to the gradient of the posterior density (following Grenander and Miller 1994 and especially Besag’s remarks!); Contents (3) the possibility of proposing addition (and deletion) of features of length 2;

JJ II (4) gradual reﬁnement of feature proposals from coarse to ﬁne levels.

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Close Figure 4.10: AVI movies of improved MCMC.

Quit 4.3.6. Reﬁning the starting state Home Page We improve matters by choosing a better starting state for the algorithm, using a variant of Kruskal’s algorithm. Title Page This generates a minimal spanning tree of a graph as follows: Let G = (V,E) be a graph with edges weighted by cost. A spanning tree is a subgraph of G which is (a) a tree and (b) spans all vertices. We wish to ﬁnd a spanning tree τ of minimal Contents cost:

def kruskal (edges): JJ II edgelist ← edges.sort () # sort so cheapest edge is ﬁrst tree ← [] J I for e ∈ edgelist: if cyclefree (tree, e): # check no cycle is created Page 121 of 163 tree.append (e) return tree

Go Back Actually we need a binary tree (so amend cyclefree accordingly!) and the cost depends on the edges to which it is proposed to join e, but a heuristic modiﬁcation

Full Screen of Kruskal’s algorithm produces a binary forest which is a good candidate for a starting position for our algorithm. Figure 4.11 shows a result; notice how now the algorithm begins with many small trees, which are progressively dropped or Close joined to other trees.

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Figure 4.11: AVI movie of MCMC using starting position provided by Kruskal’s Close algorithm.

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Contents Chapter 5

JJ II Multiresolution Ising models J I

Page 123 of 163 Contents Go Back 5.1 Generalized quad-trees ...... 125 5.2 Random clusters ...... 129 Full Screen 5.3 Percolation ...... 132 5.4 Comparison ...... 139 Close 5.5 Simulation ...... 141 5.6 Future work ...... 143

Quit The vascular problem discussed above involves multi-resolution image analysis Home Page described, for example, in Wilson and Li (2002). This represents the image on different scales: the simplest example is that of a quad-tree (a tree graph in which Title Page each node has four daughters and just one parent), for which each node carries a random value 0 (coding the colour white) and 1 (coding the colour black), and the random values interact in the manner of a Markov point process with their Contents neighbours. In the vascular case the 0/1 value is replaced by a line segment summarizing the linear structure which best explains the local data at the appropriate JJ II resolution level, but for now we prefer to analyze the simpler 0/1 case.

Conventional multi-resolution models have interactions J I only running up and down the tree; however in our case we clearly need interactions between neighbours at the Page 124 of 163 same resolution level as well. This complicates the analysis! Go Back A major question is to determine the extent to which values at high resolution

Full Screen levels are correlated with values at lower resolution levels (Kendall and Wilson 2003). This is related to much recent interest in the question of phase transitions for Ising models on non-Euclidean random graphs. We shall show how geometric Close arguments lead to a schematic phase portrait.

Quit 5.1. Generalized quad-trees Home Page d Deﬁne Qd as graph whose vertices are cells of all dyadic tessellations of R , with edges connecting each cell u to its 2d neighbours, and also its parent M(u) (cov- Title Page ering cell in tessellation at next resolution down) and its 2d daughters (cells which it covers in next resolution up).

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Case d = 1: J I

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Go Back Neighbours at same level also are connected. Remark: No spatial symmetry! Full Screen However: Qd can be constructed using an iterated function system generated by + d elements of a skew-product group R ⊗s R . Close

Quit Further deﬁne: Home Page

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• Qd;r as subgraph of Qd at resolution Contents levels of r or higher;

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Page 126 of 163 • Qd(o) as subgraph formed by o and all its descendants. Go Back

Full Screen • Remark: there are many graph-isomorphisms Su;v between Qd;r and Qd;s, with natural Zd-action; Close • Remark: there are graph homomorphisms injecting Q(o) into itself, send- ing o to x ∈ (o) (semi-transitivity). Quit Q Simplistic analysis Home Page Deﬁne Jλ to be strength of neighbour interaction, Jτ to be strength of parent interaction. If Sx = ±1 then probability of conﬁguration is proportional to exp(−H) Title Page where 1 X H = − J (S S − 1) , (5.1) 2 hx,yi x y hx,yi∈E(G) Contents

for Jhx,yi = Jλ,Jτ as appropriate. If Jλ = 0 then the free Ising model on Qd(o) is a branching process (Preston JJ II 1977; Spitzer 1975); if Jτ = 0 then the Ising model on Qd(o) decomposes into sequence of d-dimensional classical (ﬁnite) Ising models. So we know there is J I a phase change at (Jλ,Jτ ) = (0, ln(5√/3)) (results from branching processes), and expect one at (Jλ,Jτ ) = (ln(1 + 2), 0+) (results from 2-dimensional Ising model). Page 127 of 163 But is this all that there is to say? Go Back

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Quit There has been a lot of work on percolation and Ising models on non-Euclidean Home Page graphs (Grimmett and Newman 1990; Newman and Wu 1990; Benjamini and Schramm 1996; Benjamini et al. 1999; Haggstr¨ om¨ and Peres 1999; Haggstr¨ om¨ et al. 1999; Lyons 2000; Haggstr¨ om¨ et al. 2002). Mostly this concerns graphs Title Page admitting substantial symmetry (unimodular, quasi-transitive), so that one can use Haggstr¨ om’s¨ remarkable Mass Transport method: Contents

Theorem 5.1 Let Γ ⊂ Aut(G) be unimodular and quasi-transitive. Given JJ II mass transport m(x, y, ω) ≥ 0 “diagonally invariant”, set M(x, y) = R Ω m(x, y, ω) dω. Then the expected total transport out of any x equals the expected total transport into x: J I X X M(x, y) = M(y, x) . Page 128 of 163 y∈Γx y∈Γx

Go Back Sadly our graphs do not have quasi-transitive symmetry :-(. Furthermore the literature mostly deals with phase transition results “for sufﬁ-

Full Screen ciently large values of the parameter”. This is an invaluable guide, but image analysis needs actual estimates!

