Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akaike, H. (1977). On entropy maximization principle, Application of Statistics, (ed. P. R.Krishnaiah), 27–41, North Holland, Amsterdam.
Akaike, H. (1978). A new look at the Bayes procedure, Biomotrika, 65, 53–59.
Akaike, H. (1979). Likelihood and Bayes procedure, Bayesion Statisties, (eds. J. M.Bernard et al.), University Press, Valencia, Spain.
Akaike, H. and Ishiguro, M. (1983). Comparative study of the X-11 and BAYSEA procedure of seasonal adjustment, Applied Time Series Analysis of Economic Data, 17–53, U.S. Department of Commerce, Bureau of the Census.
Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussions), J. Roy. Statist. Soc. Ser. B, 36, 192–236.
Besag, J. (1986). On the statistical analysis of dirty pictures (with discussions), J. Roy. Statist. Soc. Ser. B, 48, 259–302.
Binder, K. (1986). Introduction: Theory and technical aspects of Monte Carlo simulations, Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, (ed. K.Binder), Springer-Verlag, Berlin.
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Machine Intell., 6, 721–741.
Good, I. J. (1965). The Estimation of Probabilities, M.I.T. Press, Cambridge, Massachusetts.
Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities, Biometrika, 58, 255–277.
Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods, Methuen, London.
Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.
Ishiguro, M. and Sakamoto, Y. (1983). A Bayesian approach to binary response curve estimation, Ann. Inst. Statist. Math., 35, 115–137.
Kirkpatrick, S., Gellat, C. D.Jr. and Vecchi, M. P. (1983). Optimization by simulated annealing, Science, 220, 671–680.
Kitagawa, G. (1987). Non-Gaussian state space modeling of nonstationary time series (with discussion), J. Amer. Statist. Assoc., 82, 1032–1063.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines, J. Chem. Phys., 21, 1087–1092.
Mori, M. (1986). Programming of FORTRAN77 for Numerical Analyses (in Japanese), The Iwanami Computer Science Series, Iwanami Publisher, Tokyo.
Ogata, Y. (1988). A Monte Carlo method for the objective Bayesian procedure, Research Memo. No. 347, The Institute of Statistical Mathematics, Tokyo.
Ogata, Y. (1989). A Monte Carlo method for high dimensional integration, Numer. Math., 55, 137–157.
Ogata, Y. and Katsura, K. (1988). Likelihood analysis of spatial inhomogeneity for marked point patterns, Ann. Inst. Statist. Math., 40, 29–39.
Ogata, Y. and Tanemura, M. (1981a). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure, Ann. Inst. Statist. Math., 33, 315–338.
Ogata, Y. and Tanemura, M. (1981b). Approximation of likelihood function in estimating the interaction potentials from spatial point patterns, Research Memo. No. 216, The Institute of Statistical Mathematics, Tokyo.
Ogata, Y. and Tanemura, M. (1981c). A simple simulation method for quasi-equilibrium point patterns, Research Memo. No. 210, The Institute of Statistical Mathematics, Tokyo.
Ogata, Y. and Tanemura, M. (1984a). Likelihood analysis of spatial point patterns, Research Memo. No. 241, The Institute of Statistical Mathematics, Tokyo.
Ogata, Y. and Tanemura, M. (1984b). Likelihood analysis of spatial point patterns, J. Roy. Statist. Soc. Ser. B, 46, No. 3, 496–518.
Ogata, Y. and Tanemura, M. (1989). Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns, Ann. Inst. Statist. Math., 41, 583–600.
Ogata, Y., Imoto, M. and Katsura, K. (1989). Three-dimensional spatial variation of b-values of magnitude frequeney distribution beneath the Kanto District, Japan, Research Memo. No. 369, The Institute of Statistical Mathematics, Tokyo.
Tanabe, K. and Tanaka, T. (1983). Fitting curves and surfaces by Bayesian method (in Japanese), Chikyuu (Earth), 5, No. 3, 179–186.
Wood, W. W. (1968). Monte Carlo studies of simple liquid models, Physics of Simple Liquids, (eds. H. N. V.Temperley, J. S.Rowlinson and G. S.Rushbrooke), 115–230, Chap. 5, North-Holland, Amsterdam.
Author information
Authors and Affiliations
About this article
Cite this article
Ogata, Y. A Monte Carlo method for an objective Bayesian procedure. Ann Inst Stat Math 42, 403–433 (1990). https://doi.org/10.1007/BF00049299
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049299