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Wikipedia (and different books too) seem to give two different definitions of what a regular conditional probability is. What is the correct definition and how do they relate? It seems to me that the first definition is the correct one, while the second is actually the definition of a regular conditional distribution?

Definition 1 can be found here.

Definition 1: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $\mathsf{A}\in\mathcal{F}$. Let $\mathbb{1}_{\mathsf{A}}:\Omega\to\{0, 1\}$ be the indicator random variable. A conditional probability of $\mathsf{A}$ given $\mathcal{G}$ is defined as a version of $\mathbb{E}[\mathbb{1}_{\mathsf{A}} \mid \mathcal{G}]$ and denoted $\mathbb{P}(\mathsf{A} \mid \mathcal{G})$ $$ \int_{G} \mathbb{P}(\mathsf{A}\mid \mathcal{G}) d\mathbb{P} = \mathbb{P}(\mathsf{A}\cap G) \qquad \forall \, G\in\mathcal{G}. \qquad \qquad \qquad (1) $$ A conditional probability is said to be regular if $\mathbb{P}(\cdot \mid \mathcal{G})(\omega)$ is a probability measure for any $\omega\in\Omega$. Essentially it is a markov kernel satisfying condition $(1)$.

Definition 2 can be found here. This involves an additional random variable $X$.

Definition 2: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $(E, \mathcal{E})$ be a measurable space, and $X:\Omega\to E$ be a random variable with distribution $\mathbb{P}_X = X_*\mathbb{P}$. Let $\nu:E\times\mathcal{F}\to [0, 1]$ be a Markov kernel satisfying $$ \mathbb{P}(\mathsf{A}\cap X^{-1}(\mathsf{B})) = \int_{\mathsf{B}} \nu(x, \mathsf{A}) \,d \mathbb{P}_X(x). $$

In general, I could not find anywhere the difference or relationship between regular conditional probability and regular conditional distribution

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    $\begingroup$ To relate these two, take $\mathcal{G} = \sigma(X)$ in definition 1. I think you get definition 2? $\endgroup$ Commented Apr 2, 2022 at 1:01
  • $\begingroup$ See also Durrett, page 214 (Section 4.1.3 "Regular conditional probabilties") which defines both regular conditional probabilities and regular conditional distributions: services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf $\endgroup$ Commented Apr 2, 2022 at 5:13
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    $\begingroup$ I think your equation (1) has a mistake and it should be $P[A \cap G]$ not $P[A \cap \mathcal{G}]$. $\endgroup$ Commented Apr 2, 2022 at 5:18

2 Answers 2

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The second one is a more general definition. If you take $X$ to be the identity map from the measure space $(\Omega,\mathcal{F},\mathbb{P})$ to the measure space $(\Omega,\mathcal{G},\mathbb{P})$ you recover the first definition; with $\nu(x,A)=\mathbb{P}(A\,|\,\mathcal{G})(x)$, $\mathbb{P}$ a.e $x$.

Added:

  1. (Distribution) Conditional distribution is a term that makes sense for a triple $(X,\mathcal{G},\mathbb{P})$, where $X$ is a random variable $X:(\Omega,\mathcal{F})\to (E,\mathcal{E})$, $\,\mathcal{G}\subset\mathcal{F}$ and $\mathbb{P}$ is a probability measure on $(\Omega,\mathbb{F})$. The $\mathbb{P}$-conditional distribution of the random variable $X$ is an assignment to each $B\in \mathcal{E}$, an RV represented $\mathbb{P}$ a.e by the function $\mathbb{E}[\mathbb{1}_B\circ X\,|\,\mathcal{G}]$.

  2. (Density function) For $A\in\mathcal{F}$, $$m_A(B)=\int_\Omega \mathbb{1}_A\cdot\mathbb{E}[\mathbb{1}_B\circ X\,|\,\mathcal{G}]\;d\mathbb{P},$$ where $B\in\mathcal{E}$, is a measure on $(E,\mathcal{E})$. Denote the Radon-Nikodym derivative of $m_A$ w.r.t $X_\ast \mathbb{P}$, $\nu^\mathcal{G}(\cdot,A)$. Call $\mathbb{P}$-conditional probability density function of the random variable $X$ with respect to the measure $X_\ast \mathbb{P}$, the assignment to each $A\in \mathcal{F}$, of an RV represented $X_\ast\mathbb{P}$ a.e by the function $\nu^{\mathcal{G}}(\cdot,A)$ (the kernel in the question posted if $\mathcal{G}=\sigma(X)$).

