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

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....