DOCSLIB.ORG

An Introduction to Markov Chain Monte Carlo Methods and Their Actuarial Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Markov Chain Monte Carlo Methods and Their Actuarial Applications AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS AND THEIR ACTUARIAL APPLICATIONS DAVID P. M. SCOLLNIK Department of Mathematics and Statistics University of Calgary Abstract This paper introduces the readers of the Proceed- ings to an important class of computer based simula- tion techniques known as Markov chain Monte Carlo (MCMC) methods. General properties characterizing these methods will be discussed, but the main empha- sis will be placed on one MCMC method known as the Gibbs sampler. The Gibbs sampler permits one to simu- late realizations from complicated stochastic models in high dimensions by making use of the model’s associated full conditional distributions, which will generally have a much simpler and more manageable form. In its most extreme version, the Gibbs sampler reduces the analy- sis of a complicated multivariate stochastic model to the consideration of that model’s associated univariate full conditional distributions. In this paper, the Gibbs sampler will be illustrated with four examples. The first three of these examples serve as rather elementary yet instructive applications of the Gibbs sampler. The fourth example describes a reasonably sophisticated application of the Gibbs sam- pler in the important arena of credibility for classifica- tion ratemaking via hierarchical models, and involves the Bayesian prediction of frequency counts in workers compensation insurance. 114 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 115 1. INTRODUCTION The purpose of this paper is to acquaint the readership of the Proceedings with a class of simulation techniques known as Markov chain Monte Carlo (MCMC) methods. These methods permit a practitioner to simulate a dependent sequence of ran- dom draws from very complicated stochastic models. The main emphasis will be placed on one MCMC method known as the Gibbs sampler. It is not an understatement to say that several hun- dred papers relating to the Gibbs sampling methodology have ap- peared in the statistical literature since 1990. Yet, the Gibbs sam- pler has made only a handful of appearances within the actuarial literature to date. Carlin [3] used the Gibbs sampler in order to study the Bayesian state-space modeling of non-standard actuar- ial time series, and Carlin [4] used it to develop various Bayesian approaches to graduation. Klugman and Carlin [19] also used the Gibbs sampler in the arena of Bayesian graduation, this time concentrating on a hierarchical version of Whittaker-Henderson graduation. Scollnik [24] studied a simultaneous equations model for insurance ratemaking, and conducted a Bayesian analysis of this model with the Gibbs sampler. This paper reviews the essential nature of the Gibbs sampling algorithm and illustrates its application with four examples of varying complexity. This paper is primarily expository, although references are provided to important theoretical results in the published literature. The reader is presumed to possess at least a passing familiarity with the material relating to statistical com- puting and stochastic simulation present in the syllabus for CAS Associateship Examination Part 4B. The theoretical content of the paper is mainly concentrated in Section 2, which provides a brief discussion of Markov chains and the properties of MCMC methods. Except for noting Equations 2.1 and 2.2 along with their interpretation, the reader may skip over Section 2 the first time through reading this paper. Section 3 formally introduces the Gibbs sampler and illustrates it with an example. Section 4 discusses some of the practical considerations related to the 116 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS implementation of a Gibbs sampler. In Section 5, some aspects of Bayesian inference using Gibbs sampling are considered, and two final examples are presented. The first of these concerns the Bayesian estimation of the parameter for a size of loss distribu- tion when grouped data are observed. The second addresses cred- ibility for classification ratemaking via hierarchical models and involves the Bayesian prediction of frequency counts in workers compensation insurance. In Section 6 we conclude our presen- tation and point out some areas of application to be explored in the future. Since the subject of MCMC methods is still foreign to most actuaries at this time, we will conclude this section with a simple introductory example, which we will return to in Section 3. Example 1 This example starts by recalling that a generalized Pareto dis- tribution can be constructed by mixing one gamma distribution with another gamma distribution in a certain manner. (See for example, Hogg and Klugman [16, pp. 53–54].) More precisely, if a loss random variable X has a conditional gamma (k,µ) dis- tribution with density µk f x µ xk 1 µx < x < ( ) = ¡ k ¡ exp( ), 0 , (1.1) j ( ) ¡ 1 and the mixing random variable µ has a marginal gamma (®,¸) distribution with density ¸® f µ µ® 1 ¸µ < µ < ( ) = ¡ ® ¡ exp( ), 0 , ( ) ¡ 1 then X has a marginal generalized Pareto (®,¸,k) distribution with density ¡ ® k ¸®xk 1 f x ( + ) ¡ < x < : ( ) = ¡ ® ¡ k ¸ x ® k , 0 ( ) ( )( + ) + 1 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 117 It also follows that the conditional distribution of µ given X is also given by a gamma distribution, namely, f(µ x) gamma (® + k,¸ + x), 0 < µ < : (1.2) j » 1 We will perform the following iterative sampling algorithm, which is based upon the conditional distributions appearing in Equations 1.1 and 1.2: 1. Select arbitrary starting values X(0) and µ(0). 2. Set the counter index i = 0. i i i 3. Sample X( +1) from f(x µ( )) gamma (k,µ( )). j » i i 4. Sample µ( +1) from f(µ X( +1)) gamma (® + k,¸+ i X( +1)). j » 5. Set i i + 1 and return to Step 3. à For the sake of illustration, we assigned the model parameters ® = 5, ¸ = 1000 and k = 2 so that the marginal distribution of µ is gamma (5,1000) with mean 0.005 and the marginal distribu- tion of X is generalized Pareto (5,1000,2) with mean 500. We then ran the algorithm described above on a fast computer for a total of 500 iterations and stored the sequence of generated val- ues X(0), µ(0), X(1), µ(1),:::, X(499), µ(499), X(500), µ(500). It must be emphasized that this sequence of random draws is clearly not independent, since X(1) depends upon µ(0), µ(1) depends upon X(1), and so forth. Our two starting values were arbitrarily se- lected to be X(0) = 20 and µ(0) = 10. The sequence of sampled i values for X( ) is plotted in Figure 1, along with the sequence i of sampled values for µ( ), for iterations 100 through 200. Both sequences do appear to be random, and some dependencies be- tween successive values are discernible in places. In Figure 2, we plot the histograms of the last 500 val- ues appearing in each of the two sequences of sampled values (the starting values X(0) and µ(0) were discarded at this point). 118 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS FIGURE 1 X i µ i SAMPLE PATHS FOR ( ) AND ( ) IN EXAMPLE 1 In each plot, we also overlay the actual density curve for the marginal distribution of either X or µ. Surprisingly, the depen- dent sampling scheme we implemented, which was based upon the full conditional distributions f(µ x) and f(x µ), appears to have generated random samples fromj the underljying marginal distributions. Now, notice that the marginal distribution of X may be inter- preted as the average of the conditional distribution of X given AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 119 FIGURE 2 X µ HISTOGRAMS OF SAMPLE VALUES FOR AND IN EXAMPLE 1 µ taken with respect to the marginal distribution of µ; that is, f x f x µ f µ dµ: ( ) = Z ( ) ( ) j i Since the sampled values of µ( ) appear to constitute a random sample of sorts from the marginal distribution of µ, this suggests that a naive estimate of the value of the marginal density function for X at the point x might be constructed by taking the empirical i i average of f(x µ( )) over the sampled values for µ( ). If µ(1) = j 120 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 0:0055, for example, then f(x µ(1)) = 0:00552xexp( 0:0055x): j ¡ i One does a similar computation for the other values of µ( ) and averages to get 500 1 i fˆ (x) = f(x µ( )): (1.3) 500 X i=1 j Similarly, we might construct 500 1 i fˆ (µ) = f(µ X( )) (1.4) 500 X i=1 j as a density estimate of f(µ). These estimated density functions are plotted in Figure 3 along with their exact counterparts, and it is evident that the estimated densities happen to be excellent. 2. MARKOV CHAIN MONTE CARLO In the example of the previous section, we considered an itera- tive simulation scheme that generated two dependent sequences of random variates. Apparently, we were able to use these se- quences in order to capture characteristics of the underlying joint distribution that defined the simulation scheme in the first place. In this section, we will discuss a few properties of certain sim- ulation schemes that generate dependent sequences of random variates and note in what manner these dependent sequences may be used for making useful statistical inference. The main results are given by Equations 2.1 and 2.2. Before we begin, it may prove useful to quickly, and very in- formally, review some elementary Markov chain theory. Tierney [31] provides a much more detailed and rigorous discussion of this material. A Markov chain is just a collection of random vari- X n ables n; 0 , with the distribution of the random variables k on somf e spac¸e •g R governed by the transition probabilities Xµ A X ::: X K X A Pr( n , , n) = ( n, ), +1 2 j 0 AN INTRODUCTION TO MARKOV CHAIN MONTE CARLO METHODS 121 FIGURE 3 X µ ESTIMATED AND EXACT MARGINAL DENSITIES FOR AND IN EXAMPLE 1 where A •.
