Bayes’ Theorem (also known as Bayes’ Law) is a law of probability that describes the proper way to incorporate new evidence into prior probabilities to form an updated probability estimate. It is commonly regarded as the foundation of consistent rational reasoning under uncertainty. Bayes Theorem is named after Reverend Thomas Bayes who proved the theorem in 1763.
Bayes’ theorem commonly takes the form:
where A is the proposition of interest, B is the observed evidence, P(A) and P(B) are prior probabilities, and P(A|B) is the posterior probability of A.
With the posterior odds, the prior odds and the likelihood ratio written explicitly, the theorem reads:
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From the old wiki discussion page:
I’m thinking we can leave most of the discussion of probability to Wikipedia. There might be more to say about Bayes as it applies to rationality but that might be best shoved in a separate article, like Bayesian or something. Also, I couldn’t actually find any OB or LW articles directly about Bayes’ theorem, as opposed to Bayesian rationality—if anyone can think of one, please add it. --A soulless automaton 19:31, 10 April 2009 (UTC)
I’d rather go for one article than break out a separate one for Bayesian—we can start splitting things out if the articles start to grow too long. --Paul Crowley (ciphergoth) 22:58, 10 April 2009 (UTC)
I added what I thought was the minimal technical information. I lean towards keeping separate concepts separate, even if the articles are sparse. If someone else feels it would be worthwhile to combine them though, go ahead. --BJR 23:07, 10 April 2009 (UTC)
I really would prefer to keep the maths and statistics separate from the more nebulous day-to-day rationality stuff, especially since Wikipedia already does an excellent job of covering the former, while the latter is much more OB/LW-specific. --A soulless automaton 21:59, 11 April 2009 (UTC)
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Thanks, Tofly! Flawless job with the edits.