BAYESIAN INFERENCE AND THE PARAMETRIC BOOTSTRAP.pdf

arXiv: [] 14 Jan 2013 The Annals of Applied Statistics 2012, Vol. 6, No. 4, 1971–1997 DOI: -AOAS571 cInstitute of Mathematical Statistics, 2012 BAYESIAN INFERENCE AND THE PARAMETRIC BOOTSTRAP By Bradley Efron 1 Stanford University The parametric bootstrap can be used for the puta- tion of Bayes posterior distributions. Importance sampling formulas take on an easy form relating to the deviance in exponential fami- lies and are particularly simple starting from Je?reys invariant prior. Because of the . nature of bootstrap sampling, familiar formulas describe putational accuracy of the Bayes estimates. putational methods, the theory provides a connection between Bayesian and frequentist analysis. E?cient algorithms forthe fre- quentist accuracy of Bayesian inferences are developed anddemon- strated in a model selection example. 1. article concerns the use of the parametric boot- strap to carry out Bayesian inference calculations. Two main points are made: that in paratively limited set of cases where bootstrap meth- ods apply, they o?er an e?cient putationally straightforward way pute posterior distributions and estimates, enjoying some advantages over Markov chain techniques; and, more importantly, that the parametric bootstrap helps connect Bayes and frequentist points of view. The basic idea is simple and not unfamiliar: that the bootstrap is use- ful for importance putation of Bayes posteriordistributions. An important paper by Newton and Raftery (1994) suggested a version of nonparametric bootstrapping for this purpose. By “going parametric” we can make the Bayes/bootstrap relationship more transparent. This line of thought has the advantage of linking rather than separatingfrequentist and Bayesian practices. Section2introduces the main ideas in terms of an elementary one-parame- ter example and illustrates a connection between Je?reys invariant prior den- sity and second-order accurate bootstrap con?dence lim