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What does it mean to describe a probability distribution over, say, ${\{0,1\}^\mathbb{N}}$ ? I am interested in this question because of the expert testing literature (pdf), in which an expert is supposed to provide a client with the distribution of some stochastic process, but the question is also relevant for Bayesiansts, at least those of us who think that the Bayesian paradigm captures (or should capture) the reasoning process of rational agents, in which case it makes little sense to reason about something you cannot describe.

So, what does it mean to describe your belief/theory about a stochastic process ${X_1,X_2,\dots}$ with values in ${\{0,1\}}$?  Here are some examples of descriptions:

${X_n}$ are I.I.D with ${P(X_n=0)=3/4}$.

${X_0=X_1=1}$ and ${\mathop{\mathbb P}(X_n=0|X_0,\dots,X_{n-1})=1/eX_{n-1}+0.4X_{n-2}}$

${X_n=Y_{2n}\cdot Y_{2n-1}}$ where ${Y_0,Y_1,\dots}$ are I.I.D ${(1/2,1/2)}$.

Everyone agrees that I have just described processes, and that the first and last are different descriptions of the same process. Also everyone probably agrees that there are only countably many processes I can describe, since every description is a sentence in English/Math and there are only countably many such sentences.