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SquirrelInHell comments on Geometric Bayesian Update - Less Wrong Discussion
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Two-word proof: Prin'f gurberz. (Nebhaq gur gevnatyr, plpyvpnyyl, jr unir: rivqrapr, cevbe bqqf, erpvcebpny cbfgrevbe bqqf.)
I think this would be clearer with only the triangle where all the action is happening, and without the stuff on the left whose only job is to put the whole thing into a rectangle. You can still have the prior odds and the evidence on perpendicular axes: make it a right-angled triangle and let what's now the right-hand edge of the rectangle turn into the diagonal.
You are forgetting that it makes it possible to keep the scale of all numerical input/outputs consistent.
Point taken. (I personally prefer odds ratios strongly enough for this kind of thing that keeping the scale consistent doesn't bother me.) You could fix that, kinda, by fixing the side lengths of the "prior" and "posterior" side while allowing the length of the "evidence" side to vary, but that means introducing extra not-so-visible constraints so maybe it's a bit of a cheat.