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Inferential models: A framework for prior-free posterior probabilistic inference

Ryan Martin, Chuanhai Liu

arXiv:1206.4091v3math.STstat.ME

TL;DR

Statistical inference must quantify uncertainty when observed data are limited, yet existing paradigms leave room for a prior-free approach. The paper develops inferential models through auxiliary-variable prediction and reports frequency-calibrated probabilistic uncertainty without priors.

  • Problem

    Limited observed data create uncertainty about unknown parameters, while statistical paradigms have competing advantages and disadvantages.

  • Method

    Inferential models associate observed data and unknown parameters with an unobservable auxiliary variable, predict it using a predictive random set, and condition through a three-step procedure.

  • Results

    IMs yield frequency-calibrated probabilities under very general conditions while providing measures of uncertainty about θ without a prior distribution.

  • Takeaways & Limitations

    The resulting belief and plausibility values are meaningful within and across experiments, and predictive random sets can support dimension reduction for high-dimensional auxiliary variables.

  • Takeaways & Limitations

    Exact plausibility intervals can be conservative at certain parameter values, particularly in discrete-data problems.

Abstract

from arXiv · show

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This paper presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but does so with an automatic long-run frequency calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the non-uniqueness issue. Several examples are presented to illustrate this new approach.

1 Introduction

The paper introduces inferential models (IMs) as a framework for prior-free, post-data probabilistic inference. IMs associate data and parameters with an auxiliary variable, predict it using a random set, and combine the result into calibrated uncertainty measures.

  • Inferential models target prior-free probabilistic measures of uncertainty about an unknown parameter after observing data.
  • IMs begin by associating the observable data X and unknown parameter θ with an unobservable auxiliary variable U.
  • The P-step predicts the unobserved auxiliary value with a valid predictive random set before conditioning on the observed data.
  • The C-step combines the observed data, parameter solution sets, and predictive random set to produce a random set of candidate parameter values.
  • The validity condition ensures that the resulting belief and plausibility measures have a desirable long-run frequency-calibration property.

2 Inferential models

Inferential models represent sampling through an association involving data, parameters, and an auxiliary variable, then predict the unobserved auxiliary value with a predictive random set. The resulting belief and plausibility functions summarize parameter uncertainty, subject to technical conditions and choices affecting construction.

  • The sampling model is represented through the association X = a(U, θ), where U has a specified probability measure.
  • An association can be expressed as a simulation recipe for producing X from θ and an auxiliary draw U, rather than as a formal equation.
  • Given observed x and auxiliary value u, Θx(u) is the set of parameter values satisfying the association; it contains the true parameter when u is the unobserved realization.
  • A predictive random set S is used to predict the unobserved auxiliary value, with validity intended to give S high probability of containing that value.
  • When the solution random set can be empty with positive probability, conditioning may be necessary, and the framework's modification for such cases is not discussed here.
  • The IM output reports belief and plausibility as lower and upper probabilities summarizing evidence for an assertion about θ.

3 Theoretical validity of IMs

The section establishes validity as a calibration property of predictive random sets and inferential models, then derives valid testing and plausibility procedures under mild conditions.

  • Predictive random set validity: Validity requires the predictive random set’s miss probability QS(U) to be stochastically no larger than Unif(0, 1); equality defines efficiency.This controls how often the random set misses its auxiliary target across possible auxiliary-variable values.
  • Predictive random set validity: Nested support is sufficient for predictive random set validity, although nesting is not necessary.The theorem constructs a valid predictive random set from a nested collection, while later results show non-nested sets can be improved by nested alternatives.
  • IM validity: A valid predictive random set yields a valid IM when Θx(S) is nonempty with probability one, and this result does not depend on the set’s particular form.The validity theorem transfers stochastic control of predictive-set misses to belief-function calibration for assertions.
  • IM validity: IM validity is invariant under suitable one-to-one, possibly parameter-dependent transformations of the auxiliary variable.This reparametrization can simplify examples without changing the validity property.
  • IM-based frequentist procedures: Valid IM plausibility tests control Type I error at level α, while plausibility regions achieve at least their nominal coverage probability.These results connect belief and plausibility functions to frequentist testing and confidence-region procedures.
  • IM-based frequentist procedures: Exact plausibility intervals can be conservative at some parameter values in discrete-data problems, limiting practical efficiency.In the Poisson example, the reported 90% interval has exact coverage but is described as not best possible and potentially too conservative.