Quit 5.2. Random clusters Home Page A similar problem, concerning Ising models on products of trees with Euclidean lattices, is treated by Newman and Wu (1990). We follow them by exploiting the Title Page celebrated Fortuin-Kasteleyn random cluster representation (Fortuin and Kaste- leyn 1972; Fortuin 1972a; Fortuin 1972b): Contents The Ising model is the marginal site process at q = 2 of a site/bond process derived from a dependent bond percolation model with conﬁguration probability Pq,p proportional to JJ II

C Y bhx,yi 1−bhx,yi q × (phx,yi) × (1 − phx,yi) . J I hx,yi∈E(G)

Page 129 of 163 (where bhx,yi indicates whether or not hx, yi is closed, and C is the number of connected clusters of vertices). Site spins are chosen to be the same in each cluster Go Back independently of other clusters with equal probabilities for ±1. We ﬁnd

phx,yi = 1 − exp(−Jhx,yi); (5.2) Full Screen

Argue for general q (Potts model): spot the Gibbs’ sampler! Close (I) given bonds, chose connected component colours independently and uniformly from 1,..., q;

Quit (II) given colouring, open bonds hx, yi independently, probability phx,yi. Home Page Exercise 5.2 Check step (II) gives correct conditional probabilities!

Title Page Now compute conditional probability that site o gets colour 1 given neighbours split into components C1,..., Cr with colour 1, and other neighbours in set C0. Contents

Exercise 5.3 Use inclusion-exclusion to show it is this: JJ II ! 1 Y [colour(o) = 1|neighbours] ∝ × q (1 − p ) + P q hx,yi J I y   ! Y Y Page 130 of 163 +  (1 − phx,yi) 1 − 1 − phx,zi

y∈C0 z∈C1∪...∪Cr Y Go Back = (1 − phx,yi) . y:colour(y)6=1

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Deduce interaction strength is Jhx,yi = − log(1 − phx,yi) as stated in Equation

Close (5.2) above.

Quit FK-comparison inequalities Home Page If q ≥ 1 and A is an increasing event then

Title Page Pq,p(A) ≤ P1,p(A) (5.3) Pq,p(A) ≥ P1,p0 (A) (5.4) Contents where 0 phx,yi phx,yi phx,yi = = . phx,yi + (1 − phx,yi)q q − (q − 1)phx,yi JJ II Since P1,p is bond percolation (bonds open or not independently of each other), we can ﬁnd out about phase transitions by studying independent bond percolation. J I

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Quit 5.3. Percolation Home Page

Independent bond percolation on products of trees with Euclidean lattices have Title Page been studied by Grimmett and Newman (1990), and these results were used in the Newman and Wu work on the Ising model. So we can make good progress by Contents studying independent bond percolation on Qd, using pτ for parental bonds, pλ for neighbour bonds.

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Theorem 5.4 There is almost surely no inﬁnite cluster in Qd;0 (and conse- J I quently in Qd(o)) if q Page 132 of 163 d −1 2 τXλ 1 + 1 − Xλ < 1 ,

Go Back where Xλ is the mean size of the percolation cluster at the origin for λ- percolation in Zd. Full Screen

Close q −1 Modelled on Grimmett and Newman (1990, §3 and §5). Get 1 + 1 − Xλ

Quit from matrix spectral asymptotics. 1

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Title Page λ

Contents ? JJ II

J I Infinite clusters here No (may or may not be unique) infinite Page 133 of 163 clusters here 0 −d Go Back 0 2 1 τ

Full Screen The story so far: small λ, small to moderate τ.

Quit Case of small τ Home Page Need d = 2 for mathematical convenience. Use Borel-Cantelli argument and planar duality to show, for supercritical λ > 1/2 (that is, supercritical with respect Title Page to planar bond percolation!), all but ﬁnitely many of the resolution layers Ln = n n [1, 2 ] × [1, 2 ] of Q2(o) have just one large cluster each of diameter larger than constant × n. Contents Hence . . .

JJ II Theorem 5.5 When λ > 1/2 and τ is positive there is one and only one inﬁnite cluster in Q2(o). J I

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Title Page Just one unique infinite cluster here λ

Contents 1/2

JJ II ? J I Infinite clusters here No (may or may not be unique) infinite Page 135 of 163 clusters here 0 Go Back 0 1/4 1 τ

Full Screen The story so far: adds small τ for case d = 2.

Quit Uniqueness of inﬁnite clusters Home Page The Grimmett and Newman (1990) work was remarkable in pointing out that as τ increases so there is a further phase change, from many to just one inﬁnite cluster Title Page for λ > 0. The work of Grimmett and Newman carries√ through for Qd(o). How- ever the relevant bound is improved by a factor of 2 if we take into account the hyperbolic structure of Qd(o)! Contents

Theorem 5.6 If τ < 2−(d−1)/2 and λ > 0 then there cannot be just one JJ II inﬁnite cluster in Qd;0.

