Algorithmic Randomness, Effective Disintegrations, and Rates of Convergence to the Truth

TL;DR

This paper develops a new effective probability framework to analyze the convergence of conditional expectations, identifying algorithmically random points and computable rates of convergence, with implications for Bayesian foundations.

Contribution

It introduces a novel theory of effective disintegrations and characterizes convergence points and rates within computable probability spaces, extending previous results.

Findings

Identifies points where effective random variables converge to the truth.

Establishes conditions for the existence of computable convergence rates.

Generalizes earlier convergence results within a unified framework.

Abstract

L\'evy's Upward Theorem says that the conditional expectation of an integrable random variable converges with probability one to its true value with increasing information. In this paper, we use methods from effective probability theory to characterise the probability one set along which convergence to the truth occurs, and the rate at which the convergence occurs. We work within the setting of computable probability measures defined on computable Polish spaces and introduce a new general theory of effective disintegrations. We use this machinery to prove our main results, which (1) identify the points along which certain classes of effective random variables converge to the truth in terms of certain classes of algorithmically random points, and which further (2) identify when computable rates of convergence exist. Our convergence results significantly generalize earlier results within a unifying novel abstract framework, and there are no precursors of our results on computable rates of convergence. Finally, we make a case for the importance of our work for the foundations of Bayesian probability theory.