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There's no reason why P(E|H) and P(E|~H) must sum to 1, but I can't move the lower right corner without the whole diagram rescaling.

Of course you are right, but it would just be a linear transformation of the whole diagram, so it doesn't change anything in the result. I've built the diagram starting from a square, so I can't change this easily... just imagine the whole thing scaling on the X axis, OK?

Edit: since two people asked for this, I remade the diagram and now you can put in any values of P(E|H) and P(E|~H)

When I drag the dot for P(E|~H), it only changes P(E|~H), but when I drag the dot for P(E|H), it still keeps P(E|H)+P(E|~H) conserved, which is a little weird. I think it would be better if changing either of them did not affect the other.

Agreed. The diagram strongly suggests that they do sum to one, so this geometrical method is more confusing than helpful.

[-]gjm8y00

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.

the stuff on the left whose only job is to put the whole thing into a rectangle.

You are forgetting that it makes it possible to keep the scale of all numerical input/outputs consistent.

[-]gjm8y00

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.

[-]ike8y00

Not a proof:

Gur boivbhf ohg oehgr sbepr jnl gb qb guvf jbhyq or svaqvat gur rdhngvba bs gur gjb yvarf sebz gur gjb evtug pbearef (obgu rnfl orpnhfr bs fvzvyne gevnatyrf naq/be xabja yratguf/pbbeqvangrf), svaq gur cbvag jurer gurl vagrefrpg, jevgr gur rdhngvba sbe gur yvar cnffvat guebhtu vg naq gur ybjre yrsg pbeare, fbyir sbe gung yvar uvggvat gur evtug yvar, gura fvzcyvsl. V unira'g qbar vg, ohg guvf fubhyq onfvpnyyl jbex qverpgyl.

Vf gurer fbzr oevyyvnag jnl bs fubjvat vg ol zrer fvzvynevgvrf? V qvqa'g frr bar vzzrqvngryl.

I've added a hint in the main post.