J I

Method: sum weights of “up-paths” in Qd;0 starting, Page 136 of 163 ending at level 0. For ﬁxed s and start point there are inﬁnitely many such up-paths containing s λ-bonds; d s Go Back but no more than (1 + 2d + 2 ) which cannot be reduced by “shrinking” excursions. Hence control the mean number of open up-paths stretching more than a Full Screen given distance at level 0.

Quit Contribution to upper bound on second phase transition: Home Page p Theorem 5.7 If τ > 2/3 then the inﬁnite cluster of Q2:0 is almost surely Title Page unique for all positive λ.

Contents Method: prune bonds, branching processes, 2-dim comparison . . .

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Title Page Just one unique infinite cluster here

λ Contents

1/2 JJ II ? Infinite clusters here (may or may not be unique) J I No infinite Page 138 of 163 clusters here Many infinite clusters 0 0 1 Go Back 1/4 τ 0.707 0.816 The story so far: includes uniqueness transition for case d = 2. Full Screen

Quit 5.4. Comparison Home Page We need to apply the Fortuin-Kasteleyn comparison inequalities (5.3) and (5.4). The event “just one infinite cluster” is not increasing, so we need more. Newman Title Page and Wu (1990) show it suffices to establish a finite island property for the site percolation derived under adjacency when all infinite clusters are removed. Thus:

Contents 1

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λ finite islands J I

1/2 Page 139 of 163

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0 Full Screen 0 1/4 1 τ 0.707 0.816

Quit Comparison arguments then show the following schematic phase diagram for the Home Page Ising model on Q2(o):

Title Page

All nodes substantially Free boundary condition Contents correlated with each is mixture of the two extreme other in case of free Gibbs states (spin 1 at boundary conditions boundary, spin −1 at boundary) JJ II

1.099 J I

0.693 Page 140 of 163

Go Back Unique Root influenced J Gibbs λ by wired boundary Full Screen state

J τ 0.288 0.511 1.228 2.292 Close 0.762 1.190

Quit 5.5. Simulation Home Page Approximate simulations conﬁrm the general story (compare our original anima- Title Page tion, which had Jτ = 2, Jλ = 1/4):

Contents http://www.dcs.warwick.ac.uk/˜rgw/sira/sim.html

JJ II (1) Only 200 resolution levels;

J I (2) At each level, 1000 sweeps in scan order; Page 141 of 163

(3) At each level, simulate square sub-region of 128 × 128 pixels conditioned Go Back by mother 64 × 64 pixel region;

Full Screen (4) Impose periodic boundary conditions on 128 × 128 square region;

Close (5) At the coarsest resolution, all pixels set white. At subsequent resolutions,

Quit ‘all black’ initial state. Home Page

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Contents

(a) Jλ = 1,Jτ = 0.5 (b) Jλ = 1,Jτ = 1 (c) Jλ = 1,Jτ = 2 JJ II

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Go Back (d) Jλ = 0.5,Jτ = 0.5 (e) Jλ = 0.5,Jτ = 1 (f) Jλ = 0.5,Jτ = 2

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(g) Jλ = 0.25,Jτ = 0.5 (h) Jλ = 0.25,Jτ = 1 (i) Jλ = 0.25,Jτ = 2 5.6. Future work Home Page This is about the free Ising model on Q2(o). Image analysis more naturally concerns the case of prescribed boundary conditions (say, image at finest resolution Title Page level . . . ). Question: will boundary conditions at “infinite fineness” propagate back to finite

Contents resolution? Series and Sina˘ı (1990) show answer is yes for analogous problem on hyperbolic disk (2-dim, all bond probabilities the same). JJ II Gielis and Grimmett (2002) point out (eg, in Z3 case) these boundary conditions translate to a conditioning for random cluster model, and investigate using large deviations. J I Project: do same for Q2(o) . . . and get quantitative bounds? Project: generalize from Qd to graphs with appropriate hyperbolic structure. Page 143 of 163

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Contents Appendix A

JJ II Notes on the proofs in Chapter J I 5 Page 145 of 163

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Quit A.1. Notes on proof of Theorem 5.4 Home Page Mean size of cluster at o bounded above by

∞ Title Page X X n T (t) n−T (t) Xλτ (Xλ − 1) Xλ n=0 t:|t|=n Contents ∞ X n X −1 T (t) ≤ Xλ(τXλ) (1 − Xλ ) n=0 t:|t|=n JJ II ∞ X d n X −1 T (j) ≤ Xλ(2 τXλ) (1 − Xλ ) J I n=0 j:|j|=n ∞ q n X d n −1 Page 146 of 163 ≈ Xλ(2 τXλ) 1 + 1 − Xλ . n=0

Go Back For last step, use spectral analysis of matrix representation

n X 1 1 1 (1 − X −1)T (j) = 1 1 . Full Screen λ 1 − X −1 1 1 j:|j|=n λ

Quit A.2. Notes on proof of Theorem 5.5 Home Page Uniqueness: For negative exponent ξ(1 − λ) of dual connectivity function, set

Title Page `n = (n log 4 + (2 + ) log n) ξ(1 − λ) .

