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Conditioning and Markov Properties

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Conditioning and Markov properties

Anders Rønn-Nielsen
Ernst Hansen

Department of Mathematical Sciences
University of Copenhagen
Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen

Copyright 2014 Anders Rønn-Nielsen & Ernst Hansen ISBN 978-87-7078-980-6

Contents

  • Preface
  • v

  • 1
  • Conditional distributions
  • 1

136
1.1 Markov kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Integration of Markov kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Properties for the integration measure . . . . . . . . . . . . . . . . . . . . . . 1.4 Conditional distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Existence of conditional distributions . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

23

  • Conditional distributions: Transformations and moments
  • 27

2.1 Transformations of conditional distributions . . . . . . . . . . . . . . . . . . . 27 2.2 Conditional moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

  • Conditional independence
  • 51

3.1 Conditional probabilities given a σ–algebra . . . . . . . . . . . . . . . . . . . 52 3.2 Conditionally independent events . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Conditionally independent σ-algebras . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Shifting information around . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Conditionally independent random variables . . . . . . . . . . . . . . . . . . . 61 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

  • 4
  • Markov chains
  • 71

4.1 The fundamental Markov property . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 The strong Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4 An integration formula for a homogeneous Markov chain . . . . . . . . . . . . 99 4.5 The Chapmann-Kolmogorov equations . . . . . . . . . . . . . . . . . . . . . . 100

iv 5
CONTENTS

4.6 Stationary distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

  • Ergodic theory for Markov chains on general state spaces
  • 111

5.1 Convergence of transition probabilities . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Transition probabilities with densities . . . . . . . . . . . . . . . . . . . . . . 115 5.3 Asymptotic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4 Minorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.5 The drift criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

  • 6
  • An introduction to Bayesian networks
  • 141

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Directed graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3 Moral graphs and separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.4 Bayesian networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.5 Global Markov property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

  • A Supplementary material
  • 155

A.1 Measurable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A.2 Random variables and conditional expectations . . . . . . . . . . . . . . . . . 157

  • B Hints for exercises
  • 161

B.1 Hints for chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 B.2 Hints for chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 B.3 Hints for chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 B.4 Hints for chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 B.5 Hints for chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Preface

The present lecture notes are intended for the course ”Beting”. The purpose is to give a detailed probabilistic introduction to conditional distributions and how this concept is used to define and describe Markov chains and Bayesian networks.

The chapters 1–4 are mainly based on different sets of lecture notes written by Ernst Hansen but are adapted to suit students with the knowledge obtained in the courses MI, VidSand1, and VidSand2. Parts of these chapters also contain material inspired by lecture notes written by Martin Jacobsen and Søren Tolver Jensen. Chapter 5 is an adapted version of a set of lecture notes written by Søren Tolver Jensen and Martin Jacobsen. These notes themselves were inspired by earlier notes by Søren Feodor Nielsen. Chapter 6 contains some standard results on Bayesian networks - though both formulations and proofs are formulated in the rather general framework from chapter 3 and 4.

There are exercises in the end of each chapter and hints to selected exercises in the end of the notes. Some exercises are adapted versions of exercises and exam exercises from previous lecture notes. Others are inspired by examples and results collected from a large number of monographs on Markov chains, stochastic simulation, and probability theory in general.

I am grateful to both students and the teaching assistants from the last two years, Ketil Biering Tvermosegaard and Daniele Cappelletti, who have contributed to the notes by identifying mistakes and suggesting improvements.

Anders Rønn-Nielsen København, 2014

  • vi
  • CONTENTS

Chapter 1

Conditional distributions

Let (X, E) and (Y, K) be two measurable spaces. In this chapter we shall discuss the relation between measures on the product space (X × Y, E ⊗ K) and measures on the two marginal spaces (X, E) and (Y, K). More precisely we will see that measures on the product space can be constructed from measures on the two marginal spaces. A particularly simple example is a product measure µ ⊗ ν where µ and ν are measures on (X, E) and (Y, K) respectively.

The two factors X and Y will not enter the setup symmetrically in the more general construction.

