Bayesian good-of-fit tests: past, present and future

Abstract

In this paper we will build the Bayesian version for the good-of-fitt tests Chi^2 and Kolmogorov-Smirnov. Because for the last test the theoretical distribution must be totally specified, we will divide first the sample in two parts: the first part is for inference, and the second part is for test. The completely specified theoretical cdf for the second part of the sample is the Bayesian forecasted cdf from the first part. This is unique if the prior distribution is fixed. For the Chi^2 test, we do the same Bayesian inference in the first part, and we perform the Bayesian forecasts for the probability such that X belongs to the involved intervals (the values of p_i). The parameters of the prior distribution are chosen such that the Chi^2 statistics is minimum, and the number of degrees of freedom is k-1-npar, where k is the number of intervals, and npar is the number of parameters of X. Of course, we can fix the prior distribution as for Kolmogorov-Smirnov test, but the number of degrees of freedom is k - 1. For the last test we can consider the whole sample, and the parameters that characterise the distribution of X are the Bayesian estimators. The number of degrees of freedom are the same as above, and npar is again the number of parameters of the distribution of X. When we estimate the values of forecasted cdf/ forecasted probabilities of the intervals or when we estimate the parameters for the chi square test we apply analytical formulae if they exist. Otherwise, we generate a sample according the forecasted distribution of X|S (or the posterior distribution of theta|S), and next we apply the Monte Carlo method. The way we generate the values of X is to use the mixture method: we generate theta according the posterior distribution, and X is generated forr each theta.