Advanced Markov Chain Monte Carlo Methods

Learning from Past Samples

Faming Liang, Chuanhai Liu, Raymond Carroll

• Publisher: Wiley
• Year: 2011
• ISBN:
  9781119956808
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Markov Chain Monte Carlo (MCMC) methods are now an indispensable tool in scientific computing. This book discusses recent developments of MCMC methods with an emphasis on those making use of past sample information during simulations. The application examples are drawn from diverse fields such as bioinformatics, machine learning, social science, combinatorial optimization, and computational physics.

Key Features:

• Expanded coverage of the stochastic approximation Monte Carlo and dynamic weighting algorithms that are essentially immune to local trap problems.
• A detailed discussion of the Monte Carlo Metropolis-Hastings algorithm that can be used for sampling from distributions with intractable normalizing constants.
• Up-to-date accounts of recent developments of the Gibbs sampler.
• Comprehensive overviews of the population-based MCMC algorithms and the MCMC algorithms with adaptive proposals.

This book can be used as a textbook or a reference book for a one-semester graduate course in statistics, computational biology, engineering, and computer sciences. Applied or theoretical researchers will also find this book beneficial.

Chapter 1

Bayesian Inference and Markov Chain Monte Carlo

1.1 Bayes

Bayesian inference is a probabilistic inferential method. In the last two decades, it has become more popular than ever due to affordable computing power and recent advances in Markov chain Monte Carlo (MCMC) methods for approximating high dimensional integrals.

Bayesian inference can be traced back to Thomas Bayes (1764), who derived the inverse probability of the success probability θ in a sequence of independent Bernoulli trials, where θ was taken from the uniform distribution on the unit interval (0, 1) but treated as unobserved. For later reference, we describe his experiment using familiar modern terminology as follows.

Example 1.1 The Bernoulli (or Binomial) Model With Known Prior

Suppose that θ ~ Unif(0, 1), the uniform distribution over the unit interval (0, 1), and that x₁,..., x_n is a sample from Bernoulli(θ), which has the sample space

= {0, 1} and probability mass function (pmf)

(1.1)

where X denotes the Bernoulli random variable (r.v.) with X = 1 for success and X =0 for failure. Write

the observed number of successes in the n Bernoulli trials. Then N|θ ~ Binomial(n, θ), the Binomial distribution with parameters size n and probability of success θ.

The inverse probability of θ given x₁,..., x_n, known as the posterior distribution, is obtained from Bayes’ theorem, or more rigorously in modern probability theory, the definition of conditional distribution, as the Beta distribution Beta(1 + N, 1+n − N) with probability density function (pdf)

(1.2)

where B(·, ·) stands for the Beta function.

1.1.1 Specification of Bayesian Models

Real world problems in statistical inference involve the unknown quantity θ and observed data X. For different views on the philosophical foundations of Bayesian approach, see Savage (1967a, b), Berger (1985), Rubin (1984), and Bernardo and Smith (1994). As far as the mathematical description of a Bayesian model is concerned, Bayesian data analysis amounts to

(i) specifying a sampling model for the observed data X, conditioned on an unknown quantity θ,

(1.3)

where f(X|θ) stands for either pdf or pmf as appropriate, and

(ii) specifying a marginal distribution π(θ) for θ, called the prior distribution or simply the prior for short,

(1.4)

Technically, data analysis for producing inferential results on assertions of interest is reduced to computing integrals with respect to the posterior distribution, or posterior for short,

(1.5)

where L(θ|X) ∝ f(X|θ) in θ, called the likelihood of θ given X. Our focus in this book is on efficient and accurate approximations to these integrals for scientific inference. Thus, limited discussion of Bayesian inference is necessary.

1.1.2 The Jeffreys Priors and Beyond

By its nature, Bayesian inference is necessarily subjective because specification of the full Bayesian model amounts to practically summarizing available information in terms of precise probabilities. Specification of prob...

Table of contents

• Cover
• Title
• Copyright
• Preface
• Acknowledgments
• Publisher’s Acknowledgments
• Chapter 1: Bayesian Inference and Markov Chain Monte Carlo
• Chapter 2: The Gibbs Sampler
• Chapter 3: The Metropolis-Hastings Algorithm
• Chapter 4: Auxiliary Variable MCMC Methods
• Chapter 5: Population-Based MCMC Methods
• Chapter 6: Dynamic Weighting
• Chapter 7: Stochastic Approximation Monte Carlo
• Chapter 8: Markov Chain Monte Carlo with Adaptive Proposals
• Reference
• Index