Ann. Inst. Statist. Math. Vol. 40, No. 4, 683-691 (1988) THE EXPONENTIAL RATES OF CONVERGENCE OF POSTERIOR DISTRIBUTIONS* JAMES C. FU 1 AND ROBERT E. KASS 2 J Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada 2Department of Statistics, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. (Received September 10, 1986; revised February 3, 1987) Abstract. After the observations were observed, the posterior distribu- tion under mild conditions becomes more concentrated in the neighbour- hood of the mode of the posterior distribution as sample size n increase. In this paper, the exponential rate of convergence of posterior distribu- tion around the mode is established by using the generalized Laplace method. An example is also given. Key words and phrases: Bayesian, posterior probability, exponential rate,f-uniform convergence, Q-uniform convergence. 1. Introduction After the observations were observed, from the Bayesian point of view, the statistical inference about the unknown parameter 0 is expressed by the posterior distribution only. The posterior distribution, under mild conditions becomes more concentrate as sample size increase. There is considerable literature on the asymptotic behaviour of posterior distribu- tion in a neighbourhood of the mode (or maximum likelihood estimator), for example, LeCam (1953), Freedman (1963), Lindley (1965), Johnson (1967), Walker (1969), Brenner et al. (1983) and Chen (1983). Mathemati- cally, if a and b are constant, the posterior probability (1.1) cO,+ha, 0 Pofs(On + atrn < 0 < On + ha,) =JO°+a~° f"( Is)dO, converges to ~(b)- ~(a) when n- ~, where f~(01s) is posterior density function, 0, is the mode of posterior density (or m.l.e.), s = (xl,x2 .... ) stands for observations, cr]--(-l~2)(sl0,))-I = O(n-1), 142} is the second *This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant NSERC A-9216. 683