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?
 A: 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.
A: For the case where you want to use the restriction $p_{1}, \geq 1/2$ your prior will be of the following form
$$\pi(p_{1})=\left\{\begin{matrix}
Beta(a,b), & p_{1}\geq 1/2 \\ 
0, & otherwise
\end{matrix}\right. = Trunc-Beta(a,b)$$
Hence, your posterior can be derived as
$$\pi(p_{1}|x) \propto L(p;x)*Beta(p_{1};a,b)*\mathbb{I}(p_{1}\geq 1/2) = L(p;x)*Trunc-Beta(a,b)$$
The truncated Beta distribution can be calculated in the following way restricted to the interval $[1/2,1]$
$$Trunc-Beta(a,b) = \frac{Beta(a,b)}{F(1)-F(1/2)}$$
for exact calculations you can check here https://en.wikipedia.org/wiki/Beta_distribution.
Essentially what will happen to your posterior by adding this extra information $p_{1}\geq 1/2$, is that even though your likelihood might give importance to values of $p_{1}$ less than $1/2$ your prior will make them zero, because it gives zero mass to those values less than $1/2$.
In the case where you have $p_{1}\geq p_{2}$, then your prior will be of the form
$$\pi(p_{1},p_{2}) = \left\{\begin{matrix}
f(p_{1},p_{2}), & p_{1}\geq p_{2} \\ 
 0,& otherwise
\end{matrix}\right.$$
and the posterior
$$\pi(p_{1},p_{2}|x)\propto L_{1}(p_{1};x_{1})*L_{2}(p_{2};x_{2})*f(p_{1},p_{2})*\mathbb{I}(p_{1}>p_{2})$$
similarly, the product of the likelihoods $L_{1}(p_{1};x_{1})*L_{2}(p_{2};x_{2})$ might give importance to cases where $p_{1}<p_{2}$ but your prior will give again zero mass to those choices.
