Curvature form on statistical model manifolds and its application to Bayesian analysis

In the Bayesian analysis with a statistical model, it is inevitable to determine a prior distribution of the unknown parameter. Since we encounter more and more complicated models in practical use, we need simple criteria by which we know whether there exists a certain class of prior on the statistical model. Recently, Takeuchi and Amari obtained the geometrical condition that a statistical model admits an alpha parallel prior, one generalization of well-known Jeffreys prior. Matsuzoe, Takeuchi and Amari studied extensively the geometric condition in a curved exponential family. We formulate their result in terms of differential two form called curvature form on statistical model manifolds, which seems more suitable to evaluation of global properties of statistical model. While the trace of two form vanishes in general class of statistical model including exponential family, it does not vanish in the autoregressive moving average model, which is very fundamental and practically important in time series analysis