arXiv subjects

Ryan Martin

Publications and source records attributed to Ryan Martin.

At least 19 recordsLinked to original sources

arXiv2026

Induction and the rule of succession through a possibilistic inferential model lens

Induction is the process by which empirical evidence is transformed to knowledge. Hume famously argued---and Popper and others agree---that there can be no logical justification for induction. A weaker form of induction, due to Bayes, expresses the aforementioned knowledge in terms of probabilities, and we review some well-known and not-so-well-known criticisms of the Bayesian solution. We then investigate the relatively new possibilistic inferential model (IM) framework, showing that, in addition to the IM's strong, statistical reliability guarantees that it uniquely enjoys, it is safe from those criticisms that damage the Bayesian foundations. For illustration, we reconsider the classical sunrise problem and compare our proposed solution with Laplace's famous rule of succession.

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arXiv2026

Universal Inference for model selection on networks

Model selection and hypothesis testing are important tasks on networks. A key challenge lies in the inherent dependence in network data, as well as the fact that typically only a single realization is observed. As a result, many existing methods must be carefully tailored to specific models and only come with asymptotic theoretical guarantees. In this work, however, we propose a general model selection framework using Universal Inference, making our method widely applicable to various testing scenarios. Since Universal Inference requires two sets of data, we employ edge sampling to obtain proper networks with tractable dependence. We prove that the proposed statistic is an e-value, thus controlling the type I error rate in finite samples under nearly any hypothesis test. To our knowledge, this is the first Universal Inference-type statistic constructed from dependent splits of data as well as the first finite-sample testing guarantee for hypothesis testing on networks. We also prove that the logarithm of the test statistic diverges to positive infinity under various alternative models. On simulated and real-world networks, the proposed method performs well on tasks such as choosing the random graph model and the number of communities.

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arXiv2025

Valid and efficient possibilistic structure learning in Gaussian linear regression

A crucial step in fitting a regression model to data is determining the model's structure, i.e., the subset of explanatory variables to be included. However, the uncertainty in this step is often overlooked due to a lack of satisfactory methods. Frequentists have no broadly applicable confidence set constructions for a model's structure, and Bayesian posterior credible sets do not achieve the desired finite-sample coverage. In this paper, we propose an extension of the possibility-theoretic inferential model (IM) framework that offers reliable, data-driven uncertainty quantification about the unknown model structure. This particular extension allows for the inclusion of incomplete prior information about the unknown structure that facilitates regularization. We prove that this new, regularized, possibilistic IM's uncertainty quantification is suitably calibrated relative to the set of joint distributions compatible with the data-generating process and assumed partial prior knowledge about the structure. This implies, among other things, that the derived confidence sets for the unknown model structure attain the nominal coverage probability in finite samples. We provide background and guidance on quantifying prior knowledge in this new context and analyze two benchmark data sets, comparing our results to those obtained by existing methods.

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arXiv2025

Hypothesis testing for community structure in temporal networks using e-values

Community structure in networks naturally arises in various applications. But while the topic has received significant attention for static networks, the literature on community structure in temporally evolving networks is more scarce. In particular, there are currently no statistical methods available to test for the presence of community structure in a sequence of networks evolving over time. In this work, we propose a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways. Specifically, an e-value framework retains valid testing properties even after combining dependent information, a relevant feature in the context of testing temporal networks. We apply the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

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arXiv2025

Possibilistic inferential models: a review

An inferential model (IM) is a model describing the construction of provably reliable, data-driven uncertainty quantification and inference about relevant unknowns. IMs and Fisher's fiducial argument have similar objectives, but a fundamental distinction between the two is that the former doesn't require that uncertainty quantification be probabilistic, offering greater flexibility and allowing for a proof of its reliability. Important recent developments have been made thanks in part to newfound connections with the imprecise probability literature, in particular, possibility theory. The brand of possibilistic IMs studied here are straightforward to construct, have very strong frequentist-like reliability properties, and offer fully conditional, Bayesian-like (imprecise) probabilistic reasoning. This paper reviews these key recent developments, describing the new theory, methods, and computational tools. A generalization of the basic possibilistic IM is also presented, making new and unexpected connections with ideas in modern statistics and machine learning, e.g., bootstrap and conformal prediction.

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arXiv2025

Divide-and-conquer with finite sample sizes: valid and efficient possibilistic inference

Divide-and-conquer methods use large-sample approximations to provide frequentist guarantees when each block of data is both small enough to facilitate efficient computation and large enough to support approximately valid inferences. When the overall sample size is small or moderate, likely no suitable division of the data meets both requirements, hence the resulting inference lacks validity guarantees. We propose a new approach, couched in the inferential model framework, that is fully conditional in a Bayesian sense and provably valid in a frequentist sense. The main insight is that existing divide-and-conquer approaches make use of a Gaussianity assumption twice: first in the construction of an estimator, and second in the approximation to its sampling distribution. Our proposal is to retain the first Gaussianity assumption, using a Gaussian working likelihood, but to replace the second with a validification step that uses the sampling distributions of the block summaries determined by the posited model. This latter step, a type of probability-to-possibility transform, is key to the reliability guarantees enjoyed by our approach, which are uniquely general in the divide-and-conquer literature. In addition to finite-sample validity guarantees, our proposed approach is also asymptotically efficient like the other divide-and-conquer solutions available in the literature. Our computational strategy leverages state-of-the-art black-box likelihood emulators. We demonstrate our method's performance via simulations and highlight its flexibility with an analysis of median PM2.5 in Maryborough, Queensland, during the 2023 Australian bushfire season.

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arXiv2025

No-prior Bayes reIMagined: probabilistic approximations of inferential models

When prior information is lacking, the go-to strategy for probabilistic inference is to combine a "default prior" and the likelihood via Bayes's theorem. Objective Bayes, (generalized) fiducial inference, etc. fall under this umbrella. This construction is natural, but the corresponding posterior distributions generally only offer limited, approximately valid uncertainty quantification. The present paper takes a reimagined approach that yields posterior distributions with stronger reliability properties. The proposed construction starts with an inferential model (IM), one that takes the mathematical form of a data-driven possibility measure and features exactly valid uncertainty quantification, and then returns a so-called inner probabilistic approximation thereof. This inner probabilistic approximation inherits many of the original IM's desirable properties, including credible sets with exact coverage and asymptotic efficiency. The approximation also agrees with the familiar Bayes/fiducial solution in applications where the model has a group invariance structure. A Monte Carlo method for evaluating the probabilistic approximation is presented, along with numerical illustrations.

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arXiv2025

Variational empirical Bayes variable selection in high-dimensional logistic regression

Logistic regression involving high-dimensional covariates is a practically important problem. Often the goal is variable selection, i.e., determining which few of the many covariates are associated with the binary response. Unfortunately, the usual Bayesian computations can be quite challenging and expensive. Here we start with a recently proposed empirical Bayes solution, with strong theoretical convergence properties, and develop a novel and computationally efficient variational approximation thereof. One such novelty is that we develop this approximation directly for the marginal distribution on the model space, rather than on the regression coefficients themselves. We demonstrate the method's strong performance in simulations, and prove that our variational approximation inherits the strong selection consistency property satisfied by the posterior distribution that it is approximating.

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arXiv2025

The typicality principle and its implications for statistics and data science

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in machine learning and artificial intelligence in years to come. Along these lines, the present paper advances a novel {\em typicality principle} which states, roughly, that if the observed data is sufficiently ``atypical'' in a certain sense relative to a posited theory, then that theory is unwarranted. This emphasis on typicality brings familiar but often overlooked background notions like model-checking to the inferential foreground. One instantiation of the typicality principle is in the context of parameter estimation, where we propose a new typicality-based regularization strategy that leans heavily on goodness-of-fit testing. The effectiveness of this new regularization strategy is illustrated in three non-trivial examples where ordinary maximum likelihood estimation fails miserably. We also demonstrate how the typicality principle fits within a bigger picture of reliable and efficient uncertainty quantification.

