Bayesian inference from "extra" information - Beta-binomial case Say we have two coins with unknown success probabilities $p_1$ and $p_2$. To know more about the probabilities, say that we use Bayesian approach.
To do so, we first set our prior: $P_1\sim Beta(1,1)$ and $P_2\sim Beta(1,1)$.
Tossing both of the coins together, we update each of the prior as usual.
For example, we ran 10 rounds of tossing, 4 Heads on coin 1 and 8 Heads on coin 2.
The posterior should be $P_1\sim Beta(5,7)$ and $P_2\sim Beta(9,3)$.
My question is that what happens if we receive an "extra" piece of information that says "$p_1$ is greater than $p_2$"
Is there any systematic way to accommodate this extra piece of information into the posterior?

What you're referring to in the first part of the question, is the beta-binomial model. where binomial distribution is assumed as the likelihood and beta as a prior, hence by conjugacy posterior is also a beta distribution.
Your problem description in the second part describes a different scenario because it is multivariate. If you know that $p_1 > p_2$, this means that the parameters are dependent and you are talking about some multivariate distribution for the parameters (vs two univariate beta distributions). In such a case, you cannot use two (independent) beta-binomial models. The constraint can be imposed by choosing a multivariate prior for the parameters. For such a model you won't have a closed-form solution, so you would need to use MCMC or some other kind of approximate inference.
