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Bayes Rule, Simpler Than You Think

Usually, people render Bayes Rule like this from Wikipedia:

Bayes Rule, stating that the probability of event A given event B is equal to the probability of B given A times the probability of A, divided by the probability of B

I think this formatting presumes that the most common usage of Bayes Rule is starting with all the numbers of the right, observing event B, and calculating a posterior probability.

Nice work if you can get it.

But I think a perhaps more useful formatting would be

P(A&B) = P(A|B)â‹…P(B) #

and by symmetry of &:

P(A|B)â‹…P(B) = P(A&B) = P(B&A) = P(B|A)â‹…P(A) #

or, if you're highlighting similarity to the original formula (which can be obtained by dividing by P(B)):

P(A&B) = P(A|B)â‹…P(B) = P(B|A)â‹…P(A) #

Which points out a much more interesting fact, I think, about conditional probabilities. You are equally surprised no matter the order you learn things, and there's a difference between 'and' and 'given', in that 'given' factors out the probability of the underlying circumstances.

The standard form, by baking in a specific mathematical manipulation, limits your ability to see the connection to conjunction probabilities. I think most people should be comfortable with a singular division, in terms of trying to isolate things algebraically.

Also, this is substantially easier to remember, I think. It's just so basic that you might genuinely never forget. It feels more like a definition of conditional probability.