More than one “`n-large” cluster in Ln forces existence of open path in dual lattice Contents longer than `n. Now use Borel-Cantelli . . . . On the other hand super-criticality will mean some distant points in Ln are inter- connected. JJ II n−[n/2] Existence: consider 4 points in Ln−1 and speciﬁed daughters in Ln. Study probability that J I (a) parent percolates more than `n−1, Page 147 of 163 (b) parent and child are connected,

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Quit A.3. Notes on proof of Theorem 5.6 Home Page Two relevant lemmata:

Title Page Lemma A.1 Consider u ∈ Ls+1 ⊂ Qd and v = M(u) ∈ Ls ⊂ Qd. There d are exactly 2 solutions in Ls+1 of Contents

M(x) = Su;v(x) .

JJ II One is x = u. Others are the remaining 2d −1 vertices y such that closure of cell representing y intersects vertex shared by closures of cells representing J I u and M(u). Finally, if x ∈ Ls+1 does not solve M(x) = Su;v(x) then

kSu;v(x) − Su;v(u)ks,∞ > kM(x) − M(u)ks,∞ . Page 148 of 163

Go Back Lemma A.2 Given distinct v and y in the same resolution level. Count pairs Full Screen of vertices u, x in the resolution level one step higher, such that

(a) M(u) = v; (b) M(x) = y; (c) Su;v(x) = y. Close There are at most 2d−1 such vertices.

Quit A.4. Notes on proof of Theorem 5.7 Home Page Prune! Then a direct connection is certainly established across the boundary between the cells corresponding to two neighbouring vertices u, v in L0 if Title Page (a) the τ-bond leading from u to the relevant boundary is open;

Contents (b) a τ-branching process (formed by using τ-bonds mirrored across the boundary) survives indeﬁnitely, where this branching process has family-size distribution Binomial(2, τ 2); JJ II (c) the τ-bond leading from v to the relevant boundary is open. J I Then there are inﬁnitely many Page 149 of 163 chances of making a connection across the cell boundary. Go Back

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Quit A.5. Notes on proof of inﬁnite island property Home Page Notion of “cone boundary” ∂c(S) of ﬁnite subset S of vertices: collection of daughters v of S such that Qd(v) ∩ S = ∅. Title Page Use induction on S, building it layer Ln on layer Ln−1 to obtain an isoperimetric d bound: #(∂c(S)) ≥ (2 − 1)#(S). Hence deduce Contents (2d−1)n P [S in island at u] ≤ (1 − pτ (1 − η))

JJ II where #(S) = n and η = P [u not in inﬁnite cluster of Qd(u)]. Upper bound on number N(n) of self-avoiding paths S of length n beginning at u0: J I N(n) ≤ (1 + 2d + 2d)(2d + 2d)n .

Page 150 of 163 Hence upper bound on the mean size of the island: ∞ X d d n n(1−2−d) Go Back (1 + 2d + 2 )(2d + 2 ) ηbr , n=0

d Full Screen where ηbr is extinction probability for branching process based on Binomial(2 , pτ ) family distribution.

Quit Home Page Title Page Bibliography Contents

JJ II Aldous, D. and M. T. Barlow [1981]. On vision. Journal of Applied Statis- countable dense random sets. Lec- tics 20, 233–258. J I ture Notes in Mathematics 850, 311– Baddeley, A. J. and M. N. M. 327. van Lieshout [1995]. Area- Page 151 of 163 Aldous, D. J. and P. Diaconis [1987]. interaction point processes. An- Strong uniform times and ﬁnite ran- nals of the Institute of Statistical dom walks. Advances in Applied Mathematics 47, 601–619. Go Back Mathematics 8(1), 69–97. Barndorff-Nielsen, O. E., W. S. Kendall, Baddeley, A. J. and J. Møller [1989]. and M. N. M. van Lieshout (Eds.)

