Star Ratings

A product has an unknown rating distribution p = (p₁, …, p₅): the share of everyone who would give it 1 to 5 stars. Each rating is a draw from p, so with counts n₁, …, n₅ the likelihood is ∏ p_k^n_k. The conjugate prior is Dirichlet(α₁, …, α₅), with density ∝ ∏ p_k^{α_k − 1}, and the posterior is Dirichlet(α₁ + n₁, …, α₅ + n₅): each α_k acts as k-star ratings counted before the data, and the Beta–Bernoulli update is the two-category case.

Mean rating. Quality is the average rating the product would get from everyone, μ = Σ k p_k, a number between 1 and 5. Its posterior has no simple closed form, so the page draws 40,000 rating distributions from the posterior Dirichlet (normalized Gamma draws), computes μ for each, and takes the curve, the mean, the interval and the probabilities from these samples; probabilities are accurate to about ±0.005.

Share of positive ratings. Grouping 4 and 5 stars as positive and the rest as negative reduces the model to a Bernoulli one: quality is θ = p₄ + p₅, the prior is Beta(α, β), and with h positive and t negative ratings the posterior is Beta(α + h, β + t).

Priors. Flat: every rating distribution is equally plausible, Dirichlet(1, …, 1) or Beta(1, 1). Jeffreys: Dirichlet(½, …, ½) or Beta(½, ½), the prior that does not depend on how the parameter is written down; it puts extra weight near the edges. Skeptic: ratings counted before the data that lean low, ten pseudo-ratings averaging 2.4 stars, or Beta(2, 8), two positive and eight negative. Custom: any pseudo-counts.

Comparing two products. Given the ratings, the two qualities are independent, so Pr(A is better than B) = Pr(μ_A > μ_B) is computed over all pairs of one sample from each posterior.