Bayesian Sports Models in R

Author

Andrew Mack

Bayesian reasoning

  • Learn from experience and make decisions in face of uncertainty

  • Three elements: prior belief, new evidence, updated belief

Basic probability concepts

  • P (A): probability of event A occuring

  • P (Ac): complement, probability of event A not occurring

  • P(A ∩ B): intersection, probability of event A and event B occurring

  • P (A u B): union, probability of event A or event B occurring

  • P (A | B): conditional, probability event A occurs given event B

Probability rules

  • Non-negative rule: probability of any event is a non-negative number

  • Certainty rule: probability of a certain event is 1

  • Complement rule: probability of of an event not occurring is 1 minus the probability of the event occurring

  • Addition rule: probability of A or B occurring is the sum of their individual probabilities minus the probability of of their intersection

  • Conditional probability: probability of an event occurring given another events has already occurred

  • Multiplication rule: probability of events A and B occurring, multiply two independent events to get joint probability

  • Total probability rule: probability of an event that can be broken down into several mutually exclusive events

Bayes rule

  • P(B|A) = [P(A|B) * P(A)] / P(B)

  • Updated belief or posterior probability: P(A|B)

  • New evidence or likelihood: P(B|A)

  • Prior belief or prior: P(B)

  • Marginal likelihood: P(A)

Frequentist and Bayesian approaches

  • Frequentist: long-run frequency, no priors, confidence intervals

  • Bayesian: degree of belief, use priors, posterior distributions

Probability distributions

  • Discrete distribution: