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.

The formulation of `learning to make predictions’ goes through merging, which traces back to Blackwell and Dubins. I will not give Blackwell and Dubin’s definition in this post but a weaker definition, suggested by Kalai and Lehrer.

A Bayesian agent observes an infinite sequence of outcomes from a finite set ${A}$. Let ${\mu\in\Delta(A^\mathbb{N})}$ represent the agent’s belief about the future outcomes. Suppose that before observing every day’s outcome the agent makes a probabilistic prediction about it. I denote by ${\mu(\cdot|a_0,\dots,a_{n-1})}$ the element in ${\Delta(A)}$ which represents the agent’s prediction about the outcome of day ${n}$ just after he observed the outcomes ${a_0,\dots,a_{n-1}}$ of previous days. In the following definition it is instructive to think about ${\tilde\mu}$ as the true data generating process, i.e., the process that generates the sequence of outcomes, which may be different from the agent’s belief.

Definition 1 (Kalai and Lehrer) Let ${\mu,\tilde\mu\in\Delta(A^\mathbb{N})}$. Then ${\mu}$ merges with ${\tilde\mu}$ if for ${\tilde\mu}$-almost every realization ${(a_0,\dots,a_{n-1},\dots)}$ it holds that

$\displaystyle \lim_{n\rightarrow\infty}\|\mu(\cdot|a_0,\dots,a_{n-1})-\tilde\mu(\cdot|a_0,\dots,a_{n-1})\|=0.$

Assume now that the agent’s belief ${\mu}$ is stationary, and let ${\mu=\int \theta~\lambda(\mathrm{d}\theta)}$ be its ergodic decomposition. Recall that in this decomposition ${\theta}$ ranges over ergodic beliefs and ${\lambda}$ represents structural uncertainty. Does the agent learn to make predictions ? Using the definition of merging we can ask, does ${\mu}$ merges with ${\theta}$ ? The answer, perhaps surprisingly, is no. I gave an example in my previous post.

Let me now move to a weaker definition of merging, that was first suggested by Lehrer and Smorodinsky. This definition requires the agent to make correct predictions in almost every period.

Definition 2 Let ${\mu,\tilde\mu\in\Delta(A^\mathbb{N})}$. Then ${\mu}$ weakly merges with ${\tilde\mu}$ if ${\tilde\mu}$-almost every realization ${(a_0,\dots,a_{n-1},\dots)}$ it holds that

$\displaystyle \lim_{n\rightarrow\infty,n\in T}\|\mu(\cdot|a_0,\dots,a_{n-1})-\tilde\mu(\cdot|a_0,\dots,a_{n-1})\|=0$

for a set ${T\subseteq \mathbb{N}}$ of periods of density ${1}$.

The definition of weak merging is natural: patient agents whose belief weakly merges with the true data generating process will make almost optimal decisions. Kalai, Lehrer and Smorodinsky discuss these notions of mergings and also their relationship with Dawid’s idea of calibration.

I am now in a position to state the theorem I have been talking about for two months:

Theorem 3 Let ${\mu\in\Delta(A^\mathbb{N})}$ be stationary, and let ${\mu=\int \theta~\lambda(\mathrm{d}\theta)}$ be its ergodic decomposition. Then ${\mu}$ weakly merges with ${\theta}$ for ${\lambda}$-almost every ${\theta}$.

In words: An agent who has some structural uncertainty about the data generating process will learn to make predictions in most periods as if he knew the data generating process.

Finally, here are the promised open questions. They deal with the two qualification in the theorem. The first question is about the “${\lambda}$-almost every ${\theta}$” in the theorem. As Larry Wasserman mentioned this is unsatisfactory in some senses. So,

Question 1 Does there exists a stationary ${\mu}$ (equivalently a belief ${\lambda}$ over ergodic beliefs) such that ${\mu}$ weakly merges with ${\theta}$ for every ergodic distribution ${\theta}$ ?

The second question is about strengthening weak merging to merging. We already know that this cannot be done for arbitrary belief ${\lambda}$ over ergodic processes, but what if ${\lambda}$ is concentrated on some natural family of processes, for example hidden markov processes with a bounded number of hidden states ? Here is the simplest setup for which I don’t know the answer.

Question 2 The outcome of the stock market at every day is either U or D (up or down). An agent believes that this outcome is a stochastic function of an unobserved (hidden) state of the economy which can be either G or B (good or bad): When the hidden state is B the outcome is U with probability ${q_B}$ (and D with probability ${1-q_B}$), and when the state is G the outcome is U with probability ${q_G}$. The hidden state changes according to a markov process with transition probability ${\rho(B|B)=1-\rho(G|B)=p_B}$, ${\rho(B|G)=1-\rho(G|G)=p_G}$. The parameter is ${(p_B,p_G,q_B,q_G)}$ and the agent has some prior ${\lambda}$ over the parameter. Does the agent’s belief about outcomes merge with the truth for ${\lambda}$-almost every ${(p_B,p_G,q_B,q_G)}$ ?.