1.1 Markov kernels

Definition 1.1.1. A (X, E)–Markov Kernel on (Y, K) is a family of probability measures (Px)x∈X on (Y, K) indexed by points in X such that the map

x → Px(B)

is E − B–measurable for every fixed B ∈ K. Theorem 1.1.2. Let (X, E) and (Y, K) be measurable spaces, let ν be a σ–finite measure on

+

(Y, K), and let f ∈ M (X × Y, E ⊗ K) have the property that

Z

f(x, y) dν(y) = 1

for all x ∈ X .

  • 2
  • Conditional distributions

Then (Px)x∈X given by

Z

Px(B) =

f(x, y) dν(y) for all B ∈ K, x ∈ X

B

is a (X, E)–Markov Kernel on (Y, K).

Proof. For each fixed set B ∈ K we need to argue that

Z

x →
1X×B(x, y)f(x, y) dν(y)

is an E–measurable function. That is a direct result of EH Theorem 8.7 applied to the function 1X×Bf.

Lemma 1.1.3. If (Y, K) has an intersection–stable generating system D, then (Px)x∈X is a (X, E)–Markov Kernel on (Y, K) if only

x → Px(D)

is E − B-measurable for all fixed D ∈ D. Proof. Define

H = {F ∈ K : x → Px(F) is E − B–measurable}
It is easily checked that H is a Dynkin Class. Since D ⊆ H, we have H = K.

Lemma 1.1.4. Let (Px)x∈X be a (X, E)–Markov kernel on (Y, K). For each G ∈ E ⊗ K the map

x → Px(Gx)

is E − B–measurable. Proof. Let

H = {G ∈ E ⊗ K : x → Px(Gx) is E − B–measurable} and consider a product set A × B ∈ E ⊗ K. Then

(

∅

if x ∈/ A

(A × B)x =

  • B
  • if x ∈ A

such that

(

0

if x ∈/ A

  • Px((A × B)x) =
  • = 1A(x) · Px(B)

Px(B) if x ∈ A

  • 1.2 Integration of Markov kernels
  • 3

This is a product of two E−B–measurable functions. Hence it is E−B–measurable. Thereby we conclude that H contains all product sets, and since this is a intersection stable generating system for E ⊗ K, we have H = E ⊗ K, if we can show that H is a Dynkin class:

We already have that X × Y ∈ H – it is a product set! Assume that G1 ⊆ G2 are two sets in H. Then obviously also Gx1 ⊆ G2x for all x ∈ X, and

(G2 \ G1)x = G2x \ Gx1 .

Then
Px((G2 \ G1)x) = Px(G2x) − Px(Gx1 )

which is a difference between two measurable functions. Hence G2 \ G1 ∈ H. Finally, assume that G1 ⊆ G2 ⊆ · · · is an increasing sequence of H–sets. Similarly to above

we have Gx1 ⊆ Gx2 ⊆ · · · and

  •  
  • !

x

  • ∞
  • ∞

  • [
  • [

Gn

=

Gxn .

  • n=1
  • n=1

Then

  

! !

  •  
  • !

x

  • ∞
  • ∞

  • [
  • [

  • Px
  • Gn

  • = Px
  • Gxn = lim Px(Gxn)

n→∞

  • n=1
  • n=1

This limit is E − B–measurable, since each of the functions x → Px(Gxn) are measurable.

S∞

Then n=1 Gn ∈ H.

1.2 Integration of Markov kernels

Theorem 1.2.1. Let µ be a probability measure on (X, E) and let (Px)x∈X be a (X, E)– Markov kernel on (Y, K). There exists a uniquely determined probability measure λ on (X × Y, E ⊗ K) satisfying

Z

  • λ(A × B) =
  • Px(B) dµ(x)

A

for all A ∈ E and B ∈ K.

The probability measure λ constructed in Theorem 1.2.1 is called the integration of (Px)x∈X with respect to µ. The interpretation is that λ describes an experiment on X × Y that is performed in two steps: The first step is drawing x ∈ X. The second step is drawing y ∈ Y according to a probability measure that is determined by x.

  • 4
  • Conditional distributions

Proof. The uniqueness follows, since λ is determined on all product sets and these form an intersection stable generating system for E ⊗ K.

