01
Why probability matters
Real environments rarely provide complete or perfectly reliable information. Sensors are noisy, observations are incomplete and future events are uncertain. Probability provides a formal framework for representing degrees of belief rather than forcing every proposition into a binary true-or-false state.
02
Bayes' rule
P(H|E) = P(E|H)P(H) / P(E)
Bayes' rule describes how evidence changes belief in a hypothesis. The prior expresses what was believed before the evidence, the likelihood describes how compatible the evidence is with the hypothesis, and the posterior is the updated belief.
03
Bayesian networks
A Bayesian network is a directed acyclic graph whose nodes represent variables and whose edges encode conditional relationships. The graph can compactly represent a joint probability distribution by exploiting conditional independence.
P(X₁,...,Xₙ) = ∏ᵢ P(Xᵢ | Parents(Xᵢ))
04
Inference
- Variable elimination computes exact marginals by systematically eliminating hidden variables.
- Junction-tree methods exploit graph structure to perform exact inference.
- Monte Carlo methods approximate distributions through sampling.
- Variational methods replace difficult distributions with tractable approximations.
05
Decision theory
Probability describes beliefs; decision theory adds preferences. An agent can choose the action that maximises expected utility over possible outcomes.
EU(a) = Σₛ P(s|a)U(s)