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Schnorr Randomness and Effective Bayesian Consistency and Inconsistency

Simon M. Huttegger, Sean Walsh, Francesca Zaffora Blando

TL;DR

This work analyzes Bayes-style consistency and inconsistency from computable-probability and algorithmic-randomness perspectives. It proves an effective Doob Consistency Theorem for Schnorr-random parameters, showing computable convergence of posteriors under a suitable computable map from samples to parameters, and develops a robust theory of $\mathsf{SR}^{\nu}$ via $\Sigma^{0,\nu}_2$ classes and $L_1(\nu)$ tests. It establishes identifiability-based conditions under which all $\theta$ in $\mathsf{SR}^{p}$ are computably consistent, and, conversely, constructs effective countermodels to show that Freedman-style inconsistency can be effectively generic. The paper also extends these results to the infinite-dimensional simplex $\mathbb{S}_{\infty}$, providing computable Polish-subspace structure, identifiability criteria, and an effective Freedman inconsistency theorem, thereby delivering a computability-theoretic resolution to when Bayes' rule remains consistent. Collectively, the results connect Doob's classical convergence theorems with algorithmic randomness, strengthening Takahashi's Cantor-space findings and offering a framework to distinguish parameters for which Bayesian rules are effectively reliable.

Abstract

We study Doob's Consistency Theorem and Freedman's Inconsistency Theorem from the vantage point of computable probability and algorithmic randomness. We show that the Schnorr random elements of the parameter space are computably consistent, when there is a map from the sample space to the parameter space satisfying many of the same properties as limiting relative frequencies. We show that the generic inconsistency in Freedman's Theorem is effectively generic, which implies the existence of computable parameters which are not computably consistent. Taken together, this work provides a computability-theoretic solution to Diaconis and Freedman's problem of ``know[ing] for which [parameters] the rule [Bayes' rule] is consistent'', and it strengthens recent similar results of Takahashi on Martin-Löf randomness in Cantor space.

Schnorr Randomness and Effective Bayesian Consistency and Inconsistency

TL;DR

This work analyzes Bayes-style consistency and inconsistency from computable-probability and algorithmic-randomness perspectives. It proves an effective Doob Consistency Theorem for Schnorr-random parameters, showing computable convergence of posteriors under a suitable computable map from samples to parameters, and develops a robust theory of via classes and tests. It establishes identifiability-based conditions under which all in are computably consistent, and, conversely, constructs effective countermodels to show that Freedman-style inconsistency can be effectively generic. The paper also extends these results to the infinite-dimensional simplex , providing computable Polish-subspace structure, identifiability criteria, and an effective Freedman inconsistency theorem, thereby delivering a computability-theoretic resolution to when Bayes' rule remains consistent. Collectively, the results connect Doob's classical convergence theorems with algorithmic randomness, strengthening Takahashi's Cantor-space findings and offering a framework to distinguish parameters for which Bayesian rules are effectively reliable.

Abstract

We study Doob's Consistency Theorem and Freedman's Inconsistency Theorem from the vantage point of computable probability and algorithmic randomness. We show that the Schnorr random elements of the parameter space are computably consistent, when there is a map from the sample space to the parameter space satisfying many of the same properties as limiting relative frequencies. We show that the generic inconsistency in Freedman's Theorem is effectively generic, which implies the existence of computable parameters which are not computably consistent. Taken together, this work provides a computability-theoretic solution to Diaconis and Freedman's problem of ``know[ing] for which [parameters] the rule [Bayes' rule] is consistent'', and it strengthens recent similar results of Takahashi on Martin-Löf randomness in Cantor space.
Paper Structure (9 sections, 38 theorems, 38 equations, 1 figure)

This paper contains 9 sections, 38 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sigma zero two and Schnorr randomness
  3. Computable Polish space structure on the product
  4. Proof of Theorem \ref{['thm:doob']}
  5. Limiting relative frequencies
  6. Proof of Theorem \ref{['thm:reversal']}
  7. The infinite simplex, subspaces, and identifiability
  8. Sufficient conditions for inconsistency
  9. Effective Freedman inconsistency

Key Result

Theorem 1.5

Suppose there is a sequence of uniformly computable continuous functions ${f_n:\mathcal{X}\rightarrow \Omega}$ such that both of the following happen: Then all pairs $(\theta, x)$ of parameters and samples in $\mathsf{SR}^{\mu}$ are such that, for all c.e. open $U$ in $\Omega$ with $p(U)$ computable, one has that $\lim_n p(U \space\mid\space x_{n})=I_U(\theta)$. Further, all parameters $\theta$ i

Figures (1)

  • Figure 1: The five spaces and maps between them.

Theorems & Definitions (81)

  • Definition 1.1: Effective assumptions
  • Definition 1.2: Convention on versions: differences of $L_1(p)$ Schnorr tests
  • Definition 1.3: Infinite-dimensional simplex
  • Definition 1.4: Consistency notions, classical and effective
  • Theorem 1.5: Effective Doob Consistency Theorem and Schnorr randomness
  • Definition 1.6
  • Definition 1.6
  • Proposition 1.6
  • Proposition 1.6
  • Proposition 1.6: Paradigmatic example: limiting relative frequencies
  • ...and 71 more