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A case study of the widely applicable Bayesian information criterion and its optimality. (English) Zbl 1332.62099
Summary: In Bayesian statistics, the marginal likelihood (evidence) is one of the key factors that can be used as a measure of model goodness. However, for many practical model families it cannot be computed analytically. An alternative solution is to use some approximation method or time-consuming sampling method. The widely applicable Bayesian information criterion (WBIC) was developed recently to have a marginal likelihood approximation that works also with singular models. The central idea of the approximation is to select a single thermodynamic integration term (power posterior) with the (approximated) optimal temperature \(\beta^*=1/\log (n)\), where \(n\) is the data size. We apply this new approximation to the analytically solvable Gaussian process regression case to show that the optimal temperature may depend also on data itself or other variables, such as the noise level. Moreover, we show that the steepness of a thermodynamic curve at the optimal temperature indicates the magnitude of the error that WBIC makes.

MSC:
62F15 Bayesian inference
62G08 Nonparametric regression and quantile regression
Software:
GPstuff
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