Independence and Conditional Independence

Math for CS · Axiom Academy

When events don't influence each other Two events are independent if the occurrence of one doesn't affect the probability of the other. Formally, A and B are independent if: In other words, knowing B tells us nothing about A. The conditional probability equals the unconditional probability. This multiplicative form is often easier to verify in practice. Events can be dependent overall but independent given some additional information. This is conditional independence : Conditional independence is the key insight behind Bayesian networks. Variables that are dependent overall can be treated independently given their common causes. Method 1: Check P(A ∩ B) = P(A) × P(B) Many randomized algorithms assume independence of random choices: Pairwise vs. Mutual Independence Events A, B, C are mutually independent if every pair is independent AND the triple is independent: A and B independent: P(A|B) = P(A) or P(A ∩ B) = P(A) × P(B) Independence means knowing one event tells us nothing about the other Conditional independence: independence given additional information Depends on the probability model; not just the events themselves Critical assumption in many algorithms and statistical models Axiom Academy | Probability for Computer Science

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