Bayesian Statistics

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:

The Formula

By definition…

$$ \mathbb{P}(A) = \mathbb{P}(A \cap B) + \mathbb{P}(A \cap B^{c}) $$

From conditional probability …

$$ \mathbb{P}(A) = \mathbb{P}(A|B) \ \mathbb{P}(B) + \mathbb{P}(A|B^c) \ \mathbb{P}(B^c) $$

Therefore

$$ \mathbb{P}(B|A) = \frac{ \mathbb{P}(B \cap A) }{ \mathbb{P}(A) } $$ $$ = \frac{ \mathbb{P}(A|B) \mathbb{P}(B) }{ \mathbb{P}(A|B) \ \mathbb{P}(B) + \mathbb{P}(A|B^c) \ \mathbb{P}(B^c) } $$