Load more

Recommended publications
  • The Exponential Family 1 Definition
    The Exponential Family David M. Blei Columbia University November 9, 2016 The exponential family is a class of densities (Brown, 1986). It encompasses many familiar forms of likelihoods, such as the Gaussian, Poisson, multinomial, and Bernoulli. It also encompasses their conjugate priors, such as the Gamma, Dirichlet, and beta. 1 Definition A probability density in the exponential family has this form p.x / h.x/ exp >t.x/ a./ ; (1) j D f g where is the natural parameter; t.x/ are sufficient statistics; h.x/ is the “base measure;” a./ is the log normalizer. Examples of exponential family distributions include Gaussian, gamma, Poisson, Bernoulli, multinomial, Markov models. Examples of distributions that are not in this family include student-t, mixtures, and hidden Markov models. (We are considering these families as distributions of data. The latent variables are implicitly marginalized out.) The statistic t.x/ is called sufficient because the probability as a function of only depends on x through t.x/. The exponential family has fundamental connections to the world of graphical models (Wainwright and Jordan, 2008). For our purposes, we’ll use exponential 1 families as components in directed graphical models, e.g., in the mixtures of Gaussians. The log normalizer ensures that the density integrates to 1, Z a./ log h.x/ exp >t.x/ d.x/ (2) D f g This is the negative logarithm of the normalizing constant. The function h.x/ can be a source of confusion. One way to interpret h.x/ is the (unnormalized) distribution of x when 0. It might involve statistics of x that D are not in t.x/, i.e., that do not vary with the natural parameter.
    [Show full text]
  • Markov Chain Monte Carlo Convergence Diagnostics: a Comparative Review
    Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review By Mary Kathryn Cowles and Bradley P. Carlin1 Abstract A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribu- tion of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler convergence, including applying diagnostic procedures to a small number of parallel chains, monitoring autocorrelations and cross- correlations, and modifying parameterizations or sampling algorithms appropriately. We emphasize, however, that it is not possible to say with certainty that a finite sample from an MCMC algorithm is rep- resentative of an underlying stationary distribution. 1 Mary Kathryn Cowles is Assistant Professor of Biostatistics, Harvard School of Public Health, Boston, MA 02115. Bradley P. Carlin is Associate Professor, Division of Biostatistics, School of Public Health, University of Minnesota, Minneapolis, MN 55455.