In order to prove the existence, we define

Z

λ(G) = Px(Gx) dµ(x)
For each G ∈ E⊗K the integrand is measurable according to Lemma 1.1.4. It is furthermore non–negative, such that λ(G) is well–defined with values in [0, ∞]. Now let G1, G2, . . . be a sequence of disjoint sets in E ⊗ K. Then for each x ∈ X the sets Gx1 , Gx2 , . . . are disjoint as well. Hence

  • Z
  • Z

  • ∞
  • ∞
  • ∞
  • ∞

  • ꢀ
  • ꢁ
  • ꢀꢀ

ꢁ ꢁ

  • [
  • [
  • X
  • X

x

λ

Gn

=

  • Px
  • Gn

  • dµ(x) =
  • Px(Gxn) dµ(x) =
  • λ(Gn)

  • n=1
  • n=1
  • n=1
  • n=1

In the second equality we have used that each Px is a measure, and in the third equality we have used monotone convergence to interchange integration and summation. From this we have that λ is a measure. And since

  • Z
  • Z
  • Z

λ(X × Y) = Px((X × Y)x) dµ(x) = Px(Y) dµ(x) = 1 dµ(x) = 1 we obtain, than λ is actually a probability measure. Finally, it follows that

  • Z
  • Z
  • Z

λ(A × B) = Px((A × B)x) dµ(x) = 1A(x)Px(B) dµ(x) =
Px(B) dµ(x)

A

for all A ∈ E and B ∈ K.

Corollary 1.2.2. Let µ be a probability measure on (X, E) and let (Px)x∈X be a (X, E)– Markov kernel on (Y, K). Let λ be the integration of (Px)x∈X with respect to µ. Then λ satisfies

  • λ(A × Y) =
  • µ(A)

for all A ∈ E

Z

λ(X × B) = Px(B) dµ(x)

for all B ∈ K

Proof. The second statement is obvious. For the first result just note that Px(Y) = 1 for all

x ∈ X.

The probability measure on (Y, K) defined by λ(X ×B) is called the mixture of the Markov kernel with respect to µ. Example 1.2.3. Let µ be a probability measure on (X, E) and let ν be a probability measure

  • 1.2 Integration of Markov kernels
  • 5

on (Y, K). Define Px = ν for all x ∈ X. Then, trivially, (Px)x∈X is a X–Markov kernel on Y. Let λ be the integration of this kernel with respect to λ. Then for all A ∈ E and B ∈ K

Z

  • λ(A × B) =
  • ν(B) dµ(x) = µ(A) · ν(B) .

A

The only measure satisfying this property is the product measure µ ⊗ ν, so λ = µ ⊗ ν. Hence a product measure is a particularly simple example of a measure constructed by integrating a Markov kernel.

◦

Example 1.2.4. Let µ be the Poisson distribution with parameter λ. For each x ∈ N0 we define Px to be the binomial distribution with parameters (x, p). Then it is seen that (Px)x∈N is a N0–Markov kernel on N0.

0

Let ξ be the mixture of (Px)x∈N with respect to µ. This must be a probability measure on

0

N0 and is thereby given by the probability function q. For n ∈ N0 we obtain

ꢂ ꢃ

∞

X

k

λk pn(1 − p)k−n e−λ

k! q(n) =

n

k=n

  • ꢄ
  • ꢅ

k−n

∞

(λp)n

X

(1 − p)λ
=

e−λ

  • n!
  • (k − n)!

k=n

(λp)n
=

=

e

−λe(1−p)λ

n!
(λp)n

e−λp

n!
Hence the mixture ξ is seen to be the Poisson distribution with parameter λp.

◦

Theorem 1.2.5 (Uniqueness of integration). Suppose that (Y, K) has a countable generating system that is intersection stable. Let µ and µ˜ be two probability measures on (X, E) and

˜

assume that (Px)x∈X and (Px)x∈X are two (X, E)–Markov kernels on (Y, K). Let λ be the

  • ˜
  • ˜

integration of (Px)x∈X with respect to µ, and let λ be the integration of (Px)x∈X with respect to µ˜. Define

˜

E0 = {x ∈ X : Px = Px}

˜

Then λ = λ if and only if µ = µ˜ and µ(E0) = 1.

