Bayesian inference for Poisson and multinomial log-linear models
Bayesian inference for Poisson and multinomial log-linear models
Categorical data frequently arise in applications in the social sciences. In such applications,the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data.We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters.Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper.We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a `reference' analysis, then choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.
Southampton Statistical Sciences Research Institute, University of Southampton
Forster, Jonathan J.
e3c534ad-fa69-42f5-b67b-11617bc84879
13 May 2009
Forster, Jonathan J.
e3c534ad-fa69-42f5-b67b-11617bc84879
Forster, Jonathan J.
(2009)
Bayesian inference for Poisson and multinomial log-linear models
(S3RI Methodology Working Papers, M09/11)
Southampton, UK.
Southampton Statistical Sciences Research Institute, University of Southampton
23pp.
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Monograph
(Working Paper)
Abstract
Categorical data frequently arise in applications in the social sciences. In such applications,the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data.We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters.Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper.We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a `reference' analysis, then choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.
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Published date: 13 May 2009
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Local EPrints ID: 66220
URI: http://eprints.soton.ac.uk/id/eprint/66220
PURE UUID: b0b8c3ea-fa39-428f-895a-136a9d0b5996
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Date deposited: 13 May 2009
Last modified: 14 Mar 2024 02:37
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Jonathan J. Forster
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