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arXiv2025

An efficient Monte Carlo method for valid prior-free possibilistic statistical inference

Inferential models (IMs) offer prior-free, Bayesian-like posterior degrees of belief designed for statistical inference, which feature a frequentist-like calibration property that ensures reliability of said inferences. The catch is that IMs' degrees of belief are possibilistic rather than probabilistic and, since the familiar Monte Carlo methods approximate probabilistic quantities, there are significant computational challenges associated with putting this framework into practice. The present paper overcomes these challenges by developing a new Monte Carlo method designed specifically to approximate the IM's possibilistic output. The proposal is based on a characterization of the possibilistic IM's credal set, which identifies the "best probabilistic approximation" of the IM as a mixture distribution that can be readily approximated and sampled from. These samples can then be transformed into an approximation of the possibilistic IM. Numerical results are presented highlighting the proposed approximation's accuracy and computational efficiency.

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arXiv2024

Asymptotic efficiency of inferential models and a possibilistic Bernstein--von Mises theorem

The inferential model (IM) framework offers an alternative to the classical probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. A key distinction is that classical uncertainty quantification takes the form of precise probabilities and offers only limited large-sample validity guarantees, whereas the IM's uncertainty quantification is imprecise in such a way that exact, finite-sample valid inference is possible. But is the IM's imprecision and finite-sample validity compatible with statistical efficiency? That is, can IMs be both finite-sample valid and asymptotically efficient? This paper gives an affirmative answer to this question via a new possibilistic Bernstein--von Mises theorem that parallels a fundamental Bayesian result. Among other things, our result shows that the IM solution is efficient in the sense that, asymptotically, its credal set is the smallest that contains the Gaussian distribution with variance equal to the Cramer--Rao lower bound. Moreover, a corresponding version of this new Bernstein--von Mises theorem is presented for problems that involve the elimination of nuisance parameters, which settles an open question concerning the relative efficiency of profiling-based versus extension-based marginalization strategies.

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arXiv2024

Multiple Testing in Generalized Universal Inference

Compared to p-values, e-values provably guarantee safe, valid inference. If the goal is to test multiple hypotheses simultaneously, one can construct e-values for each individual test and then use the recently developed e-BH procedure to properly correct for multiplicity. Standard e-value constructions, however, require distributional assumptions that may not be justifiable. This paper demonstrates that the generalized universal inference framework can be used along with the e-BH procedure to control frequentist error rates in multiple testing when the quantities of interest are minimizers of risk functions, thereby avoiding the need for distributional assumptions. We demonstrate the validity and power of this approach via a simulation study, testing the significance of a predictor in quantile regression.

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arXiv2024

Regularized e-processes: anytime valid inference with knowledge-based efficiency gains

Classical statistical methods have theoretical justification when the sample size is predetermined. In applications, however, it's often the case that sample sizes are data-dependent rather than predetermined. The aforementioned methods aren't reliable in this latter case, hence the recent interest in e-processes and methods that are anytime valid, i.e., reliable for any dynamic data-collection plan. But if the investigator has relevant-yet-incomplete prior information about the quantity of interest, then there's an opportunity for efficiency gain. This paper proposes a regularized e-process framework featuring a knowledge-based, imprecise-probabilistic regularization with improved efficiency. A generalized version of Ville's inequality is established, ensuring that inference based on the regularized e-process are anytime valid in a novel, knowledge-dependent sense. Regularized e-processes also facilitate possibility-theoretic uncertainty quantification with strong frequentist-like calibration properties and other Bayesian-like properties: satisfies the likelihood principle, avoids sure-loss, and offers formal decision-making with reliability guarantees.

