Prior effective sample size for exponential family distributions with multiple parameters

Abstract

The setting of priors is an important issue in Bayesian analysis. In particular, when external information is applied, a prior with too much information can dominate the posterior inferences. To prevent this effect, the effective sample size (ESS) can be used. Various ESSs have been proposed recently; however, all have the problem of limiting the applicable prior distributions. For example, one ESS can only be used with a prior that can be approximated by a normal distribution, and another ESS cannot be applied when the parameters are multidimensional. We propose an ESS to be applied to more prior distributions when the sampling model belongs to an exponential family (including the normal model and logistic regression models). This ESS has the predictive consistency and can be used with multidimensional parameters. It is confirmed from normally distributed data with the Student's‐t priors that this ESS behaves as well as an existing predictively consistent ESS for one‐parameter exponential families. As examples of multivariate parameters, ESSs for linear and logistic regression models are also discussed.