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.