Full Screen Nearest-neighbour Markov point [1999]. Stochastic Geometry: Like- processes and random sets. Int. lihood and Computation. Number 80 Statist. Rev. 57, 89–121. in Monographs on Statistics and Ap- Close Baddeley, A. J. and M. N. M. plied Probability. Boca Raton: Chap- van Lieshout [1993]. Stochastic man and Hall / CRC. geometry models in high-level Benjamini, I., R. Lyons, Y. Peres, and Quit O. Schramm [1999]. Group-invariant sification and Regression. Statis- Home Page percolation on graphs. Geom. Funct. tics/Probability. Wadsworth. Anal. 9(1), 29–66. Brooks, S. P., Y. Fan, and J. S. Rosen- Benjamini, I. and O. Schramm [1996]. thal [2002]. Perfect forward simu- d Title Page Percolation beyond Z , many ques- lation via simulated tempering. Re- tions and a few answers. Electronic search report, University of Cam- Communications in Probability 1, bridge. Contents no. 8, 71–82 (electronic). Burdzy, K. and W. S. Kendall [2000, Bhalerao, A., E. Thonnes,¨ W. S. Kendall, May]. Efficient Markovian cou- and R. G. Wilson [2001]. Inferring plings: examples and counterex- JJ II vascular structure from 2d and 3d im- amples. The Annals of Applied agery. In W. J. Niessen and M. A. Probability 10(2), 362–409. J I Viergever (Eds.), Medical Image Also University of Warwick Depart- Computing and Computer-Assisted ment of Statistics Research Report Intervention, proceedings of MIC- 331. Page 152 of 163 CAI 2001, Volume 2208 of Springer Cai, Y. and W. S. Kendall [2002, Lecture Notes in Computer Science, July]. Perfect simulation for corre- pp. 820–828. Springer-Verlag. Also: Go Back lated Poisson random variables con- University of Warwick Department ditioned to be positive. Statistics and of Statistics Research Report 392. Computing 12, 229–243. Also: Uni- Full Screen Billera, L. J. and P. Diaconis [2001]. versity of Warwick Department of A geometric interpretation of the Statistics Research Report 349. Metropolis-Hastings algorithm. Sta- Carter, D. S. and P. M. Prenter [1972]. Close tistical Science 16(4), 335–339. Exponential spaces and counting Breiman, L., J. H. Friedman, R. A. Ol- processes. Z. Wahrscheinlichkeits- shen, and C. J. Stone [1984]. Clas- theorie und Verw. Gebiete 21, 1–19. Quit Chipman, H., E. George, and R. McCul- Fill, J. [1998]. An interruptible algo- Home Page loch [2002]. Bayesian treed models. rithm for exact sampling via Markov Machine Learning 48, 299–320. Chains. The Annals of Applied Prob- Clancy, D. and P. D. O’Neill [2002]. Per- ability 8, 131–162. Title Page fect simulation for stochastic models Fill, J. A., M. Machida, D. J. Mur- of epidemics among a community of doch, and J. S. Rosenthal [2000]. Extension of Fill’s perfect rejec- Contents households. Research report, School of Mathematical Sciences, Univer- tion sampling algorithm to general sity of Nottingham. chains. Random Structures and Algo- rithms 17(3-4), 290–316. JJ II Cowan, R., M. Quine, and S. Zuyev [2003]. Decomposition of gamma- Fortuin, C. M. [1972a]. On the random- distributed domains constructed cluster model. II. The percolation J I from Poisson point processes. Ad- model. Physica 58, 393–418. vances in Applied Probability 35(1), Fortuin, C. M. [1972b]. On the random- Page 153 of 163 56–69. cluster model. III. The simple Diaconis, P. [1988]. Group Representa- random-cluster model. Physica 59, tions in Probability and Statistics, 545–570. Go Back Volume 11 of IMS Lecture Notes Se- Fortuin, C. M. and P. W. Kaste- ries. Hayward, California: Institute leyn [1972]. On the random-cluster Full Screen of Mathematical Statistics. model. I. Introduction and relation to Ferrari, P. A., R. Fernandez,´ and N. L. other models. Physica 57, 536–564. Garcia [2002]. Perfect simulation for Foss, S. G. and R. L. Tweedie [1998]. Close interacting point processes, loss net- Perfect simulation and backward works and Ising models. Stochastic coupling. Stochastic Models 14, Process. Appl. 102(1), 63–88. 187–203. Quit Gelfand, A. E. and A. F. M. Smith Grenander, U. and M. I. Miller [1994]. Home Page [1990]. Sampling-based approaches Representations of knowledge in to calculating marginal densities. complex systems (with discussion Journal of the American Statistical and a reply by the authors). Journal Title Page Association 85(410), 398–409. of the Royal Statistical Society (Se- Geman, S. and D. Geman [1984]. ries B: Methodological) 56(4), 549– 603. Contents Stochastic relaxation, Gibbs distribution, and Bayesian restoration of im- Grimmett, G. R. and C. Newman [1990]. ages. IEEE Transactions on Pattern Percolation in ∞ + 1 dimensions. In Disorder in physical systems, pp. JJ II Analysis and Machine Intelligence 6, 721–741. 