Über die Autorin bzw. den Autor

Faming Liang, Associate Professor, Department of Statistics, Texas A&M University.

Chuanhai Liu, Professor, Department of Statistics, Purdue University.

Raymond J. Carroll, Distinguished Professor, Department of Statistics, Texas A&M University.

Von der hinteren Coverseite

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.

• Accompanied by a supporting website featuring datasets used in the book, along with codes used for some simulation examples.

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.

Aus dem Klappentext

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.

• Accompanied by a supporting website featuring datasets used in the book, along with codes used for some simulation examples.

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.

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Advanced Markov Chain Monte Carlo Methods

John Wiley & Sons

Chapter One

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

Pr (X = 1|θ) = θ and Pr (X = 0|θ) = 1 - θ, (1.1)

where X denotes the Bernoulli random variable (r.v.) with X = 1 for success and X = 0 for failure. Write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 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)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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,

X ~ f(X|θ) (X [member of] X, θ [member of] 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,

(θ) ~ π(θ) (θ [member of] Θ0. (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,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)

where L(θ|X) [varies] 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 probability models is unavoidable even for frequentist methods, which requires specification of the sampling model, either parametric or non-parametric, for the observed data X. In addition to the sampling model of the observed data X for developing frequentist procedures concerning the unknown quantity θ, Bayesian inference demands a fully specified prior for θ. This is natural when prior information on θ is available and can be summarized precisely by a probability distribution. For situations where such information is neither available nor easily quantified with a precise probability distribution, especially for high dimensional problems, a commonly used method in practice is the Jeffreys method, which suggests the prior of the form

π_J (θ) [varies] |I(θ)|^1/2 (θ [member of] Θ), (1.6)

where I(θ) denotes the Fisher information

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Jeffreys priors have the appealing property that they are invariant under reparameterization. A theoretical adjustment in terms of frequency properties in the context of large samples can be found in Welch and Peers (1963). Note that prior distributions do not need to be proper as long as the posteriors are proper and produce sensible inferential results. The following Gaussian example shows that the Jeffreys prior is sensible for single parameters.

* Example 1.2 The Gaussian N(μ, 1) Model

Suppose that a sample is considered to have taken from the Gaussian population N(μ, 1) with unit variance and unknown mean μ to be inferred. The Fisher information is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where φ (x - μ) = (2π)^-1/2 exp{-1/2 (x - μ)²} is the pdf of N(μ, 1). It follows that the Jeffreys prior for θ is the flat prior

π_J (μ) [varies] 1 (-∞ < μ < ∞), (1.7)

resulting in the corresponding posterior distribution of θ given X

π_J (μ|X) = N(X, 1). (1.8)

Care must be taken when using the Jeffreys rule. For example, it is easy to show that applying the Jeffreys rule to the Gaussian model N(μ, σ²) with both mean μ and variance σ² unknown leads to the prior

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, this is not the commonly used prior that has better frequency properties (for inference about μ or σ) and is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is, μ and σ² are independent and the distributions for both μ and ln σ² are flat. For high dimensional problems with small samples, the Jeffreys rule often becomes even less appealing. There are also different perspectives, provided by the extensive work on reference priors by José Bernardo and James Berger (see, e.g., Bernardo, 1979; Berger, 1985). For more discussion of prior specifications, see Kass and Wasserman (1996).

For practical purposes, we refer to Box and Tiao (1973) and Gelman et al. (2004) for discussion on specification of prior distributions. The general guidance for specification of priors when no prior information is available, as is typical in Bayesian analysis, is to find priors that lead to posteriors having good frequency...