intro

Probability density function

A function $p : R \to R$ is a probability density function (pdf) for X if it is a nonnegative integrable function s.t.

\int_{V a l (X)} p (x) d x = 1.

the Gaussian (Normal) distribution

N (μ, σ^{2}) = \frac{1}{\sqrt{2 π} σ} e^{\frac{- (x - μ)^{2}}{2 σ^{2}}}

Conditional probability

With respect to prior probabilities P(X), conditional or posterior probabilities P(X|Evidence) represent a more informed distribution in the light of the (new) Evidence.

Definition of conditional probability: $P (a | b) = \frac{P (a \land b)}{P (b)}$ if $P (b) \neq 0$

Product rule gives an alternative formulation:

P (a \land b) = P (a | b) P (b) = P (b | a) P (a)

Probability query

A probability query P(Y|e) defines the posterior joint distribution of a set of query variables Y given specific values e for some evidence variables.

In principle, one could answer the query by summing out.

P (Y | E = e) = α P (Y, E = e) = α \sum_{h} P (Y, E = e, H = h)