In a clear and direct explanation David Robinson writes about bayesian A/B testing . His article is a must read for getting a good and concise introduction to the topic.

Understanding Bayesian A/B testing (using baseball statistics)

What if we want to compare two batters, give a probability that one is better than the other, and estimate by how much? This is a topic rather relevant to my own work and to the data science field, because understanding the difference between two proportions is important in A/B testing. One of the most common examples of A/B testing is comparing clickthrough rates (out of X impressions, there have been Y clicks)- which on the surface is similar to our batting average estimation problem (outof X at-bats, there have been Y hits).