Annals of the Institute of Statistical Mathematics

, Volume 42, Issue 3, pp 403–433

A Monte Carlo method for an objective Bayesian procedure

  • Yosihiko Ogata
Bayesian Proceduce

DOI: 10.1007/BF00049299

Cite this article as:
Ogata, Y. Ann Inst Stat Math (1990) 42: 403. doi:10.1007/BF00049299


This paper describes a method for an objective selection of the optimal prior distribution, or for adjusting its hyper-parameter, among the competing priors for a variety of Bayesian models. In order to implement this method, the integration of very high dimensional functions is required to get the normalizing constants of the posterior and even of the prior distribution. The logarithm of the high dimensional integral is reduced to the one-dimensional integration of a cerain function with respect to the scalar parameter over the range of the unit interval. Having decided the prior, the Bayes estimate or the posterior mean is used mainly here in addition to the posterior mode. All of these are based on the simulation of Gibbs distributions such as Metropolis' Monte Carlo algorithm. The improvement of the integration's accuracy is substantial in comparison with the conventional crude Monte Carlo integration. In the present method, we have essentially no practical restrictions in modeling the prior and the likelihood. Illustrative artificial data of the lattice system are given to show the practicability of the present procedure.

Key words and phrases

ABICBayesian likelihoodposterior meanϕ- and ψ-statisticGibbs distributionhyper-parametersMetropolis' algorithm, normalizing factorpotential functiontype II maximum likelihood method

Copyright information

© The Institute of Statistical Mathematics 1990

Authors and Affiliations

  • Yosihiko Ogata
    • 1
  1. 1.The Institute of Statistical MathematiesTokyoJapan