New Developments in the Foundations of Statistics

Ryan Martin

Imprecision in inferential models: why it's needed, where it comes from, and how it's beneficial

Abstract: My basic claim is that the likelihood (data + model) alone can't support reliable probabilistic inference. I'll justify this claim by reconsidering Fisher's well-known warnings about the interpretation of the likelihood.  So, to achieve a sort of middle-road -- what Basu called a "via media" -- we need to relax either the "reliability" or the "probabilistic" parts.  With today's research landscape being almost singularly focused on statistical methods and their performance, reliability is necessary, so the only option is to relax probability, i.e., to allow for the right amount of imprecision.  This is the *why*.  I'll explain the *where* by making some new connections to certain probability-to-imprecise-probability transforms, and point out how these relate to Fisher's fiducial argument.  It's important to understand that the relaxation of probability isn't a sacrifice -- it's actually beneficial in many ways.  *How* imprecision is beneficial includes the ability for the user to incorporate incomplete prior information and the flexibility to satisfy the likelihood principle (or something close to it) if he/she wants.  

Jan Hannig

A Geometric Perspective on Bayesian and Generalized Fiducial Inference

Abstract: Post-data statistical inference concerns making probability statements about model parameters conditional on observed data. When {\it a priori} knowledge about parameters is available, post-data inference can be conveniently made from Bayesian posteriors. In the absence of prior information, we may still rely on objective Bayes or generalized fiducial inference (GFI). Inspired by approximate Bayesian computation, we propose a novel characterization of post-data inference with the aid of differential geometry. Under suitable smoothness conditions, we establish that Bayesian posteriors and generalized fiducial distributions (GFDs) can be respectively characterized by absolutely continuous distributions supported on the same differentiable manifold: The manifold is uniquely determined by the observed data and the data generating equation of the fitted model. Our geometric analysis not only sheds light on the connection and distinction between Bayesian inference and GFI, but also allows us to sample from posteriors and GFDs using manifold Markov chain Monte Carlo algorithms.

Leonardo Cella

Finite sample valid probabilistic inference on quantile regression

Abstract: In most applications, uncertainty quantification in quantile regression takes the form of set estimates for the regression coefficients. However, often a more informative type of uncertainty quantification is desired where other inference-related tasks can be performed, such as the assignment of (imprecise) probabilities to assertions of interest about (any feature of) the regression coefficients. Validity of these probabilities, in the sense that their values are well-calibrated in a frequentist sense, is fundamental to the reliability of the drawn conclusions.  In this talk, I will present a nonparametric Inferential Model (IM) construction that offers provably valid probabilistic uncertainty quantification in quantile regression, even in finite sample settings. It is also shown that this IM can be used to derive finite sample confidence regions for (any feature of) the regression coefficients. The proposed IM construction is not specific to the quantile regression problem. During this talk, I will also discuss its generality and potential application to other relevant nonparametric problems.

Ruobin Gong

Differential Privacy: General Inferential Limits via Intervals of Measures:

Abstract: Differential privacy (DP) is a mathematical standard for assessing the privacy provided by a data-release mechanism. We provide formulations of pure ɛ-differential privacy first as a Lipschitz continuity condition and then using an object from the imprecise probability literature: the interval of measures. We utilize this second formulation to establish bounds on the appropriate likelihood function for ɛ-DP data -- and in turn establishing bounds on key quantities in both frequentist hypothesis testing and Bayesian inference. Under very mild conditions, these results are valid for arbitrary parameters, priors, and data generating models. These bounds are weaker than those attainable when analyzing specific data generating models or data-release mechanisms. However, they provide generally applicable limits on the ability to learn from differentially private data -- even when the analyst's knowledge of the model or mechanism is limited. They also shed light on the semantic interpretation of differential privacy, a subject of contention in the current literature. Joint work with James Bailie.