No-prior Bayes reIMagined: probabilistic approximations of inferential models

TL;DR

The work addresses inference without prior information by introducing a reIMagined approach that starts from a provably reliable possibilistic inferential model (IM) and extracts an inner probabilistic approximation to obtain a data-driven posterior. The inner approximation Q_x^* preserves probability-matching and calibration, agrees with Bayes/fiducial solutions in group-invariant settings, and enjoys a Bernstein–von Mises–type asymptotic normality and efficiency. Computationally, the method relies on Monte Carlo schemes and a two-stage sampling construction on level-set boundaries, enabling practical posterior-like inferences without specifying priors. A Behrens–Fisher demonstration shows favorable calibration and efficiency relative to standard no-prior Bayes solutions, highlighting the approach’s reliability and applicability, with potential extensions to partial priors and broader decision problems.

Abstract

When prior information is lacking, the go-to strategy for probabilistic inference is to combine a "default prior" and the likelihood via Bayes's theorem. Objective Bayes, (generalized) fiducial inference, etc. fall under this umbrella. This construction is natural, but the corresponding posterior distributions generally only offer limited, approximately valid uncertainty quantification. The present paper takes a reimagined approach that yields posterior distributions with stronger reliability properties. The proposed construction starts with an inferential model (IM), one that takes the mathematical form of a data-driven possibility measure and features exactly valid uncertainty quantification, and then returns a so-called inner probabilistic approximation thereof. This inner probabilistic approximation inherits many of the original IM's desirable properties, including credible sets with exact coverage and asymptotic efficiency. The approximation also agrees with the familiar Bayes/fiducial solution in applications where the model has a group invariance structure. A Monte Carlo method for evaluating the probabilistic approximation is presented, along with numerical illustrations.

Figures (14)

• Figure 1: Illustration of a typical possibility contour $\theta \mapsto \pi_x(\theta)$ and how the possibility measure $\mathsf{\Pi}_x$ is determined from it. Here, the hypothesis $H$ is the interval $[3,5]$ and the maximum value $\mathsf{\Pi}_x(H)$ on contour is highlighted. Horizontal dashed line at $\alpha=0.1$ determines the 90% confidence interval $C_{0.1}(x)$ in \ref{['eq:region']}.
• Figure 2: Plots of three inner probabilistic approximations for the gamma illustration with $n=7$ and $x=14$: uniform kernel (solid), non-uniform kernel with $w \approx 0.45$ (dashed), and the Bayesian posterior distribution (dotted) based on the default prior $\theta \mapsto \theta^{-1}$. Panel (a) shows the distribution function $t \mapsto \mathsf{Q}_x^\star(\Theta \leq t)$ and Panel (b) shows the non-coverage probability function $\alpha \mapsto 1 - \mathsf{Q}_x^\star\{C_\alpha(x)\}$.
• Figure 3: Plot of the directional data analyzed in Section \ref{['SS:group']}. Points on the circle determine the direction. The small closed circle denotes the sample mean in Cartesian coordinates and the corresponding angle determined by that point, $\hat{\theta}_x=1.35$, is the maximum likelihood estimator of the mean angle. The length of that vector, 0.87, is a to-be-conditioned-on ancillary statistic; see Appendix \ref{['AA:wheel']}
• Figure 4: Analysis of the directional data in Section \ref{['SS:group']}. Panel (a) shows the possibilistic IM contour and Panel (b) shows a histogram of samples from the corresponding inner probabilistic approximation with the flat-prior Bayes posterior density overlaid.
• Figure 5: Marginal inference results for $\Phi=\cos(a\Theta)$, with $a=1$, in the directional data example. Panel (a) shows the extension-based marginal IM contour for $\Phi$; Panel (b) shows the Bayesian marginal posterior distribution for $\Phi$; and Panel (c) shows the non-credibility function for the marginal Bayesian posterior (dashed) and the marginal IM's inner probabilistic approximation (solid).

Theorems & Definitions (6)

• Theorem 1: immc
• Theorem 2
• Theorem 3
• Theorem 4: imbvm.ext
• Theorem 5
• Theorem 6