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Possibilistic inferential models: a review

Ryan Martin

TL;DR

This paper surveys possibilistic inferential models (IMs), which map observed data $Z$ to a possibility-based uncertainty quantification instead of precise probabilities, thereby achieving fully Bayesian-like conditional inference with frequentist calibration and avoiding the unreliability of probabilistic methods under vacuous priors. It develops a likelihood-based construction that yields a contour $\pi_z(\theta)$, and from it upper and lower possibility measures $\Pi_z$ and $\amalg_z$, guaranteeing strong validity and asymptotic efficiency via a possibilistic Bernstein–von Mises theorem. Computation is advanced through sampling from credal sets and inner probabilistic approximations, enabling practical uncertainty quantification even in nuisance-parameter settings and for marginal/inference-on-risk problems. The framework also connects to conformal prediction for predictive inference and demonstrates compatibility with frequentist and Bayesian perspectives, while offering a principled alternative in contexts with limited prior information and complex uncertainty structures. Overall, possibilistic IMs provide a rigorous, flexible, and reliable approach to data-driven uncertainty quantification with broad applicability to risk minimization and prediction tasks in statistics and data science.

Abstract

An inferential model (IM) is a model describing the construction of provably reliable, data-driven uncertainty quantification and inference about relevant unknowns. IMs and Fisher's fiducial argument have similar objectives, but a fundamental distinction between the two is that the former doesn't require that uncertainty quantification be probabilistic, offering greater flexibility and allowing for a proof of its reliability. Important recent developments have been made thanks in part to newfound connections with the imprecise probability literature, in particular, possibility theory. The brand of possibilistic IMs studied here are straightforward to construct, have very strong frequentist-like reliability properties, and offer fully conditional, Bayesian-like (imprecise) probabilistic reasoning. This paper reviews these key recent developments, describing the new theory, methods, and computational tools. A generalization of the basic possibilistic IM is also presented, making new and unexpected connections with ideas in modern statistics and machine learning, e.g., bootstrap and conformal prediction.

Possibilistic inferential models: a review

TL;DR

This paper surveys possibilistic inferential models (IMs), which map observed data to a possibility-based uncertainty quantification instead of precise probabilities, thereby achieving fully Bayesian-like conditional inference with frequentist calibration and avoiding the unreliability of probabilistic methods under vacuous priors. It develops a likelihood-based construction that yields a contour , and from it upper and lower possibility measures and , guaranteeing strong validity and asymptotic efficiency via a possibilistic Bernstein–von Mises theorem. Computation is advanced through sampling from credal sets and inner probabilistic approximations, enabling practical uncertainty quantification even in nuisance-parameter settings and for marginal/inference-on-risk problems. The framework also connects to conformal prediction for predictive inference and demonstrates compatibility with frequentist and Bayesian perspectives, while offering a principled alternative in contexts with limited prior information and complex uncertainty structures. Overall, possibilistic IMs provide a rigorous, flexible, and reliable approach to data-driven uncertainty quantification with broad applicability to risk minimization and prediction tasks in statistics and data science.

Abstract

An inferential model (IM) is a model describing the construction of provably reliable, data-driven uncertainty quantification and inference about relevant unknowns. IMs and Fisher's fiducial argument have similar objectives, but a fundamental distinction between the two is that the former doesn't require that uncertainty quantification be probabilistic, offering greater flexibility and allowing for a proof of its reliability. Important recent developments have been made thanks in part to newfound connections with the imprecise probability literature, in particular, possibility theory. The brand of possibilistic IMs studied here are straightforward to construct, have very strong frequentist-like reliability properties, and offer fully conditional, Bayesian-like (imprecise) probabilistic reasoning. This paper reviews these key recent developments, describing the new theory, methods, and computational tools. A generalization of the basic possibilistic IM is also presented, making new and unexpected connections with ideas in modern statistics and machine learning, e.g., bootstrap and conformal prediction.

Paper Structure

This paper contains 38 sections, 6 theorems, 79 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Problem setup and notation
  4. Probabilistic uncertainty quantification
  5. Existing approaches
  6. Default-prior Bayes.
  7. Fiducial and the like.
  8. Is probability theory right for the job?
  9. If not probability theory, then what?
  10. Possibilistic inferential models
  11. Perspective
  12. Construction
  13. Sampling-based reliability properties
  14. Validity
  15. Efficiency
  16. ...and 23 more sections

Key Result

Theorem 1

Let $z \mapsto \mathsf{Q}_z$ determine a data-dependent probability distribution absolutely continuous with respect to Lebesgue measure on $\mathbb{T}$. Define the false confidence rate associated with $z \mapsto \mathsf{Q}_z$ as Then for any $(\alpha, \tau) \in (0,1)^2$, there exists hypotheses $H$ such that $\text{\sc fcr}_\mathsf{Q}(\alpha, H) > \tau$.

Figures (10)

  • Figure 1: A lower bound on the (empirical) false confidence rate $\alpha \mapsto \text{\sc fcr}_\mathsf{Q}(\alpha; H)$ of the Bayes posterior corresponding to the hypothesis $H$ stated in the text.
  • Figure 2: Illustration of a possibility contour $\theta \mapsto \pi_z(\theta)$ and how the possibility measure $\Pi_z$ is determined from it. Here, the hypothesis $H$ is the interval $[3,5]$ and the maximum value $\Pi_z(H)$ on contour is highlighted. Horizontal dashed line at $\alpha=0.1$ determines the 90% confidence interval $C_{0.1}(z)$ in \ref{['eq:region']}.
  • Figure 3: Approximating the IM contour's $\alpha$-level set $C_\alpha(z)$ by an ellipse $C_\alpha^\sigma(z)$ with a "good" choice of $\sigma$ (solid) and with a "bad" choice of $\sigma$ (dashed).
  • Figure 4: Plot of the joint IM contour for $\Theta=(\Theta_1,\Theta_2)$ in the two examples presented in Section \ref{['SS:examples1']}. In both plots, the solid line is the possibilistic IM contour and the gray points are samples from the Jeffreys prior Bayes posterior. Dashed line in Panel (b) is explained in the text.
  • Figure 5: Plots of the marginal IM contours---extension-based (dashed line) and profile-based (solid line)---for the mean $\Phi$ in the normal and gamma examples. Gray line in Panel (b) is explained in the text.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Theorem 1: balch.martin.ferson.2017
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Theorem 3: imbvm.ext---simpler version
  • Remark 1
  • Remark 2
  • Remark 3
  • Example : Behrens--Fisher
  • Example : Nonparametric quantile
  • ...and 2 more