Self-reference and self-reproduction of evidence

Continuing our election-eve counterprogramming, here’s another post with no political content.

It comes from Constantine Frangakis, who writes:

I think I have found something new and interesting.

In studying the topic of “evidence” for my class, where the typical principles are sufficiency and conditionally (for which some people are reluctant to accept), I considered instead asking for this: that an Evidence function should be able to self-reference and to self-reproduce; in other words, if Ev(. ) captures all features of an observation for distinguishing between two hypotheses, then when it is fed to itself, the result should not change:

       Evidence (Evidence (.)) must equal Evidence (.)      (1)

I have found there with this criterion alone, the dependence on the observation is unique and it is the likelihood ratio (for a proof, see this draft article, Self-reference and Self-reproduction of Evidence, coauthored with Michael Evans). This bypasses the debates on sufficiency and conditionality.

The power of (1) is that it can expose our limits, which include faulty assumptions. For example, if somewhere in a problem we use a model, and this produces an Evidence or other self-reproducing function, Ev*, then the checking of Ev*(Ev*()) = Ev*() can help correct our model.

I’ve never thought about such things before! It reminds me a bit of the martingale property of probabilistic forecasts.