    [Show full text]
  • Meta-Learning MCMC Proposals
    Meta-Learning MCMC Proposals Tongzhou Wang∗ Yi Wu Facebook AI Research University of California, Berkeley [email protected] [email protected] David A. Moorey Stuart J. Russell Google University of California, Berkeley [email protected] [email protected] Abstract Effective implementations of sampling-based probabilistic inference often require manually constructed, model-specific proposals. Inspired by recent progresses in meta-learning for training learning agents that can generalize to unseen environ- ments, we propose a meta-learning approach to building effective and generalizable MCMC proposals. We parametrize the proposal as a neural network to provide fast approximations to block Gibbs conditionals. The learned neural proposals generalize to occurrences of common structural motifs across different models, allowing for the construction of a library of learned inference primitives that can accelerate inference on unseen models with no model-specific training required. We explore several applications including open-universe Gaussian mixture models, in which our learned proposals outperform a hand-tuned sampler, and a real-world named entity recognition task, in which our sampler yields higher final F1 scores than classical single-site Gibbs sampling. 1 Introduction Model-based probabilistic inference is a highly successful paradigm for machine learning, with applications to tasks as diverse as movie recommendation [31], visual scene perception [17], music transcription [3], etc. People learn and plan using mental models, and indeed the entire enterprise of modern science can be viewed as constructing a sophisticated hierarchy of models of physical, mental, and social phenomena. Probabilistic programming provides a formal representation of models as sample-generating programs, promising the ability to explore a even richer range of models.
    [Show full text]
  • 1 Introduction to Bayesian Inference 2 Introduction to Gibbs Sampling
    MCMC I: July 5, 2016 1 MCMC I 8th Summer Institute in Statistics and Modeling in Infectious Diseases Course Time Plan July 13-15, 2016 Instructors: Vladimir Minin, Kari Auranen, M. Elizabeth Halloran Course Description: This module is an introduction to Markov chain Monte Carlo methods with some simple applications in infectious disease studies. The course includes an introduction to Bayesian inference, Monte Carlo, MCMC, some background theory, and convergence diagnostics. Algorithms include Gibbs sampling and Metropolis-Hastings and combinations. Programming is in R. Familiarity with the R statistical package or other computing language is needed. Course schedule: The course is composed of 10 90-minute sessions, for a total of 15 hours of instruction. 1 Introduction to Bayesian Inference • Overview of the course. • Bayesian inference: Likelihood, prior, posterior, normalizing constant • Conjugate priors; Beta-binomial; Poisson-gamma; normal-normal • Posterior summaries, mean, mode, posterior intervals • Motivating examples: Chain binomial model (Reed-Frost), General Epidemic Model, SIS model. • Lab: – Goals: Warm-up with R for simple Bayesian computation – Example: Posterior distribution of transmission probability with a binomial sampling distribution using a conjugate beta prior distribution – Summarizing posterior inference (mean, median, posterior quantiles and intervals) – Varying the amount of prior information – Writing an R function 2 Introduction to Gibbs Sampling • Chain binomial model and data augmentation • Brief introduction
    [Show full text]
  • Markov Chain Monte Carlo Edps 590BAY
    Markov Chain Monte Carlo Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2019 Overivew Bayesian computing Markov Chains Metropolis Example Practice Assessment Metropolis 2 Practice Overview ◮ Introduction to Bayesian computing. ◮ Markov Chain ◮ Metropolis algorithm for mean of normal given fixed variance. ◮ Revisit anorexia data. ◮ Practice problem with SAT data. ◮ Some tools for assessing convergence ◮ Metropolis algorithm for mean and variance of normal. ◮ Anorexia data. ◮ Practice problem. ◮ Summary Depending on the book that you select for this course, read either Gelman et al. pp 2751-291 or Kruschke Chapters pp 143–218. I am relying more on Gelman et al. C.J. Anderson (Illinois) MCMC Fall 2019 2.2/ 45 Overivew Bayesian computing Markov Chains Metropolis Example Practice Assessment Metropolis 2 Practice Introduction to Bayesian Computing ◮ Our major goal is to approximate the posterior distributions of unknown parameters and use them to estimate parameters. ◮ The analytic computations are fine for simple problems, ◮ Beta-binomial for bounded counts ◮ Normal-normal for continuous variables ◮ Gamma-Poisson for (unbounded) counts ◮ Dirichelt-Multinomial for multicategory variables (i.e., a categorical variable) ◮ Models in the exponential family with small number of parameters ◮ For large number of parameters and more complex models ◮ Algebra of analytic solution becomes overwhelming ◮ Grid takes too much time. ◮ Too difficult for most applications. C.J. Anderson (Illinois) MCMC Fall 2019 3.3/ 45 Overivew Bayesian computing Markov Chains Metropolis Example Practice Assessment Metropolis 2 Practice Steps in Modeling Recall that the steps in an analysis: 1. Choose model for data (i.e., p(y θ)) and model for parameters | (i.e., p(θ) and p(θ y)).
