# Uniform bounds on rate of merging for Bayesian learners

Update. Cross posted at Cross Validated.

In a well-known paper, Blackwell & Dubins (1962) show that the posterior probabilities of two Bayesian agents, whose priors agree on events of measure $0$, will become arbitrarily close to each other under an increasing stream of information.

Mathematically, the result is as follows. Let $(\Omega, \mathcal{F}, \{\mathcal{F}_n\}, Q)$ be a filtered probability space with $\mathcal{F}_n \uparrow \mathcal{F}$. Let $P$ be a probability on $(\Omega, \mathcal{F})$ with $Q \ll P$. Then, $$d(P^n, Q^n): = \sup_{A \in \mathcal{F}}|P(A \mid \mathcal{F}_n) - Q(A \mid \mathcal{F}_n)| \to 0 \text{ a.s. Q as } n \to \infty.$$ We say that $P$ and $Q$ strongly merge.

In a more recent and also very influential paper, Kalai & Lehrer (1994) introduce the notion of weak merging. The definition is as above, except the $\sup$ is taken over finite horizon events; tail events are ignored: $$w(P^n, Q^n) : = \sup_{A \in \mathcal{F}_{n+1}}|P(A \mid \mathcal{F}_n) - Q(A \mid \mathcal{F}_n)| \to 0 \text{ a.s. Q as } n \to \infty.$$

For weak merging it is possible to find uniform bounds on the rate of convergence (Fudenberg & Levine, 1992; Sorin, 1999). I am wondering if there are any results in this direction for strong merging.

• This should be moved over to Cross Validated or Mathematics. It is more likely that people on those boards would be aware of specific papers on sequences of functions converging to a limiting function. I am very interested in the answer though as this is related to a question I am working on. I am aware of none. – Dave Harris Feb 19 '17 at 21:07
• @DaveHarris Unfortunately, the folks at MSE don't seem to be too familiar with this literature. I've asked questions about Blackwell & Dubins before. Are you sure the question shouldn't be left here? Weak merging is discussed extensively in economics journals by economists. Although, I agree of course that the subject might be a bit more technical than the average question posted here. – user12162 Feb 19 '17 at 21:10
• I don't know. Its a valid question here, if a bit esoteric for this group. There is a narrow audience for this. In part, its because there are strong, implicit assumptions about information, preferences and incentives, as well as the life of a game. We have an arbitrarily large sample on both evolution and the roundness of the earth, yet both Ken Ham and the flat earth Cavalier were in the news this week. Infinity is a long time. – Dave Harris Feb 20 '17 at 1:09
• Indeed it is a long time. And that's precisely why I want to better understand the rate of merging. Anyway, I think your suggestion to post at Cross Validated is a good one, and I've done it. I suspect this is an open problem, though hopefully some leads will emerge. – user12162 Feb 20 '17 at 2:37

## 1 Answer

This paper by Acemoglu, Chernozhukov and Yildiz (2016) and the references therein may be of interest.

The results they derive are in a much more limited environment, but I think they still gesture toward the direction you're looking. Otherwise, their literature review should also prove useful.

• Apologies for the brief answer -- this topic is a bit far afield for me. However, I suspect it should still be somewhat helpful. – Theoretical Economist Feb 20 '17 at 3:04
• Thanks for this. I'll try to read it within the next few days and report on any relevant results. – user12162 Feb 20 '17 at 3:08
• Great; do let me know. I'm curious as well. And I may have spoken too soon regarding how limited their results are -- a bit more skimming suggests it's closer to Blackwell and Dubins' formulation than I initially thought. – Theoretical Economist Feb 20 '17 at 3:16
• Having looked at the model, but not all of the results, it seems they're interested in a somewhat different phenomenon, which they explain informally on p.193. Still, the paper seems interesting and I'll probably continue reading. – user12162 Feb 20 '17 at 14:38