10 Regular Conditional Probability / Distribution
(Durrett §5.1.3, p. 193)
Let
\((\Omega,\mathcal{F}_0,P),\qquad \mathcal{F}\subset \mathcal{F}_0.\)
A regular conditional probability is a function
\(\mu:\Omega\times \mathcal{F}_0 \to [0,1]\)
such that:
-
Version property:
\(\mu(\omega,A)=P_{\mathcal{F}}(A)(\omega)\quad\text{a.s., for all }A\in \mathcal{F}_0.\) -
Probability–measure property: For each fixed $\omega$,
\(A\mapsto \mu(\omega,A)\) is a probability measure on $\mathcal{F}_0$.
Application:
\(E_{\mathcal{F}}(f(X))(\omega)=\int_{-\infty}^\infty f(x)\, d\mu(\omega,x).\)
In general, regular conditional probabilities do not always exist.
They do exist in important cases, such as when:
\((\Omega,\mathcal{F}_0,P) = (\mathbb{R},\mathcal{B},P), \quad \mathcal{F}\subset\mathcal{B}.\)
Construction in $\mathbb{R}$
Let $Q = \mathbb{Q}$ (rationals).
Step 1:
Define for all $q\in Q$, \(M_q(\omega) = P_{\mathcal{F}}((-\infty,q])(\omega).\)
Step 2:
Restrict to a full-measure subset $\Omega_0\subseteq \Omega$ with $P(\Omega_0^c)=0$ such that for all $\omega\in \Omega_0$:
(a) Monotonicity:
\(q_2>q_1 \quad \Rightarrow \quad M_{q_2}(\omega)\ge M_{q_1}(\omega).\)
(b) Right-continuity:
\(M_q(\omega) = \lim_{q_n \downarrow q} M_{q_n}(\omega).\)
(c) Limits at ±∞:
\(\lim_{q\to\infty} M_q(\omega)=1, \qquad \lim_{q\to -\infty} M_q(\omega)=0.\)
Step 3:
Define the full CDF: \(M_x(\omega)=\inf_{q\ge x} M_q(\omega),\qquad \omega\in\Omega_0.\)
Then
\(P_{\mathcal{F}}((-\infty,x])(\omega)= M_x(\omega).\)
Step 4:
For any Borel set $(a,b]$, \(\mu(\omega,(a,b]) = M_b(\omega)-M_a(\omega).\)
Thus, \(P_{\mathcal{F}}(A)(\omega)=\mu(\omega,A) \qquad (\forall A\in\mathcal{B}).\)
Regular Conditional Probability for Random Variables
Given a random variable $X:\Omega\to\mathbb{R}$, \(\sigma(X)=\{X\in B : B\in\mathcal{B}(\mathbb{R})\}.\)
A regular conditional distribution of $X$ given $\mathcal{F}$ is: \(\mu(\omega,B)=P_{\mathcal{F}}(X\in B)(\omega).\)
Example: Decomposing Borel sets
Let
\(\mathbb{R}^+=[0,\infty),\qquad \mathbb{R}^- = (-\infty,0).\)
Given a Borel set $A\subset\mathbb{R}$, \(A=A^+ \cup A^-, \qquad A^+=A\cap \mathbb{R}^+, \quad A^- = A\cap\mathbb{R}^-.\)
Define a σ-field: \(\mathcal{F} = \{B,\; B\cup\mathbb{R}^- : B\in\mathcal{B}(\mathbb{R}^+)\}.\)
Then $$ P_{\mathcal{F}}(A)(\omega) = P_{\mathcal{F}}(A^+)(\omega)
- P_{\mathcal{F}}(A^-)(\omega). $$
If $X\in\mathcal{F}$:
\[\mu(\omega,A)=\mathbf{1}_{A^+}(\omega)+ \frac{P(A^-)}{P(\mathbb{R}^-)}\mathbf{1}_{\mathbb{R}^-}(\omega).\]Example: Conditional Distribution of $(X,Y)$
Suppose $(X,Y)$ has a joint density $f_{XY}(x,y)$.
Conditional density: \(f_{Y\mid X=x_0}(y)= \frac{f_{XY}(x_0,y)}{f_X(x_0)}, \qquad f_X(x_0)=\int_{-\infty}^\infty f_{XY}(x_0,y)\,dy.\)
Thus for a Borel set $D\subset \mathbb{R}^2$, \(P_{\sigma(X)}((X,Y)\in D)(\omega) = \int_{\{y:(x_0,y)\in D\}} f_{Y\mid X=x_0}(y)\,dy, \quad x_0=X(\omega).\)
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