Selection of proposal distributions for generalized importance sampling estimators
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The standard importance sampling (IS) method uses samples from a single proposal distribution and assigns weights to them, according to the ratio of the target and proposal pdfs. This naive IS estimator, generally does not work well in multiple targets examples as the weights can take arbitrarily large values making the estimator highly unstable. In such situations, alternative generalized IS estimators involving samples from multiple proposal distributions are preferred. Just like the standard IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods based on (i) a geometric space filling coverage criterion, (ii) a sequential maximum variance approach, and (iii) a maximum entropy approach. In particular, we describe these methods in the context of Doss's (2010) two-stage IS estimator, although the first two methods are applicable to any multi-proposal IS estimator. Two of the proposed approaches use estimates of asymptotic variances of Geyer's (1994) reverse logistic estimator and Doss's (2010) IS estimators. Thus, we provide consistent spectral variance estimators for these asymptotic variances. We demonstrate the performance of these spectral variance methods for estimating the size parameter of a negative binomial generalized linear model. The proposed methods for selecting proposal densities are illustrated using two detailed examples. The first example is a robust Bayesian binary regression model, where the generalized IS method is used to estimate the link function parameter. The second example involves analysis of count data using the Poisson spatial generalized linear mixed model.
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