Recent progress on conditional randomness

TL;DR

This paper reviews recent advances in the study of ML-randomness conditioned on probabilities, highlighting a new result that generalizes previous work for mutually singular measures.

Contribution

It introduces a novel result on conditional randomness with respect to mutually singular probabilities, extending Hanssen's 2010 findings for Bernoulli processes.

Findings

New result on conditional randomness for mutually singular probabilities

Generalization of Hanssen's 2010 result for Bernoulli processes

Advances in understanding ML-randomness with respect to conditional measures

Abstract

In this article, recent progress on ML-randomness with respect to conditional probabilities is reviewed. In particular a new result of conditional randomness with respect to mutually singular probabilities are shown, which is a generalization of Hanssen's result (2010) for Bernoulli processes.

Equations

$$\forall x\in S,y^{\infty }\in {\mathcal{R}}^{P_Y}P(x|y^{\infty }):=\underset{y\to y^{\infty }}{\lim }P(x|y)\text{ (the right-hand-side exist)}$$

$${\mathcal{R}}_{y^{\infty }}^P\supseteq {\mathcal{R}}^{P(\cdot |y^{\infty }),y^{\infty }}\text{ for all }{y}^{\infty }\in {\mathcal{R}}^{P_Y}.$$

$${\mathcal{R}}_{y^{\infty }}^P={\mathcal{R}}^{P(\cdot |y^{\infty }),y^{\infty }}.$$

$${\mathcal{R}}^{P(\cdot |\theta )}={\mathcal{R}}^{P(\cdot |\theta ),\theta }\text{ for all }\theta .$$

$${\mathcal{R}}_{y^{\infty }}^P\supseteq {\mathcal{R}}^{P(\cdot |y^{\infty })}\text{ for all }{y}^{\infty }\in {\mathcal{R}}^{P_Y}.$$

$${\mathcal{R}}_{y^{\infty }}^P={\mathcal{R}}^{P(\cdot |y^{\infty })}={\mathcal{R}}^{P(\cdot |y^{\infty }),y^{\infty }}.$$

Full text

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Fix $y^{\infty}\in{\cal R}^{P_{Y}}$ and suppose that $P(\cdot|y^{\infty})$ is computable with oracle $y^{\infty}$. Then

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It is known that there is a non-computable conditional probabilities [4] and in [2] Bauwens showed an example that violates the equality in (2) when the conditional probability is not computable with oracle $y^{\infty}$. In [8], an example that for all $y^{\infty}$, the conditional probabilities are not computable with oracle $y^{\infty}$ and (2) holds. A survey on the randomness for conditional probabilities is shown in [1].

Next we study mutually singular conditional probabilities. In [3], Hanssen showed that for Bernoulli model $P(\cdot|\theta)$,

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We generalize Hanssen’s theorem (3) for mutually singular conditional probabilities. In [5, 7], equivalent conditions for mutually singular conditional probabilities are shown.

(1) $P({\cdot\,}{|\,}y)\perp P({\cdot\,}{|\,}z)$ if $\Delta(y)\cap\Delta(z)=\emptyset$ for $y,z\in S$.

(2) ${\cal R}^{P({\cdot\,}{|\,}y)}\cap{\cal R}^{P({\cdot\,}{|\,}z)}=\emptyset$ if $\Delta(y)\cap\Delta(z)=\emptyset$ for $y,z\in S$.

(3) $P_{Y|X}({\cdot\,}{|\,}x)$ converges weakly to $I_{y^{\infty}}$ as $x\to x^{\infty}$ for $(x^{\infty},y^{\infty})\in{\cal R}^{P}$, where $I_{y^{\infty}}$ is the probability that has probability of 1 at $y^{\infty}$.

(4) ${\cal R}^{P}_{y^{\infty}}\cap{\cal R}^{P}_{z^{\infty}}=\emptyset$ if $y^{\infty}\neq z^{\infty}$.

(5) There exists $f:X\to Y$ such that $f(x^{\infty})=y^{\infty}$ for $(x^{\infty},y^{\infty})\in{\cal R}^{P}$.

(6) There exists $f:X\to Y$ and $Y^{\prime}\subset Y$ such that $P_{Y}(Y^{\prime})=1$ and $f=y^{\infty},\ P({\cdot\,};y^{\infty})-a.s.$ for $y^{\infty}\in Y^{\prime}$.*

Generalized form of Hanssen’s theorem (3) is as follows.

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Fix $y^{\infty}\in{\cal R}^{P_{Y}}$ and suppose that $P(\cdot|y^{\infty})$ is computable with oracle $y^{\infty}$. Then

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