Poisson Regression¶

Definition¶

In Poisson regression, conditionally independent counts satisfy $Y_i\mid \boldsymbol{x}_i\sim\operatorname{Pois}(\mu_i)$ with $\log\mu_i=\boldsymbol{x}_i^{\top}\boldsymbol{\beta}$ $x_{i}^{⊤} β$ . Thus $\mu_i=\exp(\boldsymbol{x}_i^{\top}\boldsymbol{\beta})>0$ and $\operatorname{Var}(Y_i\mid \boldsymbol{x}_i)=\mu_i$ $μ_{i}$ .

Interpretation¶

The model relates rating factors to expected claim counts through multiplicative effects. It does not directly model claim severity.

Log-link curve μ = exp(x^T β) with two points at linear predictors log 0.1 and log 0.15; a shift of log 1.5 along the axis multiplies the mean from 0.1 to 0.15.

The log link turns additive changes in the linear predictor into multiplicative changes in the mean. In the example below, the reference group has $\boldsymbol{x}_i^{\top}\boldsymbol{\beta}=\log0.1$ and mean $0.1$ ; switching the binary rating factor on adds its coefficient $\log1.5$ , which multiplies the mean by $\mathrm{e}^{\log1.5}=1.5$ to give $0.15$ . Because $\mu_i=\mathrm{e}^{\boldsymbol{x}_i^{\top}\boldsymbol{\beta}}$ is always positive, no predictor value can produce a negative expected count.

This is a generalized linear model with a logarithmic link and response law from the Poisson distribution. The link enforces a positive mean, while the Poisson law supplies equality of the conditional mean and variance and the stated likelihood.

Key Results¶

Increasing covariate $x_{ij}$ by one multiplies the mean by $\mathrm{e}^{\beta_j}$ if the model has no relevant interaction.
The independent-data log-likelihood is $\displaystyle \sum_i[y_i \boldsymbol{x}_i^{\top}\boldsymbol{\beta}-\mathrm{e}^{\boldsymbol{x}_i^{\top}\boldsymbol{\beta}}-\log(y_i!)]$ $lo g (y_{i}!)]$ .
Equality of conditional mean and variance is the Poisson assumption; heterogeneity across covariates does not by itself violate it.
Different observation durations require exposure adjustment.

Examples¶

With intercept $\log0.1$ and a binary rating coefficient $\log1.5$ , the two conditional means are 0.1 and 0.15 at the same exposure.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 6

Aliases: log-link count regression, claim-frequency regression, Poisson GLM, Poisson log-linear model Keywords: log link, μ_i = exp(x_i^T β), multiplicative effects, rate ratio e^(β_j), Poisson log-likelihood Symbols: beta, mu, sum

Last updated: 16 October 2026 22:28