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Comment by mtklein | Hacker News Reader

On the topic of finding Bayes' theorem intuitively, I can never remember it on its own, but starting from the joint probability P(A,B) for any arbitrary A and B always helps me:

    // A and B are arbitrary.
    P(A,B) = P(B,A)

    // These two make sense if I sound them out in words.
    P(A,B) = P(A) P(B|A)
    P(B,A) = P(B) P(A|B)

    // Combine to give Bayes' theorem.
    P(A) P(B|A) = P(B) P(A|B)

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The advantage of the odds formulation is that you can compute posterior odds in your head, whereas computing the posterior probabilities via the standard form of the theorem (that you wrote down here) involves an unpleasant normalization factor.

E.g. for the example in https://en.wikipedia.org/wiki/Base_rate_fallacy#Low-incidenc...:

    prior odds of infection =   2 :  98
  × likelihood ratio        = 100 :   5
  = posterior odds          ≈ 200 : 500

and thus the probabilities (if you actually need them at this point) are ≈ (2/7, 5/7).

See also 3Blue1Brown’s exposition: https://youtu.be/watch?v=lG4VkPoG3ko.

Exactly! The mathematical aspect of Bayes' Theorem is a simple algebraic manipulation of conditional probabilities.

Another way to intuitively justify your middle two equations is to visualise a Venn diagram with overlapping circles representing A and B, such that areas correspond to probabilities. Then P(B|A) is the area of the overlap - i.e. P(A,B) - divided by the area of A - i.e. P(A).

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