Abstract
In this paper we propose a conceptually straightforward method to estimate the marginal data density value (also called the marginal likelihood). We show that the marginal likelihood is equal to the prior mean of the conditional density of the data given the vector of parameters restricted to a certain subset of the parameter space, A, times the reciprocal of the posterior probability of the subset A. This identity motivates one to use Arithmetic Mean estimator based on simulation from the prior distribution restricted to any (but reasonable) subset of the space of parameters. By trimming this space, regions of relatively low likelihood are removed, and thereby the efficiency of the Arithmetic Mean estimator is improved. We show that the adjusted Arithmetic Mean estimator is unbiased and consistent.
Highlights
The marginal data densities are key quantities needed for formal Bayesian model selection and for model averaging; see, e.g. Zellner (1971)
Under additional assumptions (first, the subset A is compact; second, the likelihood is bounded on A; third, P (A|y) is known; fourth, it is possible to generate samples from the prior distribution), it is easy to show that the Corrected Arithmetic Mean (CAM)
The aim of this example is to compare estimates obtained by using the Corrected Arithmetic Mean with those obtained by the methods proposed by Bartolucci et al (2006), Friel and Pettitt (2008)
Summary
The marginal data densities (i.e. the normalizing constants of the posterior distributions of the model parameters, called the marginal likelihoods or the integrated likelihoods) are key quantities needed for formal Bayesian model selection and for model averaging; see, e.g. Zellner (1971). Given the subset A ⊆ Θ, the identities (6) and (7) naturally lead to the following estimators of the marginal data density value Various other numerical methods have been proposed in the literature to estimate the marginal data density value directly or to compute ratios of two marginal likelihoods (i.e. Bayes factors); for a review see, e.g. We propose a new class of estimators of the marginal likelihood They are based on the Arithmetic Mean of likelihoods calculated only over an arbitrary subset of the space of model parameters corrected by the reciprocal of the posterior probability of the subset. We conclude that preliminary experience with estimators introduced here is very promising
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