4 Theoretical optimality of IMs

The paper develops an optimality theory for inferential models by comparing predictive random sets through relative efficiency and showing that nested or score-balanced choices can attain optimality under specified conditions.

  • Relative efficiency: Valid predictive random sets produce belief functions no larger than fiducial probabilities, motivating efficiency comparisons against the fiducial benchmark.The relative efficiency ratio is bounded by one when the fiducial denominator is non-zero.
  • Relative efficiency: For any predictive random set, a nested predictive random set exists whose relative efficiency is at least as large for every observed data value.This complete-class result restricts attention to nested predictive random sets without sacrificing pointwise efficiency.
  • One-sided assertions: For left-sided assertions, monotonicity of the association's right endpoint determines the optimal set: S⋆ = [0, U] or S⋆ = [U, 1], with U ∼Unif(0, 1).The first form applies when the endpoint is non-decreasing; the second applies when it is non-increasing.
  • One-sided assertions: In the Gaussian mean problem, the optimal set is S⋆ = [U, 1], and the resulting IM test is the uniformly most powerful size-α Neyman–Pearson test.The rule rejects H0 when x ≤ θ0 − Φ−1(1 − α).
  • Two-sided assertions: For two-sided assertions, score-balanced predictive random sets can be optimal, yielding uniform belief under the null and relative efficiency equal to its upper bound.Under the stated condition, the score-balanced set S⋆ is optimal; in the Gaussian case the default predictive random set is also optimal.
  • Two-sided assertions: The score-balanced optimality condition is not universal, but reparameterization can produce a transformed parameter for which the condition holds.The paper notes that the condition can fail for exponential families not in natural form.

5 Two more examples

The paper applies IMs to standardized-mean and many-exponential-rates problems, illustrating how association models and predictive random sets produce plausibility-based inference and assertion-specific testing. The new IM-based test has substantially higher power in the exponential-rates simulation than the likelihood-ratio and older IM tests.

  • 5.1 A standardized mean problem: The standardized-mean analysis uses sufficient statistics and an association involving X/S, ψ, and auxiliary variables U1 and U2.A parameter-dependent change of auxiliary variables yields a transformed association with a Uniform(0, 1) marginal on V1.
  • 5.1 A standardized mean problem: For standardized-mean inference, the IM plausibility interval for ψ exactly matches the usual frequentist confidence interval and fiducial intervals.The construction handles the nuisance parameter σ through cylinder assertions.
  • 5.2 A many-exponential-rates problem: For the many-exponential-rates problem, the IM tests the assertion A = {θ1 = · · · = θn} using a transformed auxiliary variable on a Gamma–Dirichlet space.The C-step combines the observed data, the modified association, and a predictive random set to obtain a random set for θ.
  • 5.2 A many-exponential-rates problem: The assertion-specific predictive random set is efficient and provides an easy-to-compute alternative to a hierarchical predictive random set for sorted uniforms.The transformed order-statistic coordinates have the distribution of sorted Unif(0, 1) variables.
  • 5.2 A many-exponential-rates problem: In the simulation with n = n1 + n2 = 100 observations, the likelihood-ratio and old IM tests have similar power, while the new IM-based test has strikingly larger power.The authors attribute the improvement likely to the close relationship between the predictive random set and the assertion of interest, while noting the comparison is not entirely fair.

6 Discussion

The discussion presents IMs as a prior-free framework intended to combine meaningful probabilistic uncertainty with frequency calibration. It emphasizes both the framework’s broad promise and its dependence on user choices and unresolved open problems.