    [Show full text]
  • Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation
    Fast Bayesian Non-Negative Matrix Factorisation and Tri-Factorisation Thomas Brouwer Jes Frellsen Pietro Lio’ Computer Laboratory Department of Engineering Computer Laboratory University of Cambridge University of Cambridge University of Cambridge United Kingdom United Kingdom United Kingdom [email protected] [email protected] [email protected] Abstract We present a fast variational Bayesian algorithm for performing non-negative matrix factorisation and tri-factorisation. We show that our approach achieves faster convergence per iteration and timestep (wall-clock) than Gibbs sampling and non-probabilistic approaches, and do not require additional samples to estimate the posterior. We show that in particular for matrix tri-factorisation convergence is difficult, but our variational Bayesian approach offers a fast solution, allowing the tri-factorisation approach to be used more effectively. 1 Introduction Non-negative matrix factorisation methods Lee and Seung [1999] have been used extensively in recent years to decompose matrices into latent factors, helping us reveal hidden structure and predict missing values. In particular we decompose a given matrix into two smaller matrices so that their product approximates the original one. The non-negativity constraint makes the resulting matrices easier to interpret, and is often inherent to the problem – such as in image processing or bioinformatics (Lee and Seung [1999], Wang et al. [2013]). Some approaches approximate a maximum likelihood (ML) or maximum a posteriori (MAP) solution that minimises the difference between the observed matrix and the decomposition of this matrix. This gives a single point estimate, which can lead to overfitting more easily and neglects uncertainty. Instead, we may wish to find a full distribution over the matrices using a Bayesian approach, where we define prior distributions over the matrices and then compute their posterior after observing the actual data.
    [Show full text]
  • Bayesian Phylogenetics and Markov Chain Monte Carlo Will Freyman
    IB200, Spring 2016 University of California, Berkeley Lecture 12: Bayesian phylogenetics and Markov chain Monte Carlo Will Freyman 1 Basic Probability Theory Probability is a quantitative measurement of the likelihood of an outcome of some random process. The probability of an event, like flipping a coin and getting heads is notated P (heads). The probability of getting tails is then 1 − P (heads) = P (tails). • The joint probability of event A and event B both occuring is written as P (A; B). The joint probability of mutually exclusive events, like flipping a coin once and getting heads and tails, is 0. • The probability of A occuring given that B has already occurred is written P (AjB), which is read \the probability of A given B". This is a conditional probability. • The marginal probability of A is P (A), which is calculated by summing or integrating the joint probability over B. In other words, P (A) = P (A; B) + P (A; not B). • Joint, conditional, and marginal probabilities can be combined with the expression P (A; B) = P (A)P (BjA) = P (B)P (AjB). • The above expression can be rearranged into Bayes' theorem: P (BjA)P (A) P (AjB) = P (B) Bayes' theorem is a straightforward and uncontroversial way to calculate an inverse conditional probability. • If P (AjB) = P (A) and P (BjA) = P (B) then A and B are independent. Then the joint probability is calculated P (A; B) = P (A)P (B). 2 Interpretations of Probability What exactly is a probability? Does it really exist? There are two major interpretations of probability: • Frequentists believe that the probability of an event is its relative frequency over time.
