I think Laplace's Rule of succession [1] could be better here. It assumes there are binary "successes" and "failures" (e.g. thumbs up/down). Let s be the number of "successes", n be the total number of data points (successes+failures), and 1/x the prior probability of a success. Then the probability that the next data point will be a success is:
(s + 1)/(n + x)
E.g. for prior probability 1/2 (success and failure initially equally likely), x=2, so
(s + 1)/(n + x)
E.g. for prior probability 1/2 (success and failure initially equally likely), x=2, so
(s + 1)/(n + 2)
[1] https://en.wikipedia.org/wiki/Rule_of_succession