There are a lot of explanations of Bayes' Theorem, so I won't get into the technicalities. I will get into why it should change how you think. This post is pretty introductory, so free to totally skip it if you don't feel like there's anything about Bayes' Theorem that you don't understand.

For a while I was reading LessWrong and not seeing what the big deal about Bayes' Theorem was. Sure, probability is in the mind and all, but I didn't see why it was so important to insist on bayesian methods. For me they were a tool, rather than a way of thinking. This summary also helped someone in the DC group.

After using the Anki deck, a thought occurred to me:

Bayes theorem means that when seeing how likely a hypothesis is after an event, not only do I need to think about how likely the hypothesis said the event is, I need to consider everything else that could have possibly made that event more likely.

To illustrate:

pretty clearly shows how you need to consider P(e|H), but that's slightly more obvious than the rest of it.

If you write it out the way that you would compute it you get...

where h is an element of the hypothesis space.

This means that every way that e could have happened is important, on top of (or should I say under?) just how much probability the hypothesis assigned to e.

This is because P(e) comes from every hypothesis that contributes to e happening, or more mathilyeX P(e) is the sum over all possible hypotheses of the probability of the event and that hypothesis, computed by the probability of the hypothesis times the probability of the event given the hypothesis.

In LaTeX:

where h is an element of the hypothesis space.