Well, I think I've come up with something of an answer. I want a prior to be interpreted as a frequency estimate on possible worlds. This sounds funny, because we can't possibly estimate such a frequency: we only live in one world. But this is actually just fine: we shouldn't be estimating it, because it's our prior. It's what we use to estimate.

Anything more we learn, we learn using our prior. So can't improve upon our prior. If you've got a bad prior, tough luck.

A prior is an estimate of the frequency of alternative worlds. The perfect prior would contain all knowledge we ever needed; it would give our actual world-of-birth a probability of 1, and all other worlds, 0. But no two people are born to the same world, so evolution couldn't find this prior. (By the way, we could also view evolution as using a prior-- this prior is given to it by the very nature of chemistry and physics, and is not very good, but far better then it might have been.) So a slightly weaker and more useful notion of the perfect prior would be one that would do a fair job if all humans had it. Forcing all humans to have the same prior (which is close to being true) causes the perfect prior to have far more interesting structure (ie some learning occurs), although it still would have freakish foresight for things common to all humans (it would know the one true physics, for example).

Since what I'm interested in is learning, I want some way of ruling out this freakish foresight: I want to talk about a universal prior, one that will learn well no matter what the true physics turns out to be (and so on). I'm rejecting the Solomonoff prior because I think computability is too strict a requirement, but I also know that some restrictions are needed (otherwise there is no structure for the prior to take advantage of). What kind of a universal prior is this? And once I've figured that out, is it really of any use?

## Friday, February 15, 2008

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