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Hierarchical priors and mixture models, with application in regression and density estimation. (English) Zbl 0842.62001

Freeman, P. R. (ed.) et al., Aspects of uncertainty: a tribute to D. V. Lindley. Chichester: Wiley. 363-386 (1994).
In “Bayesian statistics, a review.” (1971; Zbl 0246.62009), D. V. Lindley identified as a success story for Bayesian ideas the advances made in problems of many parameters and the growth of what is now referred to as Bayesian hierarchical modelling (section 8). In that same monograph, Lindley identified non-parametrics as an area notable for lack of Bayesian progress, bemoaning the fact that non-parametric statistics was a ‘subject about which the Bayesian method is embarrasingly silent’ (section 12.1). It is our purpose to exhibit a general framework for hierarchical linear modelling and density estimation, to show how posterior computations via Markov chain simulations can be routinely applied and to provide illustrations in each context.
Section 22.2 provides a rather general theoretical setting and summarizes key features of multivariate data models with hierarchical mixture priors. Section 22.3 discusses Markov chain simulation methods in these models, with special emphasis on models centred around traditional normal structures. Section 22.4 concerns an application in hierarchical regression, highlighting the use of mixture priors for robustness and sensitivity analysis, and section 22.5 develops an application to multivariate density estimation.
For the entire collection see [Zbl 0827.00022].

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62G07 Density estimation
62A01 Foundations and philosophical topics in statistics
62H12 Estimation in multivariate analysis

Citations:

Zbl 0246.62009
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