Abstract
Data augmentation is a common tool in Bayesian statistics, especially in the application of MCMC. Data augmentation is used where direct computation of the posterior density, π(θ|x), of the parameters θ, given the observed data x, is not possible. We show that for a range of problems, it is possible to augment the data by y, such that, π(θ|x,y) is known, and π(y|x) can easily be computed. In particular, π(y|x) is obtained by collapsing π(y,θ|x) through integrating out θ. This allows the exact computation of π(θ|x) as a mixture distribution without recourse to approximating methods such as MCMC. Useful byproducts of the exact posterior distribution are the marginal likelihood of the model and the exact predictive distribution.
Highlights
A key aim of parametric Bayesian statistics is, given a model M, with parameters θ = (θ1, θ2, . . . , θd ) and observed data P
In this paper we present a generic framework for obtaining the exact posterior distribution π(θ |x) using data augmentation
In Neal and Subba Rao (2007) and Enciso-Mora et al (2009a), MCMC is used to obtain samples from the posterior distribution of the parameters of INARMA(p, q), whereas in McCabe and Martin (2005) numerical integration is used to compute the posterior distribution of integer-valued autoregressive (INAR)(1) models with Poisson, binomial and negative binomial models
Summary
The second class of model is the integer-valued autoregressive (INAR) process, see McKenzie (2003), McCabe and Martin (2005) and Neal and Subba Rao (2007). In Neal and Subba Rao (2007) and Enciso-Mora et al (2009a), MCMC is used to obtain samples from the posterior distribution of the parameters of INARMA(p, q), whereas in McCabe and Martin (2005) numerical integration is used to compute the posterior distribution of INAR(1) models with Poisson, binomial and negative binomial models. 4. We follow Neal and Subba Rao (2007), Sect. We depart from Neal and Subba Rao (2007), who used the above data augmentation within a Gibbs sampler, by integrating out θ and identifying the sufficient statistics.
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