the generalized linear model (GLM)

Ordinary regression assumes the thing you predict can be any number, drifts up and down a straight line, and scatters with constant spread. Insurance data refuses to behave: claim counts are non-negative whole numbers, claim sizes are skewed and strictly positive, and a probability must sit between 0 and 1. The generalized linear model is the elegant repair that keeps the familiar 'add up weighted predictors' machinery but adapts it to these awkward, real-world outcomes. It is, without exaggeration, the workhorse of modern actuarial pricing.

A GLM relaxes ordinary regression in two linked ways. First, it allows the outcome to follow any distribution from the exponential family — Poisson for counts, gamma for positive skewed amounts, binomial for yes/no — instead of forcing a bell curve. Second, it connects the predictors to the outcome through a link function: rather than modeling the mean directly, it models a transformed mean, g(mean) = a + b1*x1 + b2*x2 + .... The log link is the actuarial favorite because it makes the predictors multiply rather than add: each factor becomes a relativity that scales the base rate up or down. The coefficients are fitted by maximum likelihood. So a frequency model might be a Poisson GLM with a log link, and a severity model a gamma GLM with a log link; their product reconstructs the pure premium.

This is exactly how personal-lines insurers set rates: a Poisson GLM for how often you claim, a gamma GLM for how much each claim costs, with rating variables like age, vehicle, and territory entering as multiplicative relativities. GLMs are prized because they are interpretable (every coefficient has a clear meaning), statistically principled, and acceptable to regulators who demand to see why one driver pays more than another. The honest limits: a GLM still assumes the modeler picked the right distribution, link, and variables; it captures interactions only if you put them in by hand; and like all regression it shows association, not causation.