# Distribution of oddsratio after bayesian inference under binomial model

Let us have 2 groups - treatment group (1) and control group(2). Each group has survival probability $p_1$ and $p_2$ respectively. Ofc each patient survives or dies independently given $p_i$ of his group. Let us define $y_i$ as number of survivors in group i, $n_i - y_i$ - number of deceased in group i.

So, I am setting up bayesian model for this:

$p(p_1, p_2| Y) \propto p_1^{y_1}(1-p_1)^{n_1 - y_1} * p_2^{y_2}(1-p_2)^{n_2 - y_2} * 1$

1 comes from $Beta(1, 1)*Beta(1, 1)$ which I assume as prior for $p(p_1, p_2)$.

As I understand the posterior will be a product of 2 Betas?

Now, how can I summarize the posterior distribution for the odds ratio, $\frac{(p_2/(1 − p_2))}{(p_1/(1 − p_1))}$?

Also is the choice of noninformative prior correct in this case? Or is there a way to claim that this or that particular prior is better than others?