Bayesian Missing Data Problems: EM, Data Augmentation and by Ming T. Tan

By Ming T. Tan

Bayesian lacking facts difficulties: EM, info Augmentation and Noniterative Computation provides suggestions to lacking info difficulties via particular or noniterative sampling calculation of Bayesian posteriors. The equipment are according to the inverse Bayes formulae found by way of one of many writer in 1995. utilizing the Bayesian method of vital real-world difficulties, the authors concentrate on specific numerical options, a conditional sampling process through info augmentation, and a noniterative sampling procedure through EM-type algorithms.

After introducing the lacking information difficulties, Bayesian technique, and posterior computation, the ebook succinctly describes EM-type algorithms, Monte Carlo simulation, numerical suggestions, and optimization tools. It then offers targeted posterior ideas for difficulties, resembling nonresponses in surveys and cross-over trials with lacking values. It additionally offers noniterative posterior sampling recommendations for difficulties, equivalent to contingency tables with supplemental margins, aggregated responses in surveys, zero-inflated Poisson, capture-recapture types, combined results types, right-censored regression version, and restricted parameter versions. The textual content concludes with a dialogue on compatibility, a basic factor in Bayesian inference.

This publication deals a unified remedy of an array of statistical difficulties that contain lacking info and restricted parameters. It exhibits how Bayesian systems might be important in fixing those problems.

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Extra info for Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation (Chapman & Hall/CRC Biostatistics Series)

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11), then the fourth estimator of m(Ycom ) is the Rao-Blackwellized estimator given by m ˆ 4 (Ycom ) = 1 n n f (Ycom |θ (i) ). 25) i=1 As mentioned by Gelfand et al. (1992), m ˆ 4 (Ycom ) is better than the kernel density estimator under a wide range of loss functions. The disadvantages of the Rao-Blackwellized estimator are that (i) the ˆ 4 (Ycom ) is a closed form of f (Ycom |θ) must be known, and (ii) m mixture density, which is relatively difficult to treat when n is large enough. 5 The Missing Data Problems In the previous section, we assumed that the desired dataset is completely observed.

7 (Homogeneous Poisson process). Let {Yi , i ≥ 1} ∼ HPP(λ) (cf. 1) and Ycom = {Yi }ni=1 . 1, the joint pdf of the successive event times Y1 , . . , Yn is f (Ycom |λ) = λn e−λyn , 0 < y1 < · · · < yn . Let the prior of the rate λ is Gamma(a, b), then the posterior is f (λ|Ycom ) = Gamma(λ|n + a, yn + b). 13), the prior predictive distribution is given by f (Ycom ) = Γ(n + a) ba · , Γ(a) (yn + b)n+a 0 < y1 < · · · < yn . 20) f (yn+1 |Ycom , λ) = λe−λ(yn+1 −yn ) , yn+1 > yn , f (λ|Ycom , yn+1 ) = Gamma(λ|n + 1 + a, yn+1 + b).

2 for more detailed discussions) and the DA algorithm are designed for obtaining the mode of © 2010 by Taylor and Francis Group, LLC 24 1. INTRODUCTION f (θ|Yobs ) and for simulating from f (θ|Yobs ), respectively. The EM and the DA share a simple idea that rather than performing a complicated optimization or simulation, the observed data is augmented with latent data so that a series of simple optimizations or simulations can be performed. As a stochastic version of the EM algorithm, the DA algorithm was originally proposed by Tanner & Wong (1987).

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