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Oscar_Cunningham comments on Open Thread, August 2010 - Less Wrong
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I thought I'd pose an informal poll, possibly to become a top-level, in preparation for my article about How to Explain.
The question: on all the topics you consider yourself an "expert" or "very knowledgeable about", do you believe you understand them at least at Level 2? That is, do you believe you are aware of the inferential connections between your expertise and layperson-level knowledge?
Or, to put it another way, do you think that, given enough time, but using only your present knowledge, you could teach a reasonably-intelligent layperson, one-on-one, to understand complex topics in your expertise, teaching them every intermediate topic necessary for grounding the hardest level?
Edit: Per DanArmak's query, anything you can re-derive or infer from your present knowledge counts as part of your present knowledge for purposes of answering this question.
I'll save my answer for later -- though I suspect many of you already know it!
I have a (I suspect unusual) tendency to look at basic concepts and try to see them in as many ways as possible. For example, here are seven equations, all of which could be referred to as Bayes' Theorem:
However, each one is different, and forces a different intuitive understanding of Bayes' Theorem. The fourth one down is my favourite, as it makes obvious that the update depends only on the ratio of likelihoods. It also gives us our motivation for taking odds, since this clears up the 1/(1+x)ness of the equation.
Because of this way of understanding things, I find explanations easy, because if one method isn't working, another one will.
ETA: I'd love to see more versions of Bayes' Theorem, if anyone has any more to post.
Very well said, and doubles as a reply to the last part of my comment here. (When I read your comment in my inbox, I thought it was actually a reply to that one! Needless to say, I my favorite versions of the theorem are the last two you listed.)
P (H|E) = P (H and E) / P(E)
which tends to be how conditional probability is defined, and actually the first version of Bayes that I recall seeing.