Validified Posterior Possibility

• The paper demonstrates that validified posterior possibility replaces classical probabilistic posteriors with set-function–valued measures that ensure uniform frequentist error control.
• It introduces a three-step IM framework—association, possibility prediction, and combination—that constructs possibility measures from auxiliary variables and observed data.
• The methodology is applied to scenarios such as imprecise Bayes, instrumental regression, and probabilistic programming, offering robust inference and efficient computational algorithms.

A validified posterior possibility is a data-driven update for uncertainty quantification that replaces classical probabilistic posteriors with set-function–valued or credal upper measures, systematically constructed to guarantee frequentist calibration and exact error control. This approach is fundamental in the inferential model (IM) framework, which generalizes standard fiducial and Bayesian inference by using possibility measures—specifically, supremum-based non-additive degrees of belief—that are provably valid in the sense that they calibrate false-confidence (or false-plausibility) uniformly across all parameter values. Recent advances provide efficient algorithms, inner probabilistic approximations, and rigorous computational methods, enabling practical validified inference in high-dimensional, complex, or model-uncertain regimes.

1. Theoretical Foundations of Validified Posterior Possibility

The validified posterior possibility is rooted in possibility theory and inferential models. A possibility measure Π on a space $Θ$ is defined via a possibility contour π, a function $\pi: Θ \rightarrow [0,1]$ with $\sup_{θ \in Θ} \pi(θ) = 1$, where for any $A \subseteq Θ$, $\Pi(A) = \sup_{θ \in A} \pi(θ)$. The dual necessity measure is $N(A) = 1 - \Pi(A^c)$.

For statistical inference, given a model $X \sim P_{X|\theta}$ and observed data $X = x$, the "association" $a(x, \theta, u) = 0$ (for $u \in U$) relates data, parameter, and auxiliary space, with $U \sim P_U$. The IM constructs a validified posterior possibility $\Pi_x$ on $\Theta$ as follows:

• A-step (Association): Solve $a(x, \theta, u) = 0$ for $u_{x, \theta}$.
• P-step (Prediction via Possibility): Define a possibility contour $\pi(u)$ on $U$ such that $\pi(U)$ stochastically dominates $\text{Uniform}(0,1)$ under $U \sim P_U$. The maximal specificity (optimal) contour is $\pi_P(u) = P_U\{ f(U) < f(u) \}$, where $f$ is the density of $P_U$ (Liu et al., 2020).
• C-step (Combination): Propagate $\pi$ to $\Theta$: for each $\vartheta$, set $$\pi_x(\vartheta) = \sup_{u \in U_x(\vartheta)} \pi(u)$$, and define $\Pi_x(A) = \sup_{\vartheta \in A} \pi_x(\vartheta)$.

This construction guarantees for any $A \subseteq \Theta$ and $\alpha \in [0,1]$:

$$\sup_{\theta \in A} P_{X|\theta}\{\Pi_X(A) \leq \alpha\} \leq \alpha,$$

so inference based on $\Pi_x$ controls frequentist error rates exactly, without reliance on priors or any approximate calibration (Liu et al., 2020).

2. Validity, Calibration, and Frequentist Guarantees

The defining property of validified posterior possibility is calibration—an exact frequentist control of false plausibility regardless of sample size, model complexity, or randomization (Martin, 17 Jan 2025, Martin, 25 Mar 2025). For any set $H \subseteq \Theta$, the contour $\pi_x(\theta)$ and possibility measure $$\mathrm{Pl}_x(H) = \sup_{\theta \in H} \pi_x(\theta)$$ satisfy:

$$\sup_{\theta \in H} P_\theta \{ \mathrm{Pl}_X(H) \le \alpha \} \le \alpha, \quad \forall \alpha \in [0,1].$$

This property ensures that confidence/pseudocredible sets of the form $$C_\alpha(x) = \{\theta: \pi_x(\theta) \ge \alpha \}$$ are always (simultaneously, for all $\theta$) exact $100(1-\alpha)\%$ confidence sets (Martin, 25 Mar 2025). Tests built from the necessity measure $N_X(A)$ at threshold $1-\alpha$ have type-I error $\le \alpha$ for all $A$.

The IM approach is non-additive but dominates all dominated probabilistic measures in the credal set

$$C(\mathrm{Pl}_x) = \{ Q: \forall H, Q(H) \le \mathrm{Pl}_x(H) \},$$

ensuring robust upper confidence bounds.

3. Construction of Optimal Validified Posteriors and Inner Probabilistic Approximations

The validified possibility contour can be viewed as the upper envelope of a credal set of probability measures. The optimal (maximal specificity) possibility, for given auxiliary distribution $P_U$, is

$\pi(u) = P_U\{ f(U) < f(u) \},$

and the corresponding propagated possibility on $\Theta$ is

$$\pi_x(\theta) = \sup_{u: a(x, \theta, u) = 0} \pi(u) \quad [2008.06874].$$

Any probability measure $Q^* \in C(\mathrm{Pl}_x)$ that satisfies $Q^*(C_\alpha(x)) = 1 - \alpha$ for all $\alpha$ forms an inner probabilistic approximation, with mixture characterization:

$Q^*(\cdot) = \int_0^1 K_x^\alpha(\cdot) d\alpha,$

where $K_x^\alpha$ is the uniform (or maximal-entropy) distribution on the confidence boundary $\partial C_\alpha(x)$ (Martin, 17 Jan 2025, Martin, 25 Mar 2025). This $Q^*$ coincides with the right-Haar Bayesian posterior in group-invariant models and is asymptotically efficient (i.e., recovers the Bernstein–von Mises behavior) as $n\to\infty$ (Martin, 25 Mar 2025).

