Validified Posterior Possibility Functions

• Validified posterior possibility functions are data-dependent measures for prior-free inference that yield exactly calibrated, frequentist-valid uncertainty quantification.
• They transform likelihood information into possibility contours through inferential models, ensuring confidence-credibility matching with an inner probabilistic approximation.
• Their computation uses Monte Carlo and variational techniques to achieve asymptotic efficiency and robust applicability across Bayesian and robust frequentist paradigms.

Validified posterior possibility functions are a class of data-dependent mathematical objects used for prior-free statistical inference, constructed to yield exactly calibrated, frequentist-valid uncertainty quantification through possibility measures. They arise as the posterior output from inferential models (IMs), systematically transforming likelihood information via a calibration principle to deliver posterior possibility contours. These validified possibility functions also admit probabilistic inner approximations, yielding “validified posteriors” with exact confidence-credibility matching and asymptotic efficiency. The construction, calibration, and computational realization of these functions make them a central tool in emerging methodologies that bridge Bayesian, fiducial, and robust frequentist paradigms (Martin, 17 Jan 2025, Martin, 25 Mar 2025).

1. Core Definitions and Theoretical Foundations

Let $X\sim P_\theta$ denote data from a model indexed by parameter $\theta\in\Theta\subseteq\mathbb{R}^d$, with likelihood $L_x(\theta)$. The starting point is the relative likelihood

$$R(x,\theta) = \frac{L_x(\theta)}{\sup_{t\in\Theta}L_x(t)}.$$

A validified posterior possibility function, known as the IM contour or plausibility contour, is then defined for each $\theta\in\Theta$ by

$$\pi_x(\theta) = P_\theta\{R(X,\theta)\leq R(x,\theta)\}.$$

By construction, $\pi_x(\theta)\in[0,1]$ for all $\theta$ and $\sup_\theta\pi_x(\theta)=1$. The associated possibility measure for a set $H\subseteq\Theta$ is

$\Pi_x(H) = \sup_{\theta\in H} \pi_x(\theta).$

The defining property is (strong) validity: $$\sup_{\theta\in\Theta}P_\theta\{\pi_X(\theta)\leq\alpha\} \leq \alpha \quad \forall\alpha\in[0,1].$$ This ensures that, for any fixed $\alpha$, the probability under any true $\theta$ that $\pi_X(\theta)$ falls below $\alpha$ is at most $\alpha$. As a consequence, the $\alpha$-level plausibility region $C_{x,\alpha} = \{\theta:\pi_x(\theta)\geq\alpha\}$ satisfies

$$\sup_\theta P_\theta\{\Theta\notin C_{X,\alpha}\} \leq \alpha,$$

i.e., $C_{x,\alpha}$ is an honest $(1-\alpha)$ frequentist confidence region.

2. Credal Set Characterization and Inner Probabilistic Approximation

Associated with any possibility measure $\Pi_x$ is its closed convex credal set: $$\mathcal{C}(\Pi_x) = \{ Q\in\mathrm{Prob}(\Theta): Q(H)\leq\Pi_x(H),\ \forall H\}.$$ For the level sets $C_{x,\alpha} = \{\theta:\pi_x(\theta)\geq\alpha\}$, the equivalence

$$Q \in \mathcal{C}(\Pi_x) \iff Q(C_{x,\alpha}) \geq 1-\alpha\ \forall\alpha$$

holds. Every $Q$ in the credal set thus defines (at least) a confidence distribution.

The inner probabilistic approximation $Q_x^*$ (“validified posterior,” Editor's term) is the unique $Q^*$ that saturates the bounds,

$$Q_x^*(C_{x,\alpha}) = 1 - \alpha \quad \forall\alpha\in[0,1].$$

This probability measure is characterized by the mixture representation: $$Q_x^*(\cdot) = \int_{0}^1 K_x^\alpha(\cdot)d\alpha,$$ where each $K_x^\alpha$ is a probability law supported on $\partial C_{x,\alpha}$, the boundary of the $\alpha$-cut. Sampling from $Q_x^*$ thus involves randomization of the contour level and then uniform sampling on the corresponding contour boundary.