    [Show full text]
  • Bayesian Inference in the Normal Linear Regression Model
    Bayesian Inference in the Normal Linear Regression Model () Bayesian Methods for Regression 1 / 53 Bayesian Analysis of the Normal Linear Regression Model Now see how general Bayesian theory of overview lecture works in familiar regression model Reading: textbook chapters 2, 3 and 6 Chapter 2 presents theory for simple regression model (no matrix algebra) Chapter 3 does multiple regression In lecture, I will go straight to multiple regression Begin with regression model under classical assumptions (independent errors, homoskedasticity, etc.) Chapter 6 frees up classical assumptions in several ways Lecture will cover one way: Bayesian treatment of a particular type of heteroskedasticity () Bayesian Methods for Regression 2 / 53 The Regression Model Assume k explanatory variables, xi1,..,xik for i = 1, .., N and regression model: yi = b1 + b2xi2 + ... + bk xik + #i . Note xi1 is implicitly set to 1 to allow for an intercept. Matrix notation: y1 y2 y = 2 . 3 6 . 7 6 7 6 y 7 6 N 7 4 5 # is N 1 vector stacked in same way as y () Bayesian Methods for Regression 3 / 53 b is k 1 vector X is N k matrix 1 x12 .. x1k 1 x22 .. x2k X = 2 ..... 3 6 ..... 7 6 7 6 1 x .. x 7 6 N2 Nk 7 4 5 Regression model can be written as: y = X b + #. () Bayesian Methods for Regression 4 / 53 The Likelihood Function Likelihood can be derived under the classical assumptions: 1 2 # is N(0N , h IN ) where h = s . All elements of X are either fixed (i.e. not random variables). Exercise 10.1, Bayesian Econometric Methods shows that likelihood function can be written in
    [Show full text]
  • The Markov Chain Monte Carlo Approach to Importance Sampling
    The Markov Chain Monte Carlo Approach to Importance Sampling in Stochastic Programming by Berk Ustun B.S., Operations Research, University of California, Berkeley (2009) B.A., Economics, University of California, Berkeley (2009) Submitted to the School of Engineering in partial fulfillment of the requirements for the degree of Master of Science in Computation for Design and Optimization at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2012 c Massachusetts Institute of Technology 2012. All rights reserved. Author.................................................................... School of Engineering August 10, 2012 Certified by . Mort Webster Assistant Professor of Engineering Systems Thesis Supervisor Certified by . Youssef Marzouk Class of 1942 Associate Professor of Aeronautics and Astronautics Thesis Reader Accepted by . Nicolas Hadjiconstantinou Associate Professor of Mechanical Engineering Director, Computation for Design and Optimization 2 The Markov Chain Monte Carlo Approach to Importance Sampling in Stochastic Programming by Berk Ustun Submitted to the School of Engineering on August 10, 2012, in partial fulfillment of the requirements for the degree of Master of Science in Computation for Design and Optimization Abstract Stochastic programming models are large-scale optimization problems that are used to fa- cilitate decision-making under uncertainty. Optimization algorithms for such problems need to evaluate the expected future costs of current decisions, often referred to as the recourse function. In practice, this calculation is computationally difficult as it involves the evaluation of a multidimensional integral whose integrand is an optimization problem. Accordingly, the recourse function is estimated using quadrature rules or Monte Carlo methods. Although Monte Carlo methods present numerous computational benefits over quadrature rules, they require a large number of samples to produce accurate results when they are embedded in an optimization algorithm.
    [Show full text]
  • Markov Chain Monte Carlo (Mcmc) Methods
    MARKOV CHAIN MONTE CARLO (MCMC) METHODS 0These notes utilize a few sources: some insights are taken from Profs. Vardeman’s and Carriquiry’s lecture notes, some from a great book on Monte Carlo strategies in scientific computing by J.S. Liu. EE 527, Detection and Estimation Theory, # 4c 1 Markov Chains: Basic Theory Definition 1. A (discrete time/discrete state space) Markov chain (MC) is a sequence of random quantities {Xk}, each taking values in a (finite or) countable set X , with the property that P {Xn = xn | X1 = x1,...,Xn−1 = xn−1} = P {Xn = xn | Xn−1 = xn−1}. Definition 2. A Markov Chain is stationary if 0 P {Xn = x | Xn−1 = x } is independent of n. Without loss of generality, we will henceforth name the elements of X with the integers 1, 2, 3,... and call them “states.” Definition 3. With 4 pij = P {Xn = j | Xn−1 = i} the square matrix 4 P = {pij} EE 527, Detection and Estimation Theory, # 4c 2 is called the transition matrix for a stationary Markov Chain and the pij are called transition probabilities. Note that a transition matrix has nonnegative entries and its rows sum to 1. Such matrices are called stochastic matrices. More notation for a stationary MC: Define (k) 4 pij = P {Xn+k = j | Xn = i} and (k) 4 fij = P {Xn+k = j, Xn+k−1 6= j, . , Xn+1 6= j, | Xn = i}. These are respectively the probabilities of moving from i to j in k steps and first moving from i to j in k steps.
