B.3 Conditioning

B.3.3

Corollary B.55

• $(S, \mathcal{A}, \mu)$,
  • probability sp.
• $(\mathcal{Y}, \mathcal{B}_{2})$,
  • \(\forall x \in \mathcal{X}, \{x\} \in \mathcal{B}_{2}\),
• $(\mathcal{X}, \mathcal{B})$,

\[\]

proof

$\Box$

B.3.4 Conditional Independence

Definition B.57

• $I$
  • inde set
• $Y$
  • r.v.
• \(\{X_{i}\}_{i \in I}\),
  • r.v.
• \(\mathcal{A}_{i} := \sigma(X_{i})\),

\(\{X_{i}\}\) are said to be conditionally independent given $Y$ if

\[\forall n \in \mathbb{N}, \ i_{1}, \ldots_{n}, \ A_{1} \in \mathcal{A}_{i_{1}}, \ldots, A_{n} \in \mathcal{A}_{i_{n}}, \ \mu \left( \bigcup_{j=1}^{n} A_{j} \mid Y \right) = \prod_{j=1}^{n} \mu \left( A_{j} \mid Y \right) \text{-a.s.}\]

If $Y$ is constant almost surely, we say \(\{X_{i}\}\) are independent.

■

B.3.5 The Law of Total Probability

Theorem B.70 Law of total probability

• $(\mathcal{S}, \mathcal{A}, \mu)$,
  • probability sp.
• $Z$
  • r.v.
  • \(\mathrm{E}[ |Z| ] < \infty\),

proof

$\Box$

Corollary B.71

proof

$\Box$

Corollary B.72

proof

$\Box$

Theorem B.73

• $(\mathcal{S}, \mathcal{A}, \mu)$,
  • probability sp.
• $\mathcal{C} \subseteq \mathcal{B} \subseteq \mathcal{A}$,
  • $\mathcal{C}, \mathcal{B}$ are sub $\sigma$-algebra
• $Z: S \rightarrow \mathbb{R}$,
  • measurable
  • \(\mathrm{E}[ |Z| ] < \infty\),

Then following statements are equivalent;

• (i) there exists a version of \(\mathrm{E}[Z \mid \mathcal{B}]\) which is $\mathcal{C}$ measurable
• (ii)

\[\mathrm{E}[Z \mid \mathcal{B}] = \mathrm{E}[Z \mathcal{C}] \quad \mu \text{-a.s.}\]

proof

(i) $\Rightarrow$ (ii) Suppose that $W$ is a version of \(\mathrm{E}[Z \mid \mathcal{B}]\) which is $\mathcal{C}$ measurable. We have

\[\mathrm{E}[Z \mid \mathcal{B}] = W \ \mu \text{-a.s.}\]

Then

\[C \in \mathcal{C}, \ \int_{C} W(s) \ \mu(ds) = \int_{C} \mathrm{E}[Z \mid \mathcal{B}](s) \ \mu(ds) = \int_{C} Z(s) \ \mu(ds) = \int_{C} \mathrm{E}[Z \mid \mathcal{C}](s) \ \mu(ds) .\]

(i) $\Leftarrow$ (ii)

$\mathrm{E}[Z \mid C]$ is a version of $\mathrm{E}[Z \mid \mathcal{B}]$.

$\Box$