Sometimes you want to prove a theorem like “The algorithm works well.” You generally need randomness if you want to find algorithms that work without strong assumptions on the environment, whether or not there is really an adversary (who knows what kinds of correlations exist in the environment, whether or not you call them an “adversary”).
A bayesian might not like this, because they’d prefer prove theorems like “The algorithm works well on average for a random environment drawn from the prior the agents use,” for which randomness is never useful.
But specifying the true prior is generally hideously intractable. So a slightly more wise Bayesian might want to prove statements like “The algorithm well on average for a random environment drawn from the real prior” where the “real prior” is some object that we can talk about but have no explicit access to. And now the wiser Bayesian is back to needing randomness.
A bayesian might not like this, because they’d prefer prove theorems like “The algorithm works well on average for a random environment drawn from the prior the agents use,” for which randomness is never useful.
It seems like a bayesian can conclude that randomness is useful, if their prior puts significant weight on “the environment happens to contain something that iterates over my decision algorithm and returns its worst-case input, or something that’s equivalent to or approximates this” (which they should, especially after updating on their own existence). I guess right now we don’t know how to handle this in a naturalistic way (e.g., let both intentional and accidental adversaries fall out of some simplicity prior) and so are forced to explicitly assume the existence of adversaries (as in game theory and this post).
Sometimes you want to prove a theorem like “The algorithm works well.” You generally need randomness if you want to find algorithms that work without strong assumptions on the environment, whether or not there is really an adversary (who knows what kinds of correlations exist in the environment, whether or not you call them an “adversary”).
A bayesian might not like this, because they’d prefer prove theorems like “The algorithm works well on average for a random environment drawn from the prior the agents use,” for which randomness is never useful.
But specifying the true prior is generally hideously intractable. So a slightly more wise Bayesian might want to prove statements like “The algorithm well on average for a random environment drawn from the real prior” where the “real prior” is some object that we can talk about but have no explicit access to. And now the wiser Bayesian is back to needing randomness.
It seems like a bayesian can conclude that randomness is useful, if their prior puts significant weight on “the environment happens to contain something that iterates over my decision algorithm and returns its worst-case input, or something that’s equivalent to or approximates this” (which they should, especially after updating on their own existence). I guess right now we don’t know how to handle this in a naturalistic way (e.g., let both intentional and accidental adversaries fall out of some simplicity prior) and so are forced to explicitly assume the existence of adversaries (as in game theory and this post).