Proof. Let (Bn)n∈N be a countable generating system for (Y, K). Then

∞

\

˜

E0 =

{x ∈ X : Px(Bn) = Px(Bn)}

n=1

from which we can conclude that E0 ∈ E.

  • 6
  • Conditional distributions

Assume that µ = µ˜ and µ(E0) = 1. Then for all A ∈ E and B ∈ K we have

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    Interacting particle systems MA4H3 Stefan Grosskinsky Warwick, 2009 These notes and other information about the course are available on http://www.warwick.ac.uk/˜masgav/teaching/ma4h3.html Contents Introduction 2 1 Basic theory3 1.1 Continuous time Markov chains and graphical representations..........3 1.2 Two basic IPS....................................6 1.3 Semigroups and generators.............................9 1.4 Stationary measures and reversibility........................ 13 2 The asymmetric simple exclusion process 18 2.1 Stationary measures and conserved quantities................... 18 2.2 Currents and conservation laws........................... 23 2.3 Hydrodynamics and the dynamic phase transition................. 26 2.4 Open boundaries and matrix product ansatz.................... 30 3 Zero-range processes 34 3.1 From ASEP to ZRPs................................ 34 3.2 Stationary measures................................. 36 3.3 Equivalence of ensembles and relative entropy................... 39 3.4 Phase separation and condensation......................... 43 4 The contact process 46 4.1 Mean field rate equations.............................. 46 4.2 Stochastic monotonicity and coupling....................... 48 4.3 Invariant measures and critical values....................... 51 d 4.4 Results for Λ = Z ................................. 54 1 Introduction Interacting particle systems (IPS) are models for complex phenomena involving a large number of interrelated components. Examples exist within all areas of natural and social sciences, such as traffic flow on highways, pedestrians or constituents of a cell, opinion dynamics, spread of epi- demics or fires, reaction diffusion systems, crystal surface growth, chemotaxis, financial markets... Mathematically the components are modeled as particles confined to a lattice or some discrete geometry. Their motion and interaction is governed by local rules. Often microscopic influences are not accesible in full detail and are modeled as effective noise with some postulated distribution.
  • Pre-Publication Accepted Manuscript

    Pre-Publication Accepted Manuscript

    P´eterE. Frenkel Convergence of graphs with intermediate density Transactions of the American Mathematical Society DOI: 10.1090/tran/7036 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proof- read, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. This is a pre-publication version of this article, which may differ from the final published version. Copyright restrictions may apply. CONVERGENCE OF GRAPHS WITH INTERMEDIATE DENSITY PÉTER E. FRENKEL Abstract. We propose a notion of graph convergence that interpolates between the Benjamini–Schramm convergence of bounded degree graphs and the dense graph convergence developed by László Lovász and his coauthors. We prove that spectra of graphs, and also some important graph parameters such as numbers of colorings or matchings, behave well in convergent graph sequences. Special attention is given to graph sequences of large essential girth, for which asymptotics of coloring numbers are explicitly calculated. We also treat numbers of matchings in approximately regular graphs.
  • A Review on Statistical Inference Methods for Discrete Markov Random Fields Julien Stoehr

    A Review on Statistical Inference Methods for Discrete Markov Random Fields Julien Stoehr

    A review on statistical inference methods for discrete Markov random fields Julien Stoehr To cite this version: Julien Stoehr. A review on statistical inference methods for discrete Markov random fields. 2017. hal-01462078v1 HAL Id: hal-01462078 https://hal.archives-ouvertes.fr/hal-01462078v1 Preprint submitted on 8 Feb 2017 (v1), last revised 11 Apr 2017 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A review on statistical inference methods for discrete Markov random fields Julien Stoehr1 1School of Mathematical Sciences & Insight Centre for Data Analytics, University College Dublin, Ireland February 8, 2017 Abstract Developing satisfactory methodology for the analysis of Markov random field is a very challenging task. Indeed, due to the Markovian dependence structure, the normalizing con- stant of the fields cannot be computed using standard analytical or numerical methods. This forms a central issue for any statistical approach as the likelihood is an integral part of the procedure. Furthermore, such unobserved fields cannot be integrated out and the like- lihood evaluation becomes a doubly intractable problem. This report gives an overview of some of the methods used in the literature to analyse such observed or unobserved random fields.
  • CONDITIONAL EXPECTATION Definition 1. Let (Ω,F,P) Be A