    [Show full text]
  • Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling
    Accelerating Bayesian Inference on Structured Graphs Using Parallel Gibbs Sampling Glenn G. Ko∗, Yuji Chai∗, Rob A. Rutenbary, David Brooks∗, Gu-Yeon Wei∗ ∗Harvard University, Cambridge, USA yUniversity of Pittsburgh, Pittsburgh, USA [email protected], [email protected], [email protected], fdbrooks, [email protected] Abstract—Bayesian models and inference is a class of machine with substantial uncertainty and noise is normally reserved for learning that is useful for solving problems where the amount Bayesian models and inference. of data is scarce and prior knowledge about the application Bayesian models pose challenges that are different from allows you to draw better conclusions. However, Bayesian models often requires computing high-dimensional integrals and finding deep learning. They tend to have more complex structure, the posterior distribution can be intractable. One of the most that may be hierarchical with dependencies between different commonly used approximate methods for Bayesian inference distributions Without an easily-defined common computational is Gibbs sampling, which is a Markov chain Monte Carlo kernel and hardware accelerators like GPU and TPU [5], and (MCMC) technique to estimate target stationary distribution. software frameworks such as Tensorflow [6] and PyTorch [7], The idea in Gibbs sampling is to generate posterior samples by iterating through each of the variables to sample from its it is hard to build and run models fast and efficiently. This conditional given all the other variables fixed. While Gibbs work explores hardware accelerators for Bayesian models and sampling is a popular method for probabilistic graphical models inference which has growing interest and demand. such as Markov Random Field (MRF), the plain algorithm is Probabilistic graphical models are Bayesian models described slow as it goes through each of the variables sequentially.
    [Show full text]
  • Gibbs Sampling, Exponential Families and Orthogonal Polynomials
    Statistical Science 2008, Vol. 23, No. 2, 151–178 DOI: 10.1214/07-STS252 c Institute of Mathematical Statistics, 2008 Gibbs Sampling, Exponential Families and Orthogonal Polynomials1 Persi Diaconis, Kshitij Khare and Laurent Saloff-Coste Abstract. We give families of examples where sharp rates of conver- gence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, expo- nential families, conjugate priors, location families, orthogonal polyno- mials, singular value decomposition. 1. INTRODUCTION which has f as stationary density under mild con- ditions discussed in [4, 102]. The Gibbs sampler, also known as Glauber dy- The algorithm was introduced in 1963 by Glauber namics or the heat-bath algorithm, is a mainstay of [49] to do simulations for Ising models, and inde- scientific computing. It provides a way to draw sam- pendently by Turcin [103]. It is still a standard tool ples from a multivariate probability density f(x1,x2, of statistical physics, both for practical simulation ...,xp), perhaps only known up to a normalizing (e.g., [88]) and as a natural dynamics (e.g., [10]). constant, by a sequence of one-dimensional sampling The basic Dobrushin uniqueness theorem showing problems. From (X1,...,Xp) proceed to (X1′ , X2,..., existence of Gibbs measures was proved based on Xp), then (X1′ , X2′ , X3,...,Xp),..., (X1′ , X2′ ,...,Xp′ ) this dynamics (e.g., [54]). It was introduced as a where at the ith stage, the coordinate is sampled base for image analysis by Geman and Geman [46].
    [Show full text]