  • 6 Discussion: IMs provide prior-free, post-data probabilistic measures of uncertainty whose values are described as meaningful within and across experiments.The authors characterize this as accomplishing frequentist and Bayesian goals simultaneously.
  • 6 Discussion: The framework’s claimed benefit is meaningful and frequency-calibrated uncertainty measures without a prior distribution.The paper states that no other inferential framework achieves the latter property.
  • 6 Discussion: The final IM depends on the user’s choice of association and predictive random set.The paper notes that assertion- or problem-specific predictive random sets can substantially improve results.
  • 6 Discussion: For high-dimensional auxiliary variables, predictive random sets require special attention and can be constructed for functions most relevant to the assertions of interest.This approach yields a practically useful auxiliary-variable dimension reduction.
  • 6 Discussion: Compared with established Bayesian and frequentist methods, IMs still have many open theoretical and applied problems.The authors present the framework as promising while expecting further developments.

A Details from Section 4.3.2

The appendix develops a score-balance characterization of locally optimal predictive random sets under a uniform-belief assumption. It connects this condition to the behavior of plausibility-related functions near the true parameter.

  • A Details from Section 4.3.2: Under the assumption that belX({θ0}c; S) is Uniform(0, 1), the relevant coverage condition can be represented through measurable subsets X(α).These subsets satisfy PX|θ0{X(α)} = α and characterize when the belief function is at most α.
  • A Details from Section 4.3.2: The score-balance condition requires the sets X(α) to be balanced with respect to the distribution of the score Tθ0(X).The condition traces back to a corresponding property of the predictive random set.
  • A Details from Section 4.3.2: For predictive random sets satisfying the uniform-belief assumption, local validity is equivalent to score balance together with an additional stated condition.This is formalized in Proposition 3 for all θ in a neighborhood of θ0.
  • A Details from Section 4.3.2: A second-order Taylor argument shows that the first term vanishes and the second term is negative, yielding the desired local inequality for ψα(θ).Consequently, condition (4.2) holds for θ in a neighborhood of θ0.

B.1 Correction of Theorem 1

The correction clarifies that proving validity for nested predictive random sets requires additional topological and measurability assumptions beyond nestedness alone. Under these conditions, the corrected theorem establishes validity.

  • B.1 Correction of Theorem 1: The validity proof requires predictive random-set supports to be nested collections of closed measurable subsets containing both ∅ and U.The appendix says these extra requirements make the theorem and proof more transparent and cause no real loss of generality.
  • B.1 Correction of Theorem 1: The corrected theorem defines a predictive random set supported on such a nested collection and concludes that it is valid.The proof uses the smallest support set meeting the required probability threshold.
  • B.1 Correction of Theorem 1: The proof establishes validity by bounding PU{Q(U) > 1 − α} by α for arbitrary α.The bound follows from the construction of the threshold support set and limits over decreasing support sets.

B.2 Correction/extension of Theorem 3

The correction addresses measurability gaps and constructs a predictive random set that is both nested and valid under stated conditions. The resulting belief function is at least as large as that from the original valid predictive random set.

  • Motivation: The main-text nested predictive random set is more efficient but not necessarily valid, despite validity being central to IM analysis.The correction targets this conflict between efficiency and validity.
  • Measurability correction: A measurability issue affects the construction because the sets in (B.1) are not automatically measurable.The modification is unnecessary when the sampling model P_X|θ is discrete.
  • Corrected theorem: Under the stated discrete-space or topological assumptions, any valid predictive random set S admits a nested and valid S′.The construction assumes condition (2.10) and fixes a subset A of the parameter space.
  • Belief-function comparison: For every observation x, the corrected set satisfies bel_x(A; S′) ≥ bel_x(A; S).Thus, the corrected construction preserves validity while weakly improving the belief function for the specified A.
  • Construction: The construction forms S′ from level sets of b(x) = bel_x(A; S), with closed a-events ensuring measurability and nesting.If b(y) ≥ b(x), then S_y ⊇ S_x; the resulting collection serves as the support for S′.
  • Verification: The new S′ is valid, and its belief function is compared with that of S using monotonicity and the inclusion S′_x ⊆ U_x(A).The comparison combines the validity result with the construction's set inclusions.
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