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12_Bayes'_theorem.md

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Bayes' theorem

Describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

Bayes' inference

  • One of the many applications of Bayes' theorem is Bayesian inference.
  • Bayesian inference is fundamental to Bayesian statistics.

Statement of theorem

image

  • P ( A | B ) is a conditional probability: the probability of event A occurring given that B is true.
    posterior probability of A given B.
  • P ( B ∣ A ) is also a conditional probability: the probability of event B occurring given that A is true.
    the likelihood of A given a fixed B because P ( B ∣ A ) = L ( A ∣ B ).
  • P ( A ) and P ( B ) are the probabilities of observing A and B respectively without any given conditions;
    the marginal probability or prior probability.
  • A and B must be different events.
Probability vs. Likelihood
 - Probability is the percentage that a success occur. (probability of x (given θ))
 - Likelihood is the probability (conditional probability) of an event (a set of success) occur by knowing the probability of a success occur. (given that x was observed)
 * Probability attaches to possible results; likelihood attaches to hypotheses. 
 * Probability refers to the occurrence of future events, while a likelihood refers to past events with known outcomes.