Bayesian Statistics

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 $$ \mathbb{P}(B|A) = \frac{ \mathbb{P}(B \cap A) }{ \mathbb{P}(A) } $$ $$ = \frac{ \mathbb{P}(A|B) \mathbb{P}(B) }{ \mathbb{P}(A|B) \ \mathbb{P}(B) + \mathbb{P}(A|B^c) \ \mathbb{P}(B^c) } $$ Switching the roles of the events is convenient because in many problems, one of the conditional probabilities is easier to calculate....

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. \(R = \frac{1}{N} \sum_{i} n_i r_i \)...