Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference

Martin, Ryan; Liu, Chuanhai

doi:10.1080/01621459.2012.747960

Cited by 141 publications

(166 citation statements)

References 29 publications

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“…This is discussed in Martin and Liu [15] under the name of conditional IMs (CIMs). The key idea is to condition on components or (data-free) functions of a-variables that are fully observed.…”

Section: Definition 2 (Validity Of Prs)mentioning

confidence: 99%

“…The work on efficient IM inference is briefly reviewed here. Martin and Liu [15] discuss the use of assertion-specific PRS for optimal inference. They also argued that the consideration of optimal inference helps resolve the uniqueness issue regarding both the choice of the association for representing the sample model and the choice of the PRS for generating valid inference.…”

Section: Definition 2 (Validity Of Prs)mentioning

confidence: 99%

“…Martin and Liu [15] show that sufficient statistics can be used as an initial step of CIMs. The argument for the use of sufficient statistics is, however, not the same as that Fisher used (i.e., likelihood based) for establishing the concept of sufficient statistics.…”

Section: Example 3 (The Unit Gaussian Mean)mentioning

confidence: 99%

“…A PRS S is defined by (i) a pre-specified and possibly assertionspecific collection of measurable subsets, S A , in the auxiliary space U and (ii) an appropriate probability measure on S A . For simplicity without loss of efficiency [15], S A is taken to be a nested sequence of subsets in U, i.e., for all two elements S 1 and S 2 in S A , it holds that S 1 ⊆ S 2 or S 2 ⊆ S 1 . In this case, the needed probability mass can be specified as…”

Section: The P-stepmentioning

confidence: 99%

“…Section 2 provides a brief review of the IM framework formulated recently in Martin and Liu [15]. A generalized inference model for comparing two multinomials is presented in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Large Scale Multinomial Inferences and Its Applications in Genome Wide Association Studies

Liu

Xie

2012

Advances in Intelligent and Soft Computing

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Statistical analysis of multinomial counts with a large number K of categories and a small number n of sample size is challenging to both frequentist and Bayesian methods and requires thinking about statistical inference at a very fundamental level. Following the framework of Dempster-Shafer theory of belief functions, a probabilistic inferential model is proposed for this "large K and small n" problem. The inferential model produces a probability triplet (p, q, r) for an assertion conditional on observed data. The probabilities p and q are for and against the truth of the assertion, whereas r = 1 − p − q is the remaining probability called the probability of "don't know". The new inference method is applied in a genome-wide association study with very high dimensional count data, to identify association between genetic variants to the disease Rheumatoid Arthritis.

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Section: Definition 2 (Validity Of Prs)mentioning

confidence: 99%

Section: Definition 2 (Validity Of Prs)mentioning

confidence: 99%

Section: Example 3 (The Unit Gaussian Mean)mentioning

confidence: 99%

Section: The P-stepmentioning

confidence: 99%

“…Section 2 provides a brief review of the IM framework formulated recently in Martin and Liu [15]. A generalized inference model for comparing two multinomials is presented in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Large Scale Multinomial Inferences and Its Applications in Genome Wide Association Studies

Liu

Xie

2012

Advances in Intelligent and Soft Computing

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Inferential Models

Martin

2017

Wiley StatsRef: Statistics Reference Online

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Fisher's fiducial argument was intended to solve arguably the most fundamental problem in statistics, namely, probabilistic inference on unknown parameters without a prior distribution or Bayes's theorem. Despite Fisher's ingenuity, fiducial inference has, for various reasons, been largely dismissed by the mainstream of statistics, so the problem remains an open one. This article surveys a recently proposed alternative to Fisher's theory, called inferential models (IMs), with the same goal of providing prior‐free probabilistic inference. Similar to fiducial, the IM approach starts by connecting the data and unknown parameters to an unobservable auxiliary variable defined through a particular representation of the sampling model. The IM's chief novelty is the introduction of a random set for predicting the unobserved value of that auxiliary variable. This makes the IM output a belief function, rather than a probability, but an appropriate choice of the random set guarantees that this output is valid which, among other things, implies that interval estimates derived from the IM have provable bounds on the frequentist error rate.

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A Gaussian alternative to using improper confidence intervals

Plante

2020

Can J Statistics

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The problem posed by exact confidence intervals (CIs) which can be either all-inclusive or empty for a nonnegligible set of sample points is known to have no solution within CI theory. Confidence belts causing improper CIs can be modified by using margins of error from the renewed theory of errors initiated by J. W. Tukey-briefly described in the article-for which an extended Fraser's frequency interpretation is given. This approach is consistent with Kolmogorov's axiomatization of probability, in which a probability and an error measure obey the same axioms, although the connotation of the two words is different. An algorithm capable of producing a margin of error for any parameter derived from the five parameters of the bivariate normal distribution is provided. Margins of error correcting Fieller's CIs for a ratio of means are obtained, as are margins of error replacing Jolicoeur's CIs for the slope of the major axis. Margins of error using Dempster's conditioning that can correct optimal, but improper, CIs for the noncentrality parameter of a noncentral chi-square distribution are also given.

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Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference

Cited by 141 publications

References 29 publications

Large Scale Multinomial Inferences and Its Applications in Genome Wide Association Studies

Large Scale Multinomial Inferences and Its Applications in Genome Wide Association Studies

Inferential Models

A Gaussian alternative to using improper confidence intervals

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