  3. So the conditional probability density function and conditional distribution carry the same information.

  4. (Probability) Conditional probability makes sense for a pair $(\mathcal{G},\mathbb{P})$ where $\mathcal{G}\subset \mathcal{F}$ and $\mathbb{P}$ is a measure on $(\Omega,\mathcal{F})$. The $(\mathbb{P},\mathcal{G})$-conditional probability on $(\Omega,\mathcal{F})$ is the assignment of an RV represented $\mathbb{P}$ a.e by $\mathbb{E}[\mathbb{1}_A\,|\,\mathcal{G}]=:\mathbb{P}[A\,|\,\mathcal{G}]$, to each $A\in\mathcal{F}$.

  5. They are extensions respectively of the notions of $\;$"distribution of a random variable", "probability density function of a random variable" and "probability" (with the trivial conditioning $\mathcal{G}=\{\emptyset,\Omega\}$) to general $\mathcal{G}\subset \mathcal{F}$.

  6. As for all $B\in\mathcal{E}$, $$\int_\Omega \mathbb{1}_B\circ X\cdot\mathbb{E}[\mathbb{1}_A\,|\,\mathcal{G}]\;d\mathbb{P}=\int_\Omega \mathbb{1}_A\cdot\mathbb{E}[\mathbb{1}_B\circ X\,|\,\mathcal{G}]\;d\mathbb{P}=\int_E \mathbb{1}_B\cdot \nu^{\mathcal{G}}(x,A)\;d(X_\ast \mathbb{P})(x),$$ if $X=id:(\Omega,\mathcal{F})\to (\Omega,\mathcal{G})$, with $\mathcal{G}\subset\mathcal{F}$, then $$\nu^{\mathcal{G}}(\omega,A)=\mathbb{E}[\mathbb{1}_A\,|\,G](\omega),\; \mathbb{P} \;\text{a.e}\; \omega\in\Omega.$$

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  • $\begingroup$ Thank you for your answer. It makes sense that the first one is a special case of the second one. However, what is the definition of conditional probability distribution then? $\endgroup$ Commented Apr 2, 2022 at 9:26
  • $\begingroup$ I expanded the answer. $\endgroup$ Commented Apr 2, 2022 at 18:19
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When there are two random variables $X$ and $Y$ defined on the same probability space $(\Omega, \mathcal{A}, \mathbb{P})$, taking values in the measurable spaces $(S, \mathcal{S})$ and $(T, \mathcal{T})$, respectively, a (regular) conditional distribution of $X$ given $Y$ is a Markov kernel $\mu$ from $(S, \mathcal{S})$ to $(T, \mathcal{T})$ satisfies $\mu(X, B) = \mathbb{P}(Y \in B \mid X)$ almost surely for any $B \in \mathcal{T}$, this is equivalent to $\mathbb{P}(X \in A, Y \in B) = \int_{A} \mu(s, B) ~\mathrm{d} \mathbb{P}_{X}(s)$ for any $A \in \mathcal{S}$, $B \in \mathcal{T}$.

For any sub-$\sigma$-field $\mathcal{G}$ of $\mathcal{A}$, we take $(S, \mathcal{S}) = (\Omega, \mathcal{G})$ and $X$ as the identity map $\omega \in (\Omega, \mathcal{A}) \mapsto \omega \in (\Omega, \mathcal{G})$ in the definition of conditional distribution, then we get the definition of regular conditional probability: a $\mathcal{G}$-measurable Markov kernel $\mu$ such that $\mu(\cdot, B) = \mathbb{P}(Y \in B \mid \mathcal{G})$ almost surely for any $B \in \mathcal{T}$.

"Conditional probability" is defined as a special case of "conditional expectation". Although "regular conditional probability" is literally close to "conditional probability", it is a special case of "regular conditional distribution".

Reference: Kallenberg, O. (2021). Foundations of Modern Probability (3rd ed.). Springer, Chapter 8.

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