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electives/amr/parts/00_basics/00_probability.tex
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\subsection{Probability}
\shortdefinition[Sum rule] $P(X) = \sum P(X, Y) = \sum P(X \cap Y)$
\shortdefinition[Sum rule] $\P(X) = \sum \P(X, Y) = \sum \P(X \cap Y)$
\shortdefinition[Prod] $P(X, Y) = P(X | Y) P(Y) = P(Y | X) P(X)$
\shortdefinition[Prod] $\P(X \cap Y) = \P(X | Y) \P(Y) = \P(Y | X) P(X)$
\shorttheorem[Bayes] $\displaystyle P(Y_i | X) = \frac{P(X | Y_i) P(Y_i)}{\sum_{j = 1}^n P(X | Y_j) P(Y_j)}$
\shorttheorem[Bayes] $\displaystyle \P(Y_i | X) = \frac{\P(X | Y_i) \P(Y_i)}{\sum_{j = 1}^n \P(X | Y_j) \P(Y_j)}$
\shortdefinition[Cont. Var] Sums become integrals\\
e.g. $\sum_{X} P(X) = 1$ becomes $\int p(x) \dx = 1$
e.g. $\sum_{X} \P(X) = 1$ becomes $\int \P(x) \dx = 1$
\shortdefinition[Indep.] $x, y$ indep. iff $p(x, y) = p(x) p(y)$
\shortdefinition[Indep.] $x, y$ indep. iff $\P(\cX \cap \cY) = \P(\cX) \P(\cY)$
\shortdefinition[Cond. Indep.] iff $p(x, y | z) = p(x|z) p(y|z)$
\shortdefinition[Cond. Indep.] iff $\P(\cX \cap \cY | \cZ) = \P(\cX | \cY) \P(\cY | \cZ)$
\shortdefinition $E[\vec{x}] = \int_{-\8}^{\8} \vec{x} p(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$
\shortdefinition $\E[\vec{x}] = \int_{-\8}^{\8} \vec{x} \P(\vec{x}) \dx \vec{x}$, also for $\vec{x} = \vec{f(x)}$
\shortdefinition $\text{Cov}[x] = E[\vec{x} \vec{x}^\top] - E[\vec{x}]E[\vec{x}]^\top = \mat{\Sigma}$
\shortdefinition $\text{Cov}[x] = \E[\vec{x} \vec{x}^\top] - \E[\vec{x}]\E[\vec{x}]^\top = \mat{\Sigma}$
\shortdefinition[Gauss. Dist.] $\vec{x} \sim \cN(\vec{\mu}, \mat{\Sigma})$ ($\vec{\mu}$ mean, $\mat{\Sigma}$ cov.),\\
PDF: $p(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k |\mat{\Sigma}|}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
PDF: $f(\vec{x}) = \frac{1}{\sqrt{(2\pi)^k \det(\mat{\Sigma})}} \text{exp}\left( -\frac{1}{2}(\vec{x} - \vec{\mu})^\top \mat{\Sigma}^{-1} (\vec{x} - \vec{\mu}) \right)$
$\Sigma^{-1}$ for $\Sigma$ diagonal, inverse of diag els (e.g. $\sigma^{-1}$)