I mean distinguishing between hypotheses that give very similar predictions—like the difference between a coin coming up heads 50% vs. 51% of the time.
As I said in my other comment, I think the assumption that you have discrete hypotheses is what I was missing.
Though for any countable set of hypotheses, you can expand that set by prepending some finite number of deterministic outcomes for the first several actions. The limit of this expansion is still countable, and the set of hypotheses that assign probability 1 to your observations is the same at every time step. I’m confused in this case about (1) whether or not this set of hypotheses is discrete and (2) whether hypotheses with shorter deterministic prefixes assign enough probability to allow meaningful inference in this case anyway.
I may mostly be confused about more basic statistical inference things that don’t have to do with this setting.
Can you elaborate on what you meant by locally distinguishing between hypotheses?
I mean distinguishing between hypotheses that give very similar predictions—like the difference between a coin coming up heads 50% vs. 51% of the time.
As I said in my other comment, I think the assumption that you have discrete hypotheses is what I was missing.
Though for any countable set of hypotheses, you can expand that set by prepending some finite number of deterministic outcomes for the first several actions. The limit of this expansion is still countable, and the set of hypotheses that assign probability 1 to your observations is the same at every time step. I’m confused in this case about (1) whether or not this set of hypotheses is discrete and (2) whether hypotheses with shorter deterministic prefixes assign enough probability to allow meaningful inference in this case anyway.
I may mostly be confused about more basic statistical inference things that don’t have to do with this setting.