Tuesday, July 15, 2014

Mathematics of Independent Corroboration

I just worked out the mathematics of independent corroboration.

It is startlingly powerful, which makes it startlingly important to double and triple-check that the independence is truly independent.

Considering how easy this theory is to work out, I should have seen it already, but I haven't. I did a quick check to see if I could find some prior art, but as predicted by my vague impression, couldn't find any. Best is this, which has no apparent math and seems generally weak.

While it's obvious to anyone who has formally studied physics that it's incredibly unlikely that two people get the same answer from different mistakes, it's nice to have an actual function.

If you have a theory and someone corroborates your theory with no prior contact, we have a nice truth matrix:





You're both right, only you're right, only they're right, and you're both wrong.

Nail this to something concrete and assume there's a 90% chance either of you are wrong independently. Taken separately, then the relative odds are 1%, 9%, 9%, and 81%.

However, the odds of 10 and 01 are incredibly low if the theories corroborate. (If they're not logically independent the way they were temporally independent. Say you theorize that fire makes grates hot, and they find a hot grate that had fire in it.)

You have some privileged information - you did not defraud the investigation. Therefore the odds of 10 are the odds that your epistemic misconduct just happened to precisely predict what they would see anyway. (If you did defraud, you could have targeted the fraud at their experiment to make your theory look good.)

The odds of 01 are also negligible. Out of all the mistakes and frauds they could possibly commit, why the one that makes you look good in particular? Especially as, if it's truly independent, they had or still have no knowledge of your theory?

Since probability is conserved, we have some free-floating percentage. Only, the odds that you're wrong haven't gone up, so 00 is still 81%. It all settles on 11, increasing it a whopping relative 1800%.

Thus we have some simple math.

Given an independent corroboration, the odds you're right is simply one minus the odds you're wrong times the odds they're wrong. f_knowlege(U,E) = 1 - (U*E). This function is linear in the odds you're wrong, which is entirely under your control. Having thought of epistemic misconduct, you can thoroughly ream your stuff for it until it is negligible, and thus be 100% certain. (To any reasonable number of significant figures, 99.99etc rounds to 100. Just don't let that make you forget how to learn.)

That is, linear in your error odds, but scaled up by the general climate of not-error. (E is constant as far as your force of will is concerned.)

A third independent corroboration basically guarantees you're right. It is 000 vs. 111 because 010 110 etc. all fall out. At 90% error, 000 is 73%. Still linear, and the up-scaling is even more aggressive. At 50% average error, it's 88% likely you're right even without special effort.

No comments: