Bayes prior and posterior distribution Let's assume we have prior distribution beta with parameters 2,50. Let's just say it's prior knowledge of sign up rates for our product.
Then we have two binomial models A and B, which both samples from the same prior. The population is 100 and the target for A is 2, while the target for B is 20.
Now, if you check posterior distribution in your preferred tool, you can see model B has much better rates. The distribution curve of model B is shifted much more to the right of the X-axis than for model A. Does this mean there are much bigger chances for 20 people to sign up for our product than 2??? This seems very unlogical, considering our prior distribution. How should I interpret this? Thank you.
 A: To be clear - your prior assumes that you’ve observed ($2-1 = 1$) successes and ($50-1 = 49$) failures for an effective sample size of $N_1=50$. The posterior does not sample from the prior but rather combines the likelihood with the prior to arrive at the posterior distribution over $\theta$ (conversion rate in your case). Since the posterior mean is as a weighted average between the prior and observed means, the corresponding means of model A and model B are:
$\hat{\theta}_A=\frac{N_1}{(N_1+N_2)}\cdot\frac{\alpha}{\alpha+\beta} + \frac{N_2}{(N_1+N_2)}\cdot\frac{\alpha+n_A}{\alpha+n_A+\beta+(100-n_A)}$ 
$\hat{\theta}_B=\frac{N_1}{(N_1+N_2)}\cdot\frac{\alpha}{\alpha+\beta} + \frac{N_2}{(N_1+N_2)}\cdot\frac{\alpha+n_B}{\alpha+n_B+\beta+(100-n_B)}$ 
Where $N_1$ is the effective sample size of the prior, $N_2$ is the number of new observations, $n_A$ is the count of successes observed for model A, and $n_B$ is the count of successes observed for model B
As you can see, your results are not illogical - you are observing that the posterior is overwhelming the prior in the case of model B (double the data, 7x the conversion / success rate):
$\hat{\theta}_B=\frac{50}{150}\cdot\frac{1}{50} + \frac{100}{150}\cdot\frac{21}{150}$ 
