Allows us to weight by review population size.
Let \(n_i\) be the number of reviews that item \(i\) gets, and let \(r_i\) be the naive average rating of item \(i\)
Let \(N\) be the total number of reviews across brands, i.e. \(N = \sum_{i} n_i \)
Let \(R\) be the average rating over all items across brands, i.e. \(R = \frac{1}{N} \sum_{i} n_i r_i \)
Then, the Bayesian rating \(\tilde{r_i}\) is:
$$ \tilde{r_i} = \frac{ n_i r_i + N R }{ n_i + N } $$
We place more faith in \(r_i\) as \(n_i\) increases. IMDB’s top 250 movies rating uses this. Beer Advocate replaces \(N\) with \(N_{min}\).
Bayesian rating assumes a single true rating. It performs poorly for products that create bipolar responses.
Bayesian rating can only be used within a comparable family of products.
My intuition about this is off. When comparing the Medallion Fund (averaged 39.1% over 30 years) to Berkshire Hathaway (averaged 20.5% over 53 years) , I thought that Berkshire would come out on top.