@Steve: It's a good comparison, thanks. Usually, Kalman filters hold the process noise constant. The wikipedia page references “Autocovariance Least-Squares (ALS)” as a way of estimating it. I looked into those papers to try to make sense of how to do that in the simple one-dimensional case that I have. I only spent a bit of time looking at it, but it seemed like they are trying many hypothetical covariances out (or lagged copies of the input?) and then choosing the one that has the least “validation error”, which is similar to what I'm doing in my experiments.

I think you're right. I need to read about the Jeffreys prior. I don't understand why his 64-year-old paper "An Invariant Form for the Prior Probability in Estimation Problems" is not freely available.

...and we should get whatever pushforward probability measure would result given the mapping from the old sample space to the new one. (My wording might be off.)

Here's my thinking: If you know nothing about a coin, then we're told that the maximum entropy distribution over the coin's bias is Beta(1,1). So, your answer to the question, "what is the probability that the coin's bias is less than 20%?" is 20%. If someone asks, "what is the probability that the coin's bias is less than odds 1:4?", you should also answer 20%. This is true for every probability, and so it stands to reason that we should be allowed to choose whatever parametrization we want.

John, what do you mean cite sources for the confusion?

Right, thanks. So, what's the constraint in this case? Surely, our expected odds is 1, but Beta-prime(1,1) is not the same distribution as Exponential(1).

Nice, but you probably want to get rid of 1/n so that the result is as close to the smallest x_i as possible?