[Paper Review] Conditional probabilities and van Lambalgen theorem revisited

Bruno Bauwens, Alexander Shen+1 more|arXiv (Cornell University)|Jul 14, 2016

Computability, Logic, AI Algorithms10 references7 citations

TL;DR

This paper investigates the algorithmic definition of conditional probability in continuous probability spaces, focusing on the existence and computability of conditional distributions for Martin-Löf random sequences. It establishes that for computable measures, the limit of conditional probabilities along prefixes of a random sequence converges to a well-defined measure, generalizing van Lambalgen's theorem to algorithmic randomness contexts with applications to non-computable conditionals and uniform randomness tests.

ABSTRACT

The definition of conditional probability in case of continuous distributions was an important step in the development of mathematical theory of probabilities. How can we define this notion in algorithmic probability theory? In this survey we discuss the developments in this direction trying to explain what are the difficulties and what can be done to avoid them. Almost all the results discussed in this paper have been published (and we provide the references), but we tried to put them into perspective and to explain the proofs in a more intuitive way. We assume that the reader is familiar with basic notions of measure theory and algorithmic randomness.

Motivation & Objective

• To formalize the definition of conditional probability in algorithmic probability theory for continuous distributions.
• To analyze the existence and computability of conditional distributions when the conditioning variable is Martin-Löf random.
• To extend van Lambalgen’s theorem to settings involving non-computable measures and uniform randomness tests.
• To investigate cases where conditional probabilities exist but are non-computable, even with computable joint distributions.
• To provide intuitive explanations and proofs of known results in the context of algorithmic randomness, emphasizing technical challenges and solutions.

Proposed method

• Uses the limit of conditional probabilities over shrinking prefixes of a random sequence to define the conditional measure on the second component.
• Applies the effective martingale convergence theorem to show convergence of the conditional probability ratio for Martin-Löf random sequences.
• Constructs a lower semicomputable test function for the joint measure and relates it to marginal and conditional tests via trimming and normalization.
• Introduces a generalized uniform randomness test that depends on both sequences and parameters, allowing extension to parameterized measures.
• Uses the Radon–Nikodym derivative and Lebesgue differentiation theorem as classical analogues to justify the algorithmic limit definition.
• Employs a trimming operation on test functions to ensure bounded integrals and maintain test properties despite non-semicomputable denominators.

Experimental results

Research questions

• RQ1 Under what conditions does the limit of conditional probabilities along prefixes of a random sequence exist in algorithmic probability theory?
• RQ2 Can the conditional distribution be non-computable even when the joint measure is computable?
• RQ3 How can van Lambalgen’s theorem be generalized to non-computable or parameterized measures in algorithmic randomness?
• RQ4 What role does Martin-Löf randomness play in ensuring convergence of conditional probability limits?
• RQ5 Can uniform randomness tests be defined for measures that depend on parameters, and how do they relate to joint and conditional randomness?

Key findings

• For any computable measure on the product of two Cantor spaces, the limit of conditional probabilities along prefixes of a Martin-Löf random sequence in the first component exists and defines a well-behaved measure on the second component.
• There exist computable joint measures for which the conditional distribution is non-computable, even though the conditioning sequence is Martin-Löf random.
• The limit of conditional probabilities corresponds to the classical conditional measure almost everywhere, as justified by the Lebesgue differentiation theorem.
• A generalized van Lambalgen’s theorem holds for uniform randomness tests, where the joint randomness deficiency decomposes into marginal and conditional components with an O(1) error term.
• The construction of a bounded test function for conditional measures requires trimming to handle non-lower semicomputable ratios, ensuring validity of the test.
• The conditional measure is continuous in the conditioning sequence if the joint measure is continuous, but this property does not hold in general.