In an earlier paper, I described how chi-squared goodness-of-fit statistics could be defined for a broad class of Bayesian statistical models. Pivotal to the definition of these chi-squared statistics was the use of a sampled parameter value from the posterior distribution to define either a random set of bin counts or a random vector of cell probabilities. In this article, I demonstrate how chi-squared statistics based on the same posterior distribution and data vector can be combined to obtain a single chi-squared statistic for assessing model fit. A relation between the resulting statistic and the classical chi-squared statistic based on the maximum likelihood estimate is also discussed.
Statistical Methodology | Statistical Theory
Johnson, Valen, "Combining Bayesian Chi-Squared Statistics" (November 2004). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 1.