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arXiv2024

Advances in Bayesian model selection consistency for high-dimensional generalized linear models

Uncovering genuine relationships between a response variable of interest and a large collection of covariates is a fundamental and practically important problem. In the context of Gaussian linear models, both the Bayesian and non-Bayesian literature is well-developed and there are no substantial differences in the model selection consistency results available from the two schools. For the more challenging generalized linear models (GLMs), however, Bayesian model selection consistency results are lacking in several ways. In this paper, we construct a Bayesian posterior distribution using an appropriate data-dependent prior and develop its asymptotic concentration properties using new theoretical techniques. In particular, we leverage Spokoiny's powerful non-asymptotic theory to obtain sharp quadratic approximations of the GLM's log-likelihood function, which leads to tight bounds on the errors associated with the model-specific maximum likelihood estimators and the Laplace approximation of our Bayesian marginal likelihood. In turn, these improved bounds lead to significantly stronger, near-optimal Bayesian model selection consistency results, e.g., far weaker beta-min conditions, compared to those available in the existing literature. In particular, our results are applicable to the Poisson regression model, in which the score function is not sub-Gaussian.

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arXiv2024

Computationally efficient variational-like approximations of possibilistic inferential models

Inferential models (IMs) offer provably reliable, data-driven, possibilistic statistical inference. But despite the IM framework's theoretical and foundational advantages, efficient computation is a challenge. This paper presents a simple yet powerful numerical strategy for approximating the IM's possibility contour, or at least its $\alpha$-cut for a specified $\alpha \in (0,1)$. Our proposal starts with the specification of a parametric family that, in a certain sense, approximately covers the credal set associated with the IM's possibility measure. Akin to variational inference, we then propose to tune the parameters of that parametric family so that its $100(1-\alpha)\%$ credible set roughly matches the IM contour's $\alpha$-cut. This parametric $\alpha$-cut matching strategy implies a full approximation to the IM's possibility contour at a fraction of the computational cost associated with previous strategies.

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arXiv2024

Which statistical hypotheses are afflicted with false confidence?

The false confidence theorem establishes that, for any data-driven, precise-probabilistic method for uncertainty quantification, there exists (non-trivial) false hypotheses to which the method tends to assign high confidence. This raises concerns about the reliability of these widely-used methods, and shines new light on the consonant belief function-based methods that are provably immune to false confidence. But an existence result alone is insufficient. Towards a partial answer to the title question, I show that, roughly, complements of convex hypotheses are afflicted by false confidence.

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arXiv2024

Large-sample theory for inferential models: a possibilistic Bernstein--von Mises theorem

The inferential model (IM) framework offers alternatives to the familiar probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. Allowing this uncertainty quantification to be imprecise makes it possible to achieve exact validity and reliability. But is imprecision and exact validity compatible with attainment of the classical notions of statistical efficiency? The present paper offers an affirmative answer to this question via a new possibilistic Bernstein--von Mises theorem that parallels a fundamental result in Bayesian inference. Among other things, our result demonstrates that the IM solution is asymptotically efficient in the sense that its asymptotic credal set is the smallest that contains the Gaussian distribution whose variance agrees with the Cramer--Rao lower bound.

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arXiv2024

Generalized Universal Inference on Risk Minimizers

A common goal in statistics and machine learning is estimation of unknowns. Point estimates alone are of little value without an accompanying measure of uncertainty, but traditional uncertainty quantification methods, such as confidence sets and p-values, often require distributional or structural assumptions that may not be justified in modern applications. The present paper considers a very common case in machine learning, where the quantity of interest is the minimizer of a given risk (expected loss) function. We propose a generalization of universal inference specifically designed for inference on risk minimizers. Notably, our generalized universal inference attains finite-sample frequentist validity guarantees under a condition common in the statistical learning literature. One version of our procedure is also anytime-valid, i.e., it maintains the finite-sample validity properties regardless of the stopping rule used for the data collection process. Practical use of our proposal requires tuning, and we offer a data-driven procedure with strong empirical performance across a broad range of challenging statistical and machine learning examples.

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