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The differential structure shared by probability and moment matching priors on non-regular statistical models via the Lie derivative

Masaki Yoshioka, Fuyuhiko Tanaka

TL;DR

The paper addresses the challenge of deriving objective priors for non-regular Bayesian models, focusing on probability and moment matching priors within the information-geometric framework of the one-sided truncated exponential family (oTEF). By deriving asymptotic posterior expansions and associated PDEs, it unifies the two matching-prior concepts and shows that, in the oTEF, their conditions can be expressed via a shared Lie derivative along the vector field $\chi=\partial_{\gamma}+A_i^{(1,1)}g^{ij}\partial_j$. The authors provide explicit forms in key special cases (e.g., truncated exponential and truncated normal) and connect these priors to $\alpha$-parallel priors, interpreting the results in terms of generalized volume-element invariance and decompositions into 1D submodels. This geometric perspective offers a coherent framework for objective priors in non-regular Bayesian inference and suggests avenues for extending the approach to broader non-regular settings.

Abstract

In Bayesian statistics, the selection of noninformative priors is a crucial issue. There have been various discussions on theoretical justification, problems with the Jeffreys prior, and alternative objective priors. Among them, we focus on two types of matching priors consistent with frequentist theory: the probability matching priors and the moment matching priors. In particular, no clear relationship has been established between these two types of priors on non-regular statistical models, even though they share similar objectives. Considering information geometry on a one-sided truncated exponential family, a typical example of non-regular statistical models, we find that the Lie derivative along a particular vector field provides the conditions for both the probability and moment matching priors. Notably, this Lie derivative does not appear in regular models. These conditions require the invariance of a generalized volume element with respect to differentiation along the non-regular parameter. This invariance leads to a suitable decomposition of the one-sided truncated exponential family into one-dimensional submodels. This result promotes a unified understanding of probability and moment matching priors on non-regular models.

The differential structure shared by probability and moment matching priors on non-regular statistical models via the Lie derivative

TL;DR

The paper addresses the challenge of deriving objective priors for non-regular Bayesian models, focusing on probability and moment matching priors within the information-geometric framework of the one-sided truncated exponential family (oTEF). By deriving asymptotic posterior expansions and associated PDEs, it unifies the two matching-prior concepts and shows that, in the oTEF, their conditions can be expressed via a shared Lie derivative along the vector field . The authors provide explicit forms in key special cases (e.g., truncated exponential and truncated normal) and connect these priors to -parallel priors, interpreting the results in terms of generalized volume-element invariance and decompositions into 1D submodels. This geometric perspective offers a coherent framework for objective priors in non-regular Bayesian inference and suggests avenues for extending the approach to broader non-regular settings.

Abstract

In Bayesian statistics, the selection of noninformative priors is a crucial issue. There have been various discussions on theoretical justification, problems with the Jeffreys prior, and alternative objective priors. Among them, we focus on two types of matching priors consistent with frequentist theory: the probability matching priors and the moment matching priors. In particular, no clear relationship has been established between these two types of priors on non-regular statistical models, even though they share similar objectives. Considering information geometry on a one-sided truncated exponential family, a typical example of non-regular statistical models, we find that the Lie derivative along a particular vector field provides the conditions for both the probability and moment matching priors. Notably, this Lie derivative does not appear in regular models. These conditions require the invariance of a generalized volume element with respect to differentiation along the non-regular parameter. This invariance leads to a suitable decomposition of the one-sided truncated exponential family into one-dimensional submodels. This result promotes a unified understanding of probability and moment matching priors on non-regular models.
Paper Structure (13 sections, 4 theorems, 101 equations, 2 figures)

This paper contains 13 sections, 4 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. One-sided truncated family
  3. Probability matching priors for non-regular models with regular multiparameters
  4. Moment matching priors for non-regular models with regular multiparameters
  5. The Lie derivative shared by probability and moment matching priors
  6. Concluding remarks
  7. Proof of Lemma \ref{['lem:asymptotic_expansion_posterior_density']}
  8. Proof of Theorem \ref{['thm:probability_matching_prior_oTF']}
  9. Probability matching prior for the truncation parameter
  10. Probability matching prior for the regular parameter
  11. Proof of Theorem \ref{['thm:moment_matching_prior_oTF']}
  12. Moment matching prior for the truncation parameter
  13. Moment matching prior for the regular parameter

Key Result

Lemma 1

Let $\hat{\theta}_{\mathrm{ML}}$ and $\hat{\gamma}_{\mathrm{ML}}$ be the MLEs of $\theta$ and $\gamma$. With $u=\sqrt{n}\left( \theta-\hat{\theta}_{\mathrm{ML}} \right),\,t=n\hat{c}(\gamma-\hat{\gamma}_{\mathrm{ML}})$, the posterior density $\pi(u,t;\mathbb{X}_{n})$ admits the asymptotic expansion where $\hat{\pi} = \pi(\hat{\theta}_{\mathrm{ML}}, \hat{\gamma}_{\mathrm{ML}})$, and Here, $S_{6}

Figures (2)

  • Figure 1: Streamline: $Exp(\theta,\gamma)$
  • Figure 2: Streamline: $N(\alpha,1,\gamma)$

Theorems & Definitions (13)

  • Example 1: Truncated exponential distributions
  • Example 2
  • Lemma 1
  • Theorem 2
  • Example 1: Truncated exponential distributions (continued)
  • Example 2: Truncated normal distributions (continued)
  • Theorem 3
  • Example 1: Truncated exponential distributions (continued)
  • Example 2: Truncated normal distributions (continued)
  • Theorem 4
  • ...and 3 more