Conditional probability in Renyi spaces

TL;DR

This paper introduces a general concept of conditional probability within Renyi spaces, extending classical probability theory to include unbounded measures and broadening the theoretical framework.

Contribution

It develops a new axiomatic approach to conditional probability in Renyi spaces, expanding the foundational understanding of probability measures.

Findings

Defines a general concept of conditional probability in Renyi spaces

Extends classical probability theory to unbounded measures

Provides a theoretical foundation for future research in measure theory

Abstract

In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. This note introduces a general concept of conditional probability in Renyi spaces. Keywords: Measure theory; conditional probability space; conditional expectation

Equations

$$P^t(A)=P(A\mid T=t)$$

$$P(A\mid B)=\frac{\mu (AB)}{\mu (B)}$$

$$P(A\mid B)=\frac{P(AB\mid C)}{P(B\mid C)},P(B\mid C)>0$$

$$\int \varphi (t)P^t(A\mid B)P_T(dt\mid B)=P(\varphi (T)A\mid B)$$

$$P^t(AB)=P^t(A\mid B)P^t(B)$$

$$\int \varphi (t)P^t(A)\nu (dt)=P(\varphi (T)A)$$

$${\int }_C{P}^t(AB)\nu (dt)={\int }_C{P}^t(A\mid B)P^t(B)\nu (dt)$$

Full text

Keywords: Measure theory; conditional probability space; conditional expectation

Kolmogorov is known for his axioms for probability, but he himself gives much credit for this to french mathematicians, and Fréchet in particular. Kolmogorov (1933) introduced, however, the definition of the conditional expectation

[TABLE]

for the case where $\operatorname{P}$ is a probability measure, $A$ is a positive measurable function, and T:\Omega\mbox{>\rightarrow>}\Omega_{T} is a measurable mapping. His definition extends the elementary definition, and covers in particular cases with $\operatorname{P}(T=t)=0$. Kolmogorov (1933, p.v) notes that especially the theory of conditional probabilities and conditional expectations is an important novel contribution in his book. Now, 86 years after the publication by Kolmogorov (1933), it is safe to say that Kolmogorov’s note was most relevant. The main result in this note is an extension of the Kolmogorov definition of $\operatorname{P}^{t}$ to include the case where $\operatorname{P}$ is a Rényi state.

A Rényi state $\operatorname{P}$ is defined by

[TABLE]

where $\mu$ is a $\sigma$-finite measure on $\Omega$, $A$ is an event, and $B$ is an event with $0<\mu(B)<\infty$. An event is by definition a measurable set in $\Omega$. The family ${\cal B}$ of such elementary conditions $B$ obey the axioms

$\emptyset\not\in{\cal B}$ 2. 2.

$B,C\in{\cal B}\Rightarrow B\cup C\in{\cal B}$ 3. 3.

There exist $B_{1},B_{2},\ldots\in{\cal B}$ with $\cup_{i}B_{i}=\Omega$.

A family ${\cal B}$ of events that obeys these three axioms is according to Renyi (1970, p.38, Def 2.2.1) a bunch. A Rényi state obeys the consistency relation

[TABLE]

for all events $A$ and elementary conditions $B\subset C$. A conditional probability space (Renyi, 1970, p.38, Def 2.2.2) is a measurable space equipped with a consistent family of probability measures $\{\operatorname{P}(\cdot\operatornamewithlimits{{\mid}}B)\operatornamewithlimits{{\mid}}B\in{\cal B}\}$ where ${\cal B}$ is a bunch. It follows that a Rényi state on $\Omega$ defines a conditional probability space.

