# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A Monte Carlo method for an objective Bayesian procedure. (English) Zbl 0719.65098
Often a statistical model is described by the likelihood function L($\theta$,Y) for a given set of data Y and an unknown parameter vector $\theta$. In the case of an ill-conditioned situation the function $lnL\left(\theta ,Y\right)-Q\left(\theta ,\tau \right)$ is introduced, where Q represents a set of penalties and $\tau$ is the vector of respective weights for the penalties. Consider the prior distribution $\pi$ ($\theta$ /$\tau$) with $\pi \left(\theta /\tau \right)=exp\left\{-Q\left(\theta ,\tau \right)\right\}/\int exp\left\{-Q\left(\theta ,\tau \right)\right\}d\theta ·$ Then ${\Lambda }\left(\tau ;Y\right)=\int L\left(\theta ,Y\right)\pi \left(\theta ,\tau \right)d\theta$ is the Bayesian likelihood of $\tau$, and is useful to obtain the optimal hyper-parameter $\tau$ which maximizes ${\Lambda }$ or its logarithm. A Monte Carlo integration method is described and is used to determine ${\Lambda }$ ($\nabla ,Y\right)$ and some numerical examples are discussed.

##### MSC:
 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis) 62F15 Bayesian inference
##### References:
 [1] Akaike, H. (1977). On entropy maximization principle, Application of Statistics, (ed. P. R.Krishnaiah), 27-41, North Holland, Amsterdam. [2] Akaike, H. (1978). A new look at the Bayes procedure, Biomotrika, 65, 53-59. · Zbl 0373.62008 · doi:10.1093/biomet/65.1.53 [3] Akaike, H. (1979). Likelihood and Bayes procedure, Bayesion Statisties, (eds. J. M.Bernard et al.), University Press, Valencia, Spain. [4] 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. [5] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussions), J. Roy. Statist. Soc. Ser. B, 36, 192-236. [6] Besag, J. (1986). On the statistical analysis of dirty pictures (with discussions), J. Roy. Statist. Soc. Ser. B, 48, 259-302. [7] 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. [8] 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. · Zbl 0573.62030 · doi:10.1109/TPAMI.1984.4767596 [9] Good, I. J. (1965). The Estimation of Probabilities, M.I.T. Press, Cambridge, Massachusetts. [10] Good, I. J. and Gaskins, R. A. (1971). Nonparametric roughness penalties for probability densities, Biometrika, 58, 255-277. · Zbl 0221.62012 · doi:10.2307/2334515 [11] Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods, Methuen, London. [12] Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen. [13] Ishiguro, M. and Sakamoto, Y. (1983). A Bayesian approach to binary response curve estimation, Ann. Inst. Statist. Math., 35, 115-137. · Zbl 0506.62022 · doi:10.1007/BF02480969 [14] Kirkpatrick, S., Gellat, C. D.Jr. and Vecchi, M. P. (1983). Optimization by simulated annealing, Science, 220, 671-680. · Zbl 1225.90162 · doi:10.1126/science.220.4598.671 [15] Kitagawa, G. (1987). Non-Gaussian state space modeling of nonstationary time series (with discussion), J. Amer. Statist. Assoc., 82, 1032-1063. · Zbl 0644.62088 · doi:10.2307/2289375 [16] 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. · doi:10.1063/1.1699114 [17] Mori, M. (1986). Programming of FORTRAN77 for Numerical Analyses (in Japanese), The Iwanami Computer Science Series, Iwanami Publisher, Tokyo. [18] Ogata, Y. (1988). A Monte Carlo method for the objective Bayesian procedure, Research Memo. No. 347, The Institute of Statistical Mathematics, Tokyo. [19] Ogata, Y. (1989). A Monte Carlo method for high dimensional integration, Numer. Math., 55, 137-157. · Zbl 0669.65011 · doi:10.1007/BF01406511 [20] Ogata, Y. and Katsura, K. (1988). Likelihood analysis of spatial inhomogeneity for marked point patterns, Ann. Inst. Statist. Math., 40, 29-39. · Zbl 0668.62084 · doi:10.1007/BF00053953 [21] 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. · Zbl 0478.62078 · doi:10.1007/BF02480944 [22] 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. [23] 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. [24] Ogata, Y. and Tanemura, M. (1984a). Likelihood analysis of spatial point patterns, Research Memo. No. 241, The Institute of Statistical Mathematics, Tokyo. [25] Ogata, Y. and Tanemura, M. (1984b). Likelihood analysis of spatial point patterns, J. Roy. Statist. Soc. Ser. B, 46, No. 3, 496-518. [26] Ogata, Y. and Tanemura, M. (1989). Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns, Ann. Inst. Statist. Math., 41, 583-600. · Zbl 0693.62077 · doi:10.1007/BF00050670 [27] 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. [28] Tanabe, K. and Tanaka, T. (1983). Fitting curves and surfaces by Bayesian method (in Japanese), Chikyuu (Earth), 5, No. 3, 179-186. [29] 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.