When A Log Link Is Not A Canonical Link

Canonical Links Are About the Distribution

A link function transforms a model’s mean onto a scale where predictors can be added. For ratemaking, the log link is attractive because it produces positive fitted means and supports multiplicative effects. But these features do not make it canonical for every positive-valued response. Canonical status is determined by the response distribution’s exponential-family form, not by whether the link respects the range of the mean.

This distinction is especially useful when comparing Poisson claim counts with Gamma claim severities. Both distributions can be used in ratemaking, and both can be paired with a log link. The log link is canonical for Poisson counts. For Gamma severity, it is a useful noncanonical choice; the canonical link is the negative inverse mean under the natural-parameter convention used here. This topic is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

The predictor \(\eta\) is typically a linear combination of model terms and may include a known offset. An offset changes the value of the predictor; it does not change which link is canonical for a given response distribution. The earlier material on offsets distinguishes such a known predictor component from an estimated coefficient.

Why the Poisson Log Link Is Canonical

For a Poisson response \(Y\) with mean \(\mu>0\), the probability mass function is \(P(Y=y)=e^{-\mu}\mu^y/y!\), for nonnegative integer \(y\). Taking logarithms and collecting the terms involving \(y\) gives \(y