Conditional probability (incomplete)

Dependencies: (incomplete)

  1. Probability
  2. Measure
  3. /sets-and-relations/de-morgan-laws
  4. σ-algebra is closed under countable intersections

Let $\Pr$ be a probability measure over the $\sigma$-algebra $(\Omega, \mathcal{F})$. For $B \in \mathcal{F}$ such that $\Pr(B) > 0$, define $\Pr_{|B}: \Omega \mapsto [0, 1]$ as \[ \Pr_{|B}(X) = \frac{\Pr(X \cap B)}{\Pr(B)} \] We'll prove that $\Pr_{|B}$ is also a probability measure over $(\Omega, \mathcal{F})$.

When $B = \{\}$, $\Pr_{|B}$ is not defined (or invalid). It is sometimes possible to extend the definition of $\Pr_{|B}$ to the case where $B \neq \{\}$ and $\Pr(B) = 0$ and prove that $\Pr_{|B}$ is a probability measure, but I don't know when it's possible and how it is defined in that case (maybe it's done using regular conditional probability).

$\Pr_{|B}(X)$ is usually denoted as $\Pr(X \mid B)$ and is called 'probability of $X$ given $B$' or 'probability of $X$ conditioned on $B$'. Often, $\Pr(A \mid B_1 \cap B_2 \cap \ldots \cap B_m)$ is denoted as $\Pr(A \mid B_1, B_2, \ldots, B_m)$. $\Pr(A) = \Pr(A \mid \Omega)$.

If $\Pr(B \cap C) > 0$, then \[ \Pr(A \mid B \cap C) = \frac{\Pr(A \cap B \cap C)}{\Pr(B \cap C)} = \frac{\Pr(A \cap B \mid C)}{\Pr(B \mid C)} \]

Proof that $\Pr_{|B}$ is a probability measure when $\Pr(B) > 0$

$\Pr$ is non-negative implies $\Pr_{|B}$ is non-negative. $\Pr_{|B}({\{\}}) = 0$. $\Pr_{|B}(\Omega) = \frac{\Pr(\Omega \cap B)}{\Pr(B)} = 1$.

$\sigma$-additivity: Let $S$ be a countable subset of $\mathcal{F}$ such that all sets in $S$ are pairwise-disjoint. Let $S' = \{A \cap B: A \in S\}$. Since a $\sigma$-algebra is closed under finite intersections, $S' \in \mathcal{F}$. All sets in $S'$ are also pairwise disjoint. \begin{align} \Pr_{|B}\left(\bigcup_{A \in S} A\right) &= \frac{1}{\Pr(B)}\Pr\left(\left(\bigcup_{A \in S} A\right) \cap B\right) \\ &= \frac{1}{\Pr(B)}\Pr\left(\bigcup_{A \in S} (A \cap B) \right) \tag{De Morgan's laws} \\ &= \frac{1}{\Pr(B)}\left(\sum_{A \in S} \Pr(A \cap B) \right) \tag{$S' \in \mathcal{F}$ and $S'$ is pairwise-disjoint} \\ &= \sum_{A \in S} \Pr_{|B}(A) \end{align} Therefore, $\Pr_{|B}$ is $\sigma$-additive.

Dependency for:

  1. Conditional expectation
  2. Conditioning over random variable
  3. Conditional variance
  4. Independence of events
  5. Markov chain

Info:

Transitive dependencies:

  1. /sets-and-relations/countable-set
  2. /sets-and-relations/de-morgan-laws
  3. σ-algebra
  4. Measure
  5. σ-algebra is closed under countable intersections
  6. Probability