The structure theorem of Renyi (1970, p.40, Thm 2.2.1) gives that any conditional probability space can be represented by a $\sigma$-finite measure $\mu$ as in equation (2). The original bunch ${\cal B}^{\prime}$ of the conditional probability space is contained in the maximal bunch ${\cal B}$ defined just after equation (2). Equation (2) gives then also a maximal extension of the initial family of conditional probabilities since ${\cal B}^{\prime}\subset{\cal B}$. It follows also that a Rényi state $\operatorname{P}$ can be identified with an equivalence class $[\mu]=\operatorname{P}=\{c\mu\operatornamewithlimits{{\mid}}c>0\}$ of $\sigma$-finite measures. It will be convenient in the following to use the same symbol $\operatorname{P}$ for both a Rényi state and also for a $\sigma$-finite measure $\operatorname{P}$ that represents the Rényi state. This is similar to the use of the symbol $f$ for a function and also a corresponding equivalence class $f$ of measurable functions. The study of spaces of Rényi states is still in its infancy, but we believe it will be important and interesting. This is demonstrated by the study of convergence of Radon Rényi states by Bioche and Druilhet (2016), and also - hopefully - by the constructive definition of conditional Rényi states presented next.

We introduce now a definition of $\operatorname{P}^{t}$ by defining $\operatorname{P}^{t}(A\operatornamewithlimits{{\mid}}B)$ for all positive measurable functions $A$ and all elementary conditions $B$. It is, by the Radon-Nikodym theorem, uniquely determined by requiring

[TABLE]

to hold for all positive measurable functions $\phi$ using the definition $\operatorname{P}_{T}(C\operatornamewithlimits{{\mid}}B)=\operatorname{P}(T\in C\operatornamewithlimits{{\mid}}B)$. This is a natural generalization of the definition used by Kolmogorov (1933, p.47, eq (1)).

We will next define a conditional Rényi state $\operatorname{P}^{t}$ as an equivalence class $\operatorname{P}^{t}=[\mu^{t}]=\{c(t)\mu^{t}\operatornamewithlimits{{\mid}}\operatorname{P}_{T}(c\leq 0)=0\}$ such that

[TABLE]

holds for all positive measurable functions $A$ and all elementary conditions $B$. Choose a $\sigma$-finite measure $\nu$ that dominates $\operatorname{P}_{T}$. The proof of existence of $\nu$ is left to the reader. A representative $\operatorname{P}^{t}$ is now defined uniquely by requiring

[TABLE]

to hold for all positive $\operatorname{P}_{T}$ measurable functions $\phi$. The result is a generalization of the disintegration of a measure relative to a pseudo-image as discussed by Bourbaki (2003, INT VI.45, published in 1959). Taraldsen et al. (2017) provide an alternative route by constructing $\operatorname{P}^{t}$ from $\operatorname{P}^{t}(\cdot\operatornamewithlimits{{\mid}}\cdot)$.

It remains only to prove that equation (5) holds. Observe first that $\operatorname{P}_{T}(C\operatornamewithlimits{{\mid}}B)=\operatorname{P}((T\in C)B)/\operatorname{P}(B)$ and $\operatorname{P}((T\in C)B)=\int_{C}\operatorname{P}^{t}(B)\nu(dt)$ give $\operatorname{P}_{T}(dt\operatornamewithlimits{{\mid}}B)=(\operatorname{P}^{t}(B)/\operatorname{P}(B))\,\nu(dt)$. Using this gives

$\int_{C}\operatorname{P}^{t}(AB)\nu(dt)/\operatorname{P}(B)$ $=$ $\operatorname{P}((T\in C)A\operatornamewithlimits{{\mid}}B)$ $=$ $\int_{C}\operatorname{P}^{t}(A\operatornamewithlimits{{\mid}}B)\operatorname{P}_{T}(dt\operatornamewithlimits{{\mid}}B)$ $=$ $\int_{C}\operatorname{P}^{t}(A\operatornamewithlimits{{\mid}}B)\operatorname{P}^{t}(B)\nu(dt)/\operatorname{P}(B)$, so

[TABLE]

and equation (5) is proved since this holds for all events $C$ in $\Omega_{T}$. This proof is as given by Taraldsen et al. (2018, Thm 1) who also give a more detailed presentation with examples and motivation given by statistical inference including cases with improper priors and posteriors. The results are key generalizations of the theory presented by Taraldsen and Lindqvist (2010). It cannot be concluded, in the generality in this note, that a representative $\operatorname{P}^{t}$ and $\operatorname{P}^{t}(\cdot\operatornamewithlimits{{\mid}}B)$ are measures for almost all $t$. This was observed already by Kolmogorov (1933, p.50,p.55).