3. Monte Carlo Construction and Practical Computation

In all but trivial models, the contour $\pi_x$ and inner approximation $Q_x^*$ lack closed-form expressions. Efficient computation proceeds as follows (Martin, 17 Jan 2025, Martin, 25 Mar 2025):

1. For each $\alpha$ (typically on a grid), construct a variational family $R_x^{\xi}(\cdot)$ (e.g., Gaussian law centered at the maximum likelihood estimator) with inflation parameter $\xi=\xi(x,\alpha)$ chosen so that $R_x^{\xi}(C_{x,\alpha})\approx1-\alpha$.
2. Sample $A\sim\mathrm{Uniform}(0,1)$, set $\xi=\xi(x,A)$, and draw $\theta\sim R_x^{\xi}$, conditioning on $\theta$ lying on $\partial C_{x,A}$.
3. Repeat for $j=1,\dots,N$ to obtain Monte Carlo particles $\{\theta_j\}$.
4. The possibility function is reconstructed by ranking: $$\hat\pi_x(\theta) = \frac{1}{N} \sum_{j=1}^N 1\{ r_x(\theta_j)\leq r_x(\theta) \},$$ where $r_x$ may be (for instance) the $Q_x^*$ density estimate or the original likelihood.

This procedure efficiently delivers a sample-based approximation $\hat\pi_x$ that is itself a valid possibility contour, enjoying parallel scalability and rapid convergence as $N\uparrow$.

4. Properties, Calibration, and Asymptotic Behavior

Validified posterior possibility functions exhibit several rigorous guarantees:

• Validity: $$\sup_{\theta}P_\theta\{\hat\pi_X(\theta)\leq\alpha\} \leq \alpha + \varepsilon_n + O(N^{-1/2})$$ (with $\varepsilon_n\to0$ as $n\to\infty$).
• Uniform Approximation: $$\|\hat\pi_x - \pi_x\|_\infty = O_p(\delta_{\text{var}} + N^{-1/2})$$, where $\delta_{\text{var}}$ is the variational approximation error, vanishing under regularity.
• Asymptotic Normality: As $n\to\infty$, the IM contour $\pi_x$ approaches a multivariate Gaussian possibility contour, and $Q_x^*$ becomes asymptotically efficient (i.e., matching Fisher information covariance).

When the statistical model admits a group invariance structure with a right-Haar measure, the constructed $Q_x^*$ coincides exactly with the right-Haar prior Bayes posterior and the fiducial distribution (Martin, 25 Mar 2025).

5. Interplay Between Credible Sets and Plausibility Regions

By design, plausibility regions $C_{x,\alpha}$ for the IM are also $100(1-\alpha)\%$ credible sets under the inner approximation $Q_x^*$: $$Q_x^*\{C_{x,\alpha}\} = 1 - \alpha, \quad P_\theta\{\Theta\in C_{X,\alpha}\} = 1 - \alpha.$$ This duality means that the same region enjoys simultaneous frequentist and Bayesian (credibility) interpretation, without reliance on a subjective or default prior.

6. Applications and Illustrative Examples

Applied studies showcase the efficiency and honesty of validified posterior possibility functions:

• In the Behrens-Fisher problem, the IM-based $Q_x^*$ posterior achieves empirical coverage rates at the nominal level (e.g., $0.904\approx0.90$ at $90\%$ coverage), improving on Jeffreys prior Bayes and Welch $t$ intervals (Martin, 25 Mar 2025).
• For inference on the bivariate normal correlation coefficient, the $Q_x^*$-based posterior contour closely matches grid-based IM solutions, achieving $\sup_\psi\pi_x(\psi)=1$ and absolute error $\approx0.01$ versus baseline (Martin, 17 Jan 2025).
• Possibilistic approaches to instrumental variable regression allow coherent sensitivity analysis by replacing the exogeneity assumption with a user-defined set $A$ of admissible violations of instrument validity. Posterior possibility for the treatment effect $\beta$ is constructed so that if $\alpha_0\in A$, the resulting interval enjoys type-I error control: for any $\delta$, $$\sup_\beta P_\beta[\pi_W(\beta|A)\leq\delta]\leq\delta$$ (Steiner et al., 20 Nov 2025).

7. Connections, Extensions, and Contemporary Research

Recent work emphasizes that the IM framework unifies and extends Bayesian, fiducial, and confidence-based inference. In group-invariant models, validified posteriors agree exactly with standard Bayes and fiducial benchmarks, while providing stronger frequentist validity. Computational advances using variational families and Monte Carlo enable practically scalable inference for high-dimensional models (Martin, 17 Jan 2025, Martin, 25 Mar 2025).

A plausible implication is that validified posterior possibility functions may serve as a foundational tool for robust prior-free inference, especially in domains requiring exact calibration, model invariance, or sensitivity to subjective modeling assumptions. Ongoing research targets broader classes of models, hybrid imprecise-probabilistic frameworks, and further efficiency improvements in the variational sampling schemes.