By Peter D. Congdon
The use of Markov chain Monte Carlo (MCMC) tools for estimating hierarchical versions comprises advanced facts constructions and is usually defined as a innovative improvement. An intermediate-level therapy of Bayesian hierarchical versions and their purposes, Applied Bayesian Hierarchical Methods demonstrates some great benefits of a Bayesian method of info units concerning inferences for collections of comparable devices or variables and in equipment the place parameters may be handled as random collections.
Emphasizing computational concerns, the publication presents examples of the next program settings: meta-analysis, info established in house or time, multilevel and longitudinal information, multivariate facts, nonlinear regression, and survival time information. For the labored examples, the textual content commonly employs the WinBUGS package deal, permitting readers to discover substitute chance assumptions, regression buildings, and assumptions on past densities. It additionally accommodates BayesX code, that is really priceless in nonlinear regression. to illustrate MCMC sampling from first ideas, the writer contains labored examples utilizing the R package.
Through illustrative info research and a spotlight to statistical computing, this publication specializes in the sensible implementation of Bayesian hierarchical tools. It additionally discusses numerous concerns that come up whilst making use of Bayesian thoughts in hierarchical and random results models.
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Additional resources for Applied Bayesian Hierarchical Methods
For example, Rodrigues and Assuncao (2008) demonstrate propriety in the posterior of spatially varying regression parameter models under a class of improper priors. More generally, Markov random ﬁeld (MRF) priors, such as random walks in time or spatial conditional autoregressive priors (Chapter 4), may have joint forms that are improper, with a singular covariance matrix—see for example, the discussion by Sun et al. (2000, 28–30). The joint prior only identiﬁes differences between pairs of eﬀects and unless additional constraints are applied to the random eﬀects, this may cause issues with posterior propriety.
Then the ratio of posterior model probabilities is obtained as, p(m = 2|y) p(y|m = 2) = = B21 , p(m = 1|y) p(y|m = 1) where B21 is the Bayes factor. Kass and Raftery (1995) provide guidelines for interpreting B21 . If 2 loge B21 is larger than 10 the evidence for model 2 is very strong, while values of 2 loge B21 < 2 are inconclusive as evidence in favor of one or other model. Note that such criteria are inﬂuenced by the prior adopted. In general, diﬀuse priors (whether on ﬁxed eﬀect parameters or variances) are to be avoided as they tend to favor the selection of the simpler model.
6. Seeds Data: Mixture Prior for Random Eﬀects Variance To illustrate a simple discrete mixture to the prior for a random eﬀects variance, consider the seeds data from Crowder (1978), with model, yi ∼ Binomial(Si , pi ), logit(pi ) = β1 + β2 x1i + β3 x2i + β4 x1i x2i + ui , ui ∼ N (0, σu2 ). 001) prior on the precision τ = 1/σu2 . However, instead of a single form of prior for σu2 , one may instead average over two or more plausible alternatives, either plausible mathematically, or in terms of the level of evidence they include (their informativeness using, say, previous studies).