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This post describes the main theorem in my new paper with Nabil. Scroll down for open questions following this theorem. The theorem asserts that a Bayesian agent in a stationary environment will learn to make predictions as if he knew the data generating process, so that the as time goes by structural uncertainty dissipates. The standard example is when the sequence of outcomes is i.i.d. with an unknown parameter. As times goes by the agent learns the parameter.
In the lasts posts I talked about a Bayesian agent in a stationary environment. The flagship example was tossing a coin with uncertainty about the parameter. As time goes by, he learns the parameter. I hinted about the distinction between `learning the parameter’, and `learning to make predictions about the future as if you knew the parameter’. The former seems to imply the latter almost by definition, but this is not so.
Because of its simplicity, the i.i.d. example is in fact somewhat misleading for my purposes in this post. If you toss a coin then your belief about the parameter of the coin determines your belief about the outcome tomorrow: if at some point your belief about the parameter is given by some then your prediction about the outcome tomorrow will be the expectation of . But in a more general stationary environment, your prediction about the outcome tomorrow depends on your current belief about the parameter and also on what you have seen in the past. For example, if the process is Markov with an unknown transition matrix then to make a probabilistic prediction about the outcome tomorrow you first form a belief about the transition matrix and then uses it to predict the outcome tomorrow given the outcome today. The hidden markov case is even more complicated, and it gives rise to the distinction between the two notions of learning.
The formulation of the idea of `learning to make predictions’ goes through merging. The definition traces back at least to Blackwell and Dubins. It was popularized in game theory by the Ehuds, who used Blackwell and Dubins’ theorem to prove that rational players will end up playing approximate Nash Equilibrium. In this post I will not explicitly define merging. My goal is to give an example for the `weird’ things that can happen when one moves from the i.i.d. case to an arbitrary stationary environment. Even if you didn’t follow my previous posts, I hope the following example will be intriguing for its own sake.