# Definitions and useful formulas

### Joint Conditional Probability

The probability of conditions $A$ and $B$ occuring at the same time

$P(A,B) \equiv P(A \text{ and } B)$

The probability of $A$ or $B$ occuring is given by

$P(A \cup B) = P(A)+P(B)-P(A,B)$

### Conditional Probability

The conditional Probability, of an event $A$ given a condition $B$, denoted $P(A|B)$ is defined as

$P(A|B) = \frac{P(A,B)}{P(B)}$

### Chain Rule

$P(A,B) = P(A|B) P(B)$

### Marginalization

\begin{align} P(A) &= \sum\limits_{i=0}^N P(A,B_i) \\ &= \sum\limits_{i=0}^N P(A|B_i)P(B_i) \end{align}

In the continuous case we have

\begin{align} P(A) &= \int d\theta \ P(A,\theta) \\ &= \int d\theta \ P(A|\theta)P(\theta) \end{align}

### Bayes Theorem

$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$