Abstract
In this paper, We estimate the probability density function for the stochastic process whose probability density function belongs to the exponential family. First, we hypothesize the candidate probability density functions on basis of a priori information, and then estimate the statistics of the probability density function by the maximum likelihood method. Next, a posteriori probability of each candidate density function can be calculated by the Bayesian theorem. After enough observation data were obtained, we may select the probability density function whose probability is the highest among the candidate functions.
Furthermore, we investigate the asymptotic property of a posteriori probability. A posteriori probability of the density function which is the most closed to the true probability function in the sense of Kullback's information, approaches to one as data increase enough.
