Response to the comment Confidence in confidence distributions!

Martin, Ryan; Balch, Michael; Ferson, Scott

doi:10.1098/rspa.2020.0579

Cited by 6 publications

(3 citation statements)

References 27 publications

(41 reference statements)

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“…Indeed, fiducial or default-prior Bayes solutions often have the property that credible regions derived from the posterior are (approximate) confidence regions. However, according to the false confidence theorem in Balch et al (2019), no IM whose output is countably additive can be valid; see, also, Martin (2019) and Martin et al (2021). So validity requires something beyond what these more familiar statistical frameworks can provide.…”

Section: Vacuous-prior Validitymentioning

confidence: 99%

Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information

Martin¹

2022

Preprint

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Bayesian inference quantifies uncertainty directly and formally using classical probability theory, while frequentist inference does so indirectly and informally through the use of procedures with error rate control. Both have merits in the appropriate context, but the context isn't binary. There's an entire spectrum of contexts depending on what, if any, partial prior information is available to incorporate, with "Bayesian" and "frequentist" sitting on opposite ends of the spectrum corresponding to a complete, fully precise prior distribution and a vacuous, fully imprecise prior distribution, respectively. Common examples between the two extremes include those high-dimensional problems where, e.g., sparsity assumptions are relevant but fall short of determining a complete prior distribution. This paper ties the two frameworks together by treating those cases where only partial prior information is available using the theory of imprecise probability. The end result is a unified framework of (imprecise-probabilistic) statistical inference with a new validity condition that implies both frequentist-style error rate control for derived procedures and Bayesian-style no-sure-loss properties, relative to the given partial prior information. This theory contains both the classical "Bayesian" and "frequentist" frameworks as special cases, since they're both valid in this new sense relative to their respective partial priors. Different constructions of these valid inferential models are considered, and compared based on their efficiency.

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Section: Vacuous-prior Validitymentioning

confidence: 99%

Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information

Martin¹

2022

Preprint

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“…Here the message is similar but arguably goes further. First, IMs are specifically designed to achieve valid inference about unknowns, which is stronger than the confidence property achieved by fiducial methods (e.g., Martin 2021a; Martin et al 2021). Second, the conservativeness that's built in to the IM to achieve this validity property appears in the decision decision-making context 23) with a = −1 (dotted) and a = 0 (dashed).…”

Section: Connectionsmentioning

confidence: 99%

Inferential models and the decision-theoretic implications of the validity property

Martin¹

2021

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Inferential models (IMs) are data-dependent, probability-like structures designed to quantify uncertainty about unknowns. As the name suggests, the focus has been on uncertainty quantification for inference, and on establishing a validity property that ensures the IM is reliable in a specific sense. The present paper develops an IM framework for decision problems and, in particular, investigates the decision-theoretic implications of the aforementioned validity property. I show that a valid IM's assessment of an action's quality, defined by a Choquet integral, will not be too optimistic compared to that of an oracle. This ensures that a valid IM tends not to favor actions that the oracle doesn't also favor, hence a valid IM is reliable for decision-making too. In a certain special class of structured statistical models, further connections can be made between the valid IM's favored actions and those favored by other more familiar frameworks, from which certain optimality conclusions can be drawn. An important step in these decision-theoretic developments is a characterization of the valid IM's credal set in terms of confidence distributions, which may be of independent interest.

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“…I've already responded to some of these critiques (Martin 2021a;Martin et al 2021), but I have some new insights to share that will help clear up this confusion about confidence limits versus (imprecise) probabilistic inference. Specifically, my goal here is to give a complete characterization of the relationship between the fiducial solutionwhich agrees with the default-prior Bayes solution, among others, in the context I'm considering-and the IM solution.…”

Section: Introductionmentioning

confidence: 99%

Fiducial inference viewed through a possibility-theoretic inferential model lens

Martin¹

2023

Preprint

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Fisher's fiducial argument is widely viewed as a failed version of Neyman's theory of confidence limits. But Fisher's goal-Bayesian-like probabilistic uncertainty quantification without priors-was more ambitious than Neyman's, and it's not out of reach. I've recently shown that reliable, prior-free probabilistic uncertainty quantification must be grounded in the theory of imprecise probability, and I've put forward a possibility-theoretic solution that achieves it. This has been met with resistance, however, in part due to statisticians' singular focus on confidence limits. Indeed, if imprecision isn't needed to perform confidence-limit-related tasks, then what's the point? In this paper, for a class of practically useful models, I explain specifically why the fiducial argument gives valid confidence limits, i.e., it's the "best probabilistic approximation" of the possibilistic solution I recently advanced. This sheds new light on what the fiducial argument is doing and on what's lost in terms of reliability when imprecision is ignored and the fiducial argument is pushed for more than just confidence limits.

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Response to the comment Confidence in confidence distributions!

Cited by 6 publications

References 27 publications

Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information

Valid and efficient imprecise-probabilistic inference across a spectrum of partial prior information

Inferential models and the decision-theoretic implications of the validity property

Fiducial inference viewed through a possibility-theoretic inferential model lens

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