Proof mining and probability theory

TL;DR

This paper extends proof mining techniques to probability theory, enabling the extraction of effective bounds from proofs involving measure and integration, and providing a logical foundation for uniformity in probabilistic results.

Contribution

It develops logical systems that formalize probability proofs and guarantee the extraction of uniform, effective bounds, expanding proof mining to a new mathematical domain.

Findings

Logical systems formalize probability measure proofs.

Extraction of uniform bounds independent of probability space parameters.

Transfer principle relates convergence modes for real sequences and random variables.

Abstract

We extend the theoretical framework of proof mining by establishing general logical metatheorems that allow for the extraction of the computational content of theorems with prima facie "non-computational" proofs from probability theory, thereby unlocking a major branch of mathematics as a new area of application for these methods. Concretely, we devise proof-theoretically tame logical systems that, for one, allow for the formalization of proofs involving algebras of sets together with probability contents as well as associated Lebesgue integrals on them and which, for another, are amenable to proof-theoretic metatheorems in the style of proof mining that guarantee the extractability of effective and tame bounds from large classes of ineffective existence proofs in probability theory. Moreover, these extractable bounds are guaranteed to be highly uniform in the sense that they will be independent of all parameters relating to the underlying probability space, particularly regarding events or measures of them. As such, these results, in particular, provide the first logical explanation for the success and the observed uniformities of the previous ad hoc case studies of proof mining in these areas and further illustrate their extent. Beyond these systems, we provide extensions for the proof-theoretically tame treatment of $\sigma$-algebras and associated probability measures using an intensional approach to infinite unions. Lastly, we establish a general proof-theoretic transfer principle that allows for the lift of quantitative information on a relationship between different modes of convergence for sequences of real numbers to sequences of random variables.