Why does a rating-prediction factorization model need global, user and item bias terms?

## Decomposing a rating When someone gives a film four stars, at least four things went into that number: 1. **The overall level of the rating scale as people actually use it.** Ratings cluster around 3.5-3.7 on a five-point scale in most catalogues, not around 3.0. That is the global mean `mu`. 2. **How this particular reviewer uses the scale.** Some reviewers rate almost everything 4 or 5; some reserve 5 for a handful of films a decade. That is the user bias `b_u`, the reviewer's typical offset from `mu`. 3. **How this particular film is received in general.** A widely-loved film pulls ratings up from nearly everyone. That is the item bias `b_i`. 4. **What is left over: whether *this* reviewer, specifically, is a match for *this* film.** That is the interaction, and it is the only part the latent factors should have to model. So the model is ``` r_hat(u, i) = mu + b_u + b_i + dot(p_u, q_i) ``` and the biases are learned parameters, not fixed preprocessing constants. ## Why not let the factors handle it They can, badly. A dot product *can* represent a per-user constant — reserve one latent dimension, set that coordinate of every item vector to 1, and put the user's offset in the matching coordinate of the user vector. But that spends a whole factor dimension on something one scalar per user encodes exactly, and it does so through a product of two learned quantities, which is a much harder thing to estimate than a single additive number. On a model with `k = 20`, giving up two dimensions to popularity and generosity is a 10% cut in the capacity available for real taste. Worse, without biases the factor vectors are contaminated. A generous reviewer's vector acquires a large positive component along the "everything is popular" direction, which then makes them look similar in factor space to every other generous reviewer — regardless of whether their tastes match at all. Biases keep the latent space about taste. ## A concrete pair of cases Take a reviewer who has rated 300 films and given all of them a 4. Their user bias is roughly `4 - mu`, a few tenths of a star, and their factor vector should shrink towards zero because their ratings carry no information about *which* films they prefer — there is no variation to explain. That is exactly what a bias term plus L2 regularization produces: the prediction for this user on any film becomes `mu + b_u + b_i`, that is, the general opinion of the film shifted by the reviewer's generosity. Sensible. Now take a film every reviewer rates highly. Its item bias is large and positive; its factor vector should be near zero, because it does not divide opinion. The model then predicts a high rating for everyone, which is right — universally liked films are the case where personalisation has nothing to add. Both are absorbed by the bias terms *before any factor is learned*, and that is the point. ## Regularize the biases too A film with two ratings, both 5, would get an item bias near `5 - mu` if left unpenalised, and the model would then confidently recommend it to everyone. The same L2 penalty applied to the factors is applied to `b_u` and `b_i`, which shrinks thinly-supported offsets towards zero — the shrunken estimate is close to a damped mean, where the item's observed mean is pulled towards the global mean in proportion to how few ratings support it. This is one of the highest-value details in the whole model, because the long tail of items with a handful of ratings is where an unregularized bias does its worst damage. ## The baseline you must beat `mu + b_u + b_i` alone, with no factors at all, is a genuinely strong predictor — it explains a large share of the variance in rating data. Two practical consequences. First, always report it: if the factor model's held-out error is barely better than the bias-only baseline, the factors are noise and you should question the rank, the regularization or the data. Second, it is your degraded-mode answer. A user or item too new to have a meaningful factor vector still has a bias (or falls back to `mu`), so the system always returns something defensible. ## A caveat worth voicing Biases are stated here as static parameters, but rating behaviour drifts — a reviewer's standards harden over time, and a film's reception changes after it becomes a classic or a punchline. Production models often let `b_u` and `b_i` vary with time, which typically buys more accuracy than adding factor dimensions does.