    CONDITIONAL EXPECTATION Definition 1. Let (Ω,F,P) Be A

    CONDITIONAL EXPECTATION STEVEN P.LALLEY 1. CONDITIONAL EXPECTATION: L2 THEORY ¡ Definition 1. Let (­,F ,P) be a probability space and let G be a σ algebra contained in F . For ¡ any real random variable X L2(­,F ,P), define E(X G ) to be the orthogonal projection of X 2 j onto the closed subspace L2(­,G ,P). This definition may seem a bit strange at first, as it seems not to have any connection with the naive definition of conditional probability that you may have learned in elementary prob- ability. However, there is a compelling rationale for Definition 1: the orthogonal projection E(X G ) minimizes the expected squared difference E(X Y )2 among all random variables Y j ¡ 2 L2(­,G ,P), so in a sense it is the best predictor of X based on the information in G . It may be helpful to consider the special case where the σ algebra G is generated by a single random ¡ variable Y , i.e., G σ(Y ). In this case, every G measurable random variable is a Borel function Æ ¡ of Y (exercise!), so E(X G ) is the unique Borel function h(Y ) (up to sets of probability zero) that j minimizes E(X h(Y ))2. The following exercise indicates that the special case where G σ(Y ) ¡ Æ for some real-valued random variable Y is in fact very general. Exercise 1. Show that if G is countably generated (that is, there is some countable collection of set B G such that G is the smallest σ algebra containing all of the sets B ) then there is a j 2 ¡ j G measurable real random variable Y such that G σ(Y ).
  • Lecture Notes on Markov and Feller Property

    Lecture Notes on Markov and Feller Property

    AN INTRODUCTION TO MARKOV PROCESSES AND THEIR APPLICATIONS IN MATHEMATICAL ECONOMICS UMUT C¸ETIN 1. Markov Property During the course of your studies so far you must have heard at least once that Markov processes are models for the evolution of random phenomena whose future behaviour is independent of the past given their current state. In this section we will make precise the so called Markov property in a very general context although very soon we will restrict our selves to the class of `regular' Markov processes. As usual we start with a probability space (Ω; F;P ). Let T = [0; 1) and E be a locally compact separable metric space. We will denote the Borel σ-field of E with E . Recall that the d-dimensional Euclidean space Rd is a particular case of E that appears very often in applications. For each t 2 T, let Xt(!) = X(t; !) be a function from Ω to E such that it is F=E - −1 measurable, i.e. Xt (A) 2 F for every A 2 E . Note that when E = R this corresponds to the familiar case of Xt being a real random variable for every t 2 T. Under this setup X = (Xt)t2T is called a stochastic process. We will now define two σ-algebras that are crucial in defining the Markov property. Let 0 0 Ft = σ(Xs; s ≤ t); Ft = σ(Xu; u ≥ t): 0 0 Observe that Ft contains the history of X until time t while Ft is the future evolution of X 0 after time t.
  • Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance

    Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance

    mathematics Article Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance Samuel Livingstone Department of Statistical Science, University College London, London WC1E 6BT, UK; [email protected] Abstract: We consider a Metropolis–Hastings method with proposal N (x, hG(x)−1), where x is the current state, and study its ergodicity properties. We show that suitable choices of G(x) can change these ergodicity properties compared to the Random Walk Metropolis case N (x, hS), either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of G(x) can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis. Keywords: Monte Carlo; MCMC; Markov chains; computational statistics; bayesian inference 1. Introduction Markov chain Monte Carlo (MCMC) methods are techniques for estimating expecta- Citation: Livingstone, S. Geometric tions with respect to some probability measure p(·), which need not be normalised. This is Ergodicity of the Random Walk done by sampling a Markov chain which has limiting distribution p(·), and computing Metropolis with Position-Dependent empirical averages. A popular form of MCMC is the Metropolis–Hastings algorithm [1,2], Proposal Covariance.