\section{Probability Theory}

\begin{definition}[Probability Model $\langle \Omega, P \rangle$]\label{def:prob-model}
  consists of a countable sample space $\Omega$ and a
  probability function $P : \Omega \rightarrow [0;1]$,
  s.t. $\sum_{\omega \in \Omega} P(\omega) = 1$
\end{definition}

\begin{definition}[Event]\label{def:event}
  When a random variable $X$ takes on a value $x$.
\end{definition}

\begin{definition}[Conditional/Posterior
  Probability]\label{def:cond-prob} The probability
  \[ P(a \mid b) = \frac{P(a \land b)}{P(b)}, \] i.e.\ the chance that
  \hyperref[def:event]{event} ``a'' takes place, given the event
  ``b''.
\end{definition}

\begin{definition}[Conditional Independence]\label{def:cond-indep}
  Two \hyperref[def:event]{events} $a$ and $b$ are conditionally
  independent, if $P(a \land b \mid c) = P(a \mid c) P(b \mid c)$.
\end{definition}

\begin{definition}[Probability Distribution]\label{def:prob-dist}
  A vector for $\mathbf{P}(X)$ relating each element of the
  \hyperref[def:prob-model]{sample space} to a probability:
  \[ \langle P(\omega_1), \dots, P(\omega_n) \rangle. \] Related
  concepts:
  \begin{description}
  \item[Joint PD]\label{def:jpd} Given
    $Z \subseteq \{ X_1, \dots, X_n \},$ results in a array the
    probabilities of all \hyperref[def:event]{events}.
  \item[Full joint PD]\label{def:fjdp}
    Joint PD for all random variables.
  \item[Conditional PD]\label{def:cpd}
    Given $X$ and $Y$, results in a table for
    every probability $P(X \mid Y)$.
  \end{description}
\end{definition}

\begin{definition}[Product Rule]\label{def:prod-rule}
  $P(a \land b) = P(a \mid b) P(b)$
\end{definition}

\begin{definition}[Chain Rule]\label{def:chain-rule} Extension of the
  \hyperref[def:prod-rule]{product rule},
  \[
    P(X_1, \dots, X_n) = P(X_n \mid X_{n-1}, \dots, X_1) \dots P(X_2 |
    X_1) P(X_1)
  \]
\end{definition}

\begin{definition}[Marginalisation]\label{def:margin}
  $\mathbf{P}(\mathbf{X}) = \sum_{y \in \mathbf{Y}}
  \mathbf{P}(\mathbf{X}, y)$
\end{definition}

\begin{definition}[Normalisation]\label{def:normalis}
  Given $\mathbf{P}(X \mid e)$, and a normalization constant \[
    \alpha = \frac{1}{ P(x_1 \mid e) + \dots + P(x_n \mid e)}, \]
  normalization scales each element of the probability distribution
  s.t. $\sum \alpha \mathbf{P}(X \mid e) = 1$
\end{definition}

\begin{definition}[Bayes' Rule]\label{def:bayes-rule}
  Given two propositions $a$ and $b$,
  \[ P(a \mid b) = \frac{P(b \mid a) P(a)}{P(b)}, \] where
  $P(a) \neq 0$ and $P(b) \neq 0$.
\end{definition}

\begin{definition}[Naive Bayes' Model]\label{def:naive-bayes-model}
  In this model, the \hyperref[def:fjdp]{full joint probability
    distribution} is
  \[ \mathbf{P}(c \mid e_1, \dots, e_n) = \mathbf{P}(c) \prod_i
    \mathbf{P}(e_i \mid c), \] i.e.\ a \emph{single} cause $c$
  influences a number of \hyperref[def:cond-indep]{cond. independent}
  effects $e_i$.
\end{definition}

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