167–190. New York: The Clarendon Press Oxford University Press. Also Geyer, C. J. and J. Møller [1994]. Sim- at J I ulation and likelihood inference for Hadwiger, H. [1957]. Vorlesungen uber¨ spatial point processes. Scand. J. Inhalt, Oberflache¨ und Isoperime- Statist. 21, 359–373. Page 154 of 163 trie. Berlin: Springer-Verlag. Gielis, G. and G. R. Grimmett [2002]. Haggstr¨ om,¨ O., J. Jonasson, and Rigidity of the interface for per- Go Back R. Lyons [2002]. Explicit isoperi- colation and random-cluster metric constants and phase transi- models. Journal of Statistical tions in the random-cluster model. Full Screen Physics 109, 1–37. See also The Annals of Probability 30(1), arXiv:math.PR/0109103., 443–473. Green, P. J. [1995]. Reversible jump Haggstr¨ om,¨ O. and Y. Peres [1999]. Close Markov chain Monte Carlo compu- Monotonicity of uniqueness for per- tation and Bayesian model determi- colation on Cayley graphs: all in- nation. Biometrika 82, 711–732. finite clusters are born simultane- Quit ously. Probability Theory and Re- ematical Statistics. New York: John Home Page lated Fields 113(2), 273–285. Wiley & Sons. Haggstr¨ om,¨ O., Y. Peres, and R. H. Hastings, W. K. [1970]. Monte Carlo Schonmann [1999]. Percolation on sampling methods using Markov Title Page transitive graphs as a coalescent pro- chains and their applications. cess: relentless merging followed by Biometrika 57, 97–109. simultaneous uniqueness. In Perplex- Contents Himmelberg, C. J. [1975]. Measurable ing problems in probability, pp. 69– relations. Fundamenta mathemati- 90. Boston, MA: Birkhauser¨ Boston. cae 87, 53–72. Haggstr¨ om,¨ O. and J. E. Steif [2000]. JJ II Huber, M. [2003]. A bounding chain Propp-Wilson algorithms and fini- for Swendsen-Wang. Random Struc- tary codings for high noise Markov tures and Algorithms 22(1), 43–59. J I random fields. Combin. Probab. Computing 9, 425–439. Hug, D., M. Reitzner, and R. Schnei- Haggstr¨ om,¨ O., M. N. M. van Lieshout, der [2003]. The limit shape of Page 155 of 163 and J. Møller [1999]. Characterisa- the zero cell in a stationary pois- tion results and Markov chain Monte son hyperplane tessellation. Preprint 03-03, Albert-Ludwigs-Universitat¨ Go Back Carlo algorithms including exact simulation for some spatial point Freiburg. processes. Bernoulli 5(5), 641–658. Jeulin, D. [1997]. Conditional simulation Full Screen Was: Aalborg Mathematics Depart- of object-based models. In D. Jeulin ment Research Report R-96-2040. (Ed.), Advances in Theory and Ap- Hall, P. [1988]. Introduction to the the- plications of Random Sets, pp. 271– Close ory of coverage processes. Wiley Se- 288. World Scientific. ries in Probability and Mathemati- Kallenberg, O. [1977]. A counterexam- cal Statistics: Probability and Math- ple to R. Davidson’s conjecture on Quit line processes. Math. Proc. Camb. Kendall, W. S. [1998]. Perfect simulation Home Page Phil. Soc. 82(2), 301–307. for the area-interaction point pro- Kelly, F. P. and B. D. Ripley [1976]. A cess. In L. Accardi and C. C. Heyde note on Strauss’s model for cluster- (Eds.), Probability Towards 2000, Title Page ing. Biometrika 63, 357–360. New York, pp. 218–234. Springer- Verlag. Also: University of War- Kendall, D. G. [1974]. Foundations of wick Department of Statistics Re- Contents a theory of random sets. In E. F. search Report 292. Harding and D. G. Kendall (Eds.), Kendall, W. S. [2000, March]. Station- Stochastic Geometry. John Wiley & ary countable dense random sets. Ad- Sons. JJ II vances in Applied Probability 32(1), Kendall, W. S. [1997a]. On some 86–100. Also University of J I weighted Boolean models. In Warwick Department of Statistics D. Jeulin (Ed.), Advances in Theory Research Report 341. and Applications of Random Sets, Kendall, W. S. and J. Møller [2000, Page 156 of 163 Singapore, pp. 105–120. World Sci- September]. Perfect simulation us- entific. Also: University of Warwick ing dominating processes on ordered Department of Statistics Research Go Back state spaces, with application to lo- Report 295. cally stable point processes. Ad- Kendall, W. S. [1997b]. Perfect simula- vances in Applied Probability 32(3), Full Screen tion for spatial point processes. In 844–865. Also University of Bulletin ISI, 51st session proceed- Warwick Department of Statistics ings, Istanbul (August 1997), Vol- Research Report 347. Close ume 3, pp. 163–166. Also: Uni- Kendall, W. S. and G. Montana [2002, versity of Warwick Department of May]. Small sets and Markov tran- Statistics Research Report 308. sition densities. Stochastic Processes Quit and Their Applications 99(2), 177– ies in Probability. New York: The Home Page 194. Also University of Warwick De- Clarendon