You are talking about a shrinkage estimator. Imdb is possibly the most famous example of this, how they calculate which movies will make it onto the top 250. It relies on the equation,
weighted rating (WR) = (v ÷ (v+m)) × R + (m ÷ (v+m)) × C , where:
* R = average for the movie (mean) = (Rating)
* v = number of votes for the movie = (votes)
* m = minimum votes required to be listed in the Top 250 (currently 3000)
* C = the mean vote across the whole report (currently 6.9)
They call this a "true bayesian rating" and that's true in the sense that our prior for the parameter "average rating" is that it is the same as for all other movies. This prior is then updated based on the "likelihood," which is the average rating for that movie, which has more strength if it has more votes. But I'm not sure whether this technically qualifies as bayesian, because neither the prior nor the posterior is a distribution... Can anyone clarify this?