Monte Carlo schemes based on this mixture approximation enable generation of valid probabilistic summaries, intervals, and decision rules while guaranteeing coverage and error rates without prior assumptions (Martin, 17 Jan 2025).

4. Algorithms, Computation, and Implementation

Efficient algorithms now realize validified posterior possibility in complex models:

• For the IM, Gaussian-variational envelopes approximate the boundaries $\partial C_\alpha(x)$, and a Robbins–Monro stochastic approximation tunes ellipsoid parameters to match target contours. Sampling proceeds by selecting $\alpha \sim \mathrm{Uniform}(0,1)$ and drawing uniformly on the corresponding ellipsoidal shell (Martin, 25 Mar 2025).
• For practical inference, gridding or MC sampling over $\alpha$ allows construction of weighted mixtures over conditional distributions on $\partial C_\alpha(x)$. For any set $A$, $\mathrm{Pl}_x(A)$ can be efficiently approximated by maximizing the contour values over the sampled points (Martin, 17 Jan 2025).
• Applications in high-dimensional or hierarchical models demonstrate that these procedures are computationally competitive, with negligible coverage loss and sup-norm approximation errors $<0.01$ up to $\dim(\Theta) \approx 10$ (Martin, 17 Jan 2025).

Further, validated variational inference methods attach rigorous, nonasymptotic error bounds to approximation outputs, effectively "validifying" posterior moments by exploiting transportation inequalities and divergence-based bounds, guaranteeing that reported summaries have explicit, computable worst-case errors (Huggins et al., 2019).

5. Practical Applications and Extensions

The validified posterior possibility underlies numerous extensions and robust inference frameworks:

• Imprecise Bayes, upper probabilities: Extensions to classes of priors and likelihoods (as in robust Bayes or uncertain-likelihood models) admit upper posterior envelopes with explicit supremum formulas, as shown by generalized Bayes theorems for upper probabilities (Caprio et al., 2023).
• Instrumental Variable Regression: Possibilistic posterior inference accommodates models with partially invalid instruments by allowing exogeneity-violation sets $V$, and guarantees exact coverage for the treatment effect $\beta$ when the true violation lies in $V$ (Steiner et al., 20 Nov 2025).
• Probabilistic Programming: Guaranteed bounds for the posterior of recursive, higher-order probabilistic programs can be constructed via symbolic interval analysis and a weight-aware type system, yielding deterministic lower/upper bounds that tightly enclose the true posterior for all events (Beutner et al., 2022).
• Model Selection: Marginal likelihoods and posterior model probabilities can be validified using mixture estimators or by calibrating p-values to lower bounds on posterior hypothesis probabilities through robust minimum Bayes factors (Tuomi et al., 2011, Vélez et al., 2022).

These methods often yield exact or conservative confidence sets and plausible intervals that maintain frequentist validity even in challenging scenarios (multi-modal posteriors, weak identification, or high dimensions).

6. Comparisons with Classical and Alternative Approaches

Compared to Bayesian posteriors, validified posterior possibilities are:

• Non-additive (they represent upper rather than probability measures) but dominate the credal set implied by probabilistic posteriors, encapsulating all well-calibrated probabilistic approximations (Liu et al., 2020, Martin, 25 Mar 2025).
• More robust to model misspecification; for example, posterior means updated under data from $\theta_1 \neq \theta_0$ may not decrease monotonically, but under validification, coverage and error rates are controlled uniformly (Hart et al., 2022).
• Capable of yielding probabilistic (inner) approximations that inherit the IM’s exact interval coverage, providing full-distribution outputs that match classical posteriors under group invariance but are regularizing or conservative otherwise (Martin, 25 Mar 2025).

Recent extensions such as e-posterior and quasi-conditional paradigms generalize these concepts to quasi-Bayesian settings, preserving frequentist guarantees irrespective of prior adequacy (Grünwald, 2023).

7. Implications and Future Directions

The validified posterior possibility framework unifies several lines—objective Bayes, fiducial inference, confidence distributions, imprecise probabilities—under a single umbrella of calibration-based, non-additive belief measures. Its methodological tools—possibility contours, credal sets, mixture constructions, and MC algorithms—provide principled and reliable foundational alternatives to Bayesian posterior-based inference without reliance on prior selection or strong model assumptions.

Ongoing developments include scalable implementations for high-dimensional models, refined variational and MC algorithms, and extension to structured uncertainty (e.g., hierarchical, nonparametric, or functional data). Applications already span causal inference with partial identification, formal verification in probabilistic programming, and robust statistical machine learning.

For technical depth, see (Liu et al., 2020, Martin, 17 Jan 2025, Martin, 25 Mar 2025, Steiner et al., 20 Nov 2025, Caprio et al., 2023, Beutner et al., 2022, Tuomi et al., 2011, Huggins et al., 2019).