Press Oxford University partment of Statistics Research Re- Press. Oxford Science Publications. port 371. Kolmogorov, A. N. [1936]. Zur theorie Title Page Kendall, W. S. and E. Thonnes¨ [1999]. der markoffschen ketten. Math. An- Perfect simulation in stochastic ge- nalen 112, 155–160. ometry. Pattern Recognition 32(9), Kovalenko, I. [1997]. A proof of a con- Contents 1569–1586. Also: University of jecture of David Kendall on the Warwick Department of Statistics shape of random polygons of large Research Report 323. JJ II area. Kibernet. Sistem. Anal. (4), 3– Kendall, W. S., M. van Lieshout, 10, 187. English translation: Cyber- and A. Baddeley [1999]. Quermass- net. Systems Anal. 33 461-467. J I interaction processes: conditions for stability. Advances in Applied Proba- Kovalenko, I. N. [1999]. A simpli- bility 31(2), 315–342. Also Uni- fied proof of a conjecture of D. Page 157 of 163 versity of Warwick Department of G. Kendall concerning shapes of Statistics Research Report 293. random polygons. J. Appl. Math. Stochastic Anal. 12(4), 301–310. Go Back Kendall, W. S. and R. G. Wilson [2003, March]. Ising models and multires- Kumar, V. S. A. and H. Ramesh olution quad-trees. Advances in Ap- [2001]. Coupling vs. conductance for Full Screen plied Probability 35(1), 96–122. the Jerrum-Sinclair chain. Random Also University of Warwick De- Structures and Algorithms 18(1), 1– partment of Statistics Research Re- 17. Close port 402. Kurtz, T. G. [1980]. The optional sam- Kingman, J. F. C. [1993]. Poisson pro- pling theorem for martingales in- cesses, Volume 3 of Oxford Stud- dexed by directed sets. The Annals of Quit Probability 8, 675–681. continuous-time renewal processes. Home Page Lee, A. B., D. Mumford, and J. Huang Journal of Applied Probability 19(1), [2001]. Occlusion models for nat- 82–89. ural images: A statistical study of Lindvall, T. [2002]. Lectures on the cou- Title Page a scale-invariant dead leaves model. pling method. Mineola, NY: Dover International Journal of Computer Publications Inc. Corrected reprint of Vision 41, 35–59. the 1992 original. Contents Li, W. and A. Bhalerao [2003]. Model Liu, J. S., F. Liang, and W. H. Wong based segmentation for retinal fun- [2001, June]. A theory for dynamic weighting in Monte Carlo computa- JJ II dus images. In Proceedings of Scan- dinavian Conference on Image Anal- tion. Journal of the American Statis- ysis SCIA’03,Goteborg,¨ Sweden, pp. tical Association 96(454), 561–573. J I to appear. Loizeaux, M. A. and I. W. McKeague [2001]. Perfect sampling for poste- Liang, F. and W. H. Wong [2000]. Evolu- rior landmark distributions with an Page 158 of 163 tionary Monte Carlo sampling: Ap- application to the detection of dis- plications to C model sampling p ease clusters. In Selected Proceed- and change-point problem. Statistica ings of the Symposium on Inference Go Back Sinica 10, 317–342. for Stochastic Processes, Volume 37 Liang, F. and W. H. Wong [2001, of Lecture Notes - Monograph Se- Full Screen June]. Real-parameter evolutionary ries, pp. 321–331. Institute of Math- Monte Carlo with applications to ematical Statistics. Bayesian mixture models. Journal Loynes, R. [1962]. The stability of a Close of the American Statistical Associa- queue with non-independent inter- tion 96(454), 653–666. arrival and service times. Math. Proc. Lindvall, T. [1982]. On coupling of Camb. Phil. Soc. 58(3), 497–520. Quit Lyons, R. [2000]. Phase transitions certain large random polygons. Ad- Home Page on nonamenable graphs. J. Math. vances in Applied Probability 27(2), Phys. 41(3), 1099–1126. Probabilis- 397–417. tic techniques in equilibrium and Minlos, R. A. [1999]. Introduction Title Page nonequilibrium statistical physics. to Mathematical Statistical Physics, Matheron, G. [1975]. Random Sets and Volume 19 of University Lecture Se- Integral Geometry. Chichester: John ries. American Mathematical Soci- Contents Wiley & Sons. ety. Meester, R. and R. Roy [1996]. Con- Mira, A., J. Møller, and G. O. Roberts JJ II tinuum percolation, Volume 119 of [2001]. Perfect slice samplers. Jour- Cambridge Tracts in Mathematics. nal of the Royal Statistical Soci- Cambridge: Cambridge University ety (Series B: Methodological) 63(3), J I Press. 593–606. See (Mira, Møller, and Metropolis, N., A. V. Rosenbluth, Roberts 2002) for corrigendum. Page 159 of 163 M. N. Rosenbluth, A. H. Teller, Mira, A., J. Møller, and G. O. Roberts and E. Teller [1953]. Equation of [2002]. Corrigendum: perfect slice state calculations by fast comput- samplers. Journal of the Royal Sta- Go Back ing machines. Journal of Chemical tistical Society (Series B: Method- Physics 21, 1087–1092. ological) 64(3), 581.

Full Screen Meyn, S. P. and R. L. Tweedie [1993]. Molchanov, I. S. [1996a]. A limit Markov Chains and Stochastic Sta- theorem for scaled vacancies of bility. New York: Springer-Verlag. the Boolean model. Stochastics and Close Miles, R. E. [1995]. A heuristic proof Stochastic Reports 58(1-2), 45–65. of a long-standing conjecture of D. Molchanov, I. S. [1996b]. Statistics of G. Kendall concerning the shapes of the Boolean Models for Practition- Quit ers and Mathematicians. Chichester: of Statistics Theory and Applica- Home Page John Wiley & Sons. tions 25, 483–502. Møller, J. (Ed.) [2003]. Spatial Statistics Mykland, P., L. Tierney, and B. Yu and Computational Methods, Vol- [1995]. Regeneration in Markov Title Page ume 173 of Lecture Notes in Statis- chain samplers. Journal of the Amer- tics. Springer-Verlag. ican Statistical Association 90(429), 233–241. Contents Møller, J. and G. Nicholls [1999]. Perfect simulation for sample-based infer- Newman, C. and C. Wu [1990]. Markov ence. Technical Report R-99-2011, fields on branching planes. Probabil- JJ II Department of Mathematical Sci- ity Theory and Related Fields 85(4), ences, Aalborg University. 539–552. Møller, J. and R. Waagepetersen [2003]. Nummelin, E. [1984]. General irre- J I Statistics inference and simulation ducible Markov chains and nonneg- for spatial point processes. Mono- ative operators. Cambridge: Cam- Page 160 of 163 graphs on Statistics and Applied bridge University Press. Probability. Boca Raton: Chapman O’Neill, P. D. [2001]. Perfect simula- and Hall / CRC. tion for Reed-Frost epidemic mod- Go Back Møller, J. and S. Zuyev [1996]. Gamma- els. Statistics and Computing 13, 37– type results and other related proper- 44. Full Screen ties of Poisson processes. Advances Parkhouse, J. G. and A. Kelly [1998]. in Applied Probability 28(3), 662– The regular packing of fibres in three 673. dimensions. Philosophical Transac- Close Murdoch, D. J. and P. J. Green [1998]. tions of the Royal Society, London, Exact sampling from a continuous Series A 454(1975), 1889–1909. state space. Scandinavian Journal Peskun, P. H. [1973]. Optimum Monte Quit Carlo sampling using Markov Robbins, H. E. [1944]. On the measure of Home Page chains. Biometrika 60, 607–612. a random set I. Annals of Mathemat- Preston, C. J. [1977]. Spatial birth-and- ical Statistics 15, 70–74. death processes. Bull. Inst. Internat. Robbins, H. E. [1945]. On the measure of Title Page Statist. 46, 371–391. a random set II. Annals of Mathemat- Propp, J. G. and D. B. Wilson [1996]. Ex- ical Statistics 16, 342–347. Contents act sampling with coupled Markov Roberts, G. O. and N. G. Polson [1994]. chains and applications to statistical On the geometric convergence of mechanics. Random Structures and the Gibbs sampler. Journal of the JJ II Algorithms 9, 223–252. Royal Statistical Society (Series B: Quine, M. P. and D. F. Watson [1984]. Methodological) 56(2), 377–384. Radial generation of n-dimensional Roberts, G. O. and J. S. Rosenthal J I Poisson processes. Journal of Ap- [1999]. Convergence of slice sam- plied Probability 21(3), 548–557. pler Markov chains. Journal of the Page 161 of 163 Ripley, B. D. [1977]. Modelling spatial Royal Statistical Society (Series B: patterns (with discussion). Journal of Methodological) 61(3), 643–660. the Royal Statistical Society (Series Roberts, G. O. and J. S. Rosenthal Go Back B: Methodological) 39, 172–212. [2000]. Recent progress on com- Ripley, B. D. [1979]. Simulating spatial putable bounds and the simple slice Full Screen patterns: dependent samples from a sampler. In Monte Carlo methods multivariate density. Applied Statis- (Toronto, ON, 1998), pp. 123–130. tics 28, 109–112. Providence, RI: American Mathe- Close Ripley, B. D. and F. P. Kelly [1977]. matical Society. Markov point processes. J. Lond. Roberts, G. O. and R. L. Tweedie Math. Soc. 15, 188–192. [1996a]. Exponential convergence of Quit Langevin diffusions and their dis- Verlag, Berlin). Home Page crete approximations. Bernoulli 2, Strauss, D. J. [1975]. A model for clus- 341–364. tering. Biometrika 63, 467–475. Roberts, G. O. and R. L. Tweedie Thonnes,¨ E. [1999]. Perfect simulation of Title Page [1996b]. Geometric convergence and some point processes for the impa- central limit theorems for multidi- tient user. Advances in Applied Prob- mensional Hastings and Metropolis Contents ability 31(1), 69–87. Also University algorithms. Biometrika 83, 96–110. of Warwick Statistics Research Re- port 317. JJ II Rosenthal, J. S. [2002]. Quantitative con- Thonnes,¨ E., A. Bhalerao, W. S. Kendall, vergence rates of Markov chains: A and R. G. Wilson [2002]. A Bayesian simple account. Electronic Commu- approach to inferring vascular tree J I nications in Probability 7, no. 13, structure from 2d imagery. In W. J. 123–128 (electronic). Niessen and M. A. Viergever (Eds.), Page 162 of 163 Series, C. and Y. Sina˘ı [1990]. Ising Proceedings IEEE ICIP-2002, Vol- models on the Lobachevsky plane. ume II, Rochester, NY, pp. 937–939. Comm. Math. Phys. 128(1), 63–76. Also: University of Warwick Depart- Go Back Spitzer, F. [1975]. Markov random fields ment of Statistics Research Report on an infinite tree. The Annals of 391. Full Screen Probability 3(3), 387–398. Thonnes,¨ E., A. Bhalerao, W. S. Kendall, Stoyan, D., W. S. Kendall, and J. Mecke and R. G. Wilson [2003]. Bayesian [1995]. Stochastic geometry and its analysis of vascular structure. In Close applications (Second ed.). Chich- LASR 2003, Volume to appear. ester: John Wiley & Sons. (First edi- Thorisson, H. [2000]. Coupling, station- tion in 1987 joint with Akademie arity, and regeneration. New York: Quit Springer-Verlag. on CFTP). In N. Madras (Ed.), Home Page Tierney, L. [1998]. A note on Metropolis- Monte Carlo Methods, Volume 26 Hastings kernels for general state of Fields Institute Communications, spaces. The Annals of Applied Prob- pp. 143–179. American Mathemati- Title Page ability 8(1), 1–9. cal Society. van Lieshout, M. N. M. [2000]. Markov Wilson, R., A. Calway, and E. R. S. Pear- son [March 1992]. A Generalised Contents point processes and their applications. Imperial College Press. Wavelet Transform for Fourier Anal- ysis: The Multiresolution Fourier van Lieshout, M. N. M. and A. J. Bad- Transform and Its Application to JJ II deley [2001, October]. Extrapolat- Image and Audio Signal Analysis. ing and interpolating spatial patterns. IEEE Transactions on Information CWI Report PNA-R0117, CWI, Am- Theory 38(2), 674–690. J I sterdam. Wilson, R. G. and C. T. Li [2002]. A class Widom, B. and J. S. Rowlinson [1970]. of discrete multiresolution random Page 163 of 163 New model for the study of liquid- fields and its application to image vapor phase transitions. Journal of segmentation. IEEE Transactions on Chemical Physics 52, 1670–1684. Go Back Pattern Analysis and Machine Intel- Wilson, D. B. [2000a]. How to cou- ligence 25, 42–56. ple from the past using a read- Zuyev, S. [1999]. Stopping sets: gamma- Full Screen once source of randomness. Random type results and hitting proper- Structures and Algorithms 16(1), ties. Advances in Applied Probabil- 85–113. ity 31(2), 355–366. Close Wilson, D. B. [2000b]. Layered Mul- tishift Coupling for use in Perfect

Quit Sampling Algorithms (with a primer