Abstract

We investigate the extension of information geometry to one of the non-regular statistical models, a one-sided truncated exponential family (oTEF), and derive α-parallel prior distributions in Bayesian statistics. Information geometry has been developed mainly for regular models, and in the process, It proposed α-parallel priors as an extension of the Jeffreys prior in Bayesian statistics. It is also known that the conditional equations for some uninformative prior can be described by tensors and the α-connection coefficients. However, the theory of information geometry strongly depends on the regularity conditions, and extending it to non-regular models with different statistical behaviors remains a challenge. In this presentation, we focus on an oTEF, a non-regular model, and provide a geometric interpretation of statistical theory from the perspective of Bayesian statistics. First, we give a Riemannian metric and a one-parameter family of affine connections for oTEF and show the existence of α-parallel priors. Next, we describe the conditional equations for the moment matching priors proposed by Hashimoto (2019) using connection coefficients to derive their relationship with the α-parallel prior distributions. 
A one-sided truncated exponential family is a family of distributions on the real line with two types of parameters, d-dimensional natural parameters θ and a 1-dimensional truncation parameter γ. The density functions of an oTEF are similar to those of exponential families, but one end of the support depends on a truncation parameter γ. For example, Pareto distributions, truncated exponential distributions, and truncated normal distributions belong to an oTEF. An oTEF is one of the non-regular statistical models. Non-regular means statistical models do not satisfy the regularity conditions for maximum likelihood estimators. The regularity conditions impose that supports of the density of distributions are independent of parameters, but the support depends on a truncation parameter γ in the case of an oTEF.
We show the existence of α-parallel priors on an oTEF and its relationship with other prior distributions. We define a Riemannian metric and a one-parameter family of affine connections called the α-connections on an oTEF, as in Yoshioka and Tanaka (2023). Note that these are extensions of the Fisher metric and the α-connections on regular models. In this case, there are α-parallel priors for all real numbers α on an oTEF, and the priors have the explicit form. Other extensions of the α-connection coefficients to non-regular models have been considered, but such extensions do not guarantee the above results. In the case of oTEF, the conditional expression of moment matching prior by Hashimoto (2019) can be expressed by the extended α-connection coefficient. Then, when the truncation parameter γ is a parameter of interest, we obtain that the moment matching prior for an oTEF coincides with the square of the 1/2-parallel prior. Furthermore, when the natural parameters θ are parameters of interest, it agrees with the 0-parallel prior (the Jeffreys prior).