Uniform bounds on rate of merging for Bayesian learners Update. Cross posted at Cross Validated.

In a well-known paper, Blackwell &amp; 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 &amp; 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 &amp; Levine, 1992; Sorin, 1999). I am wondering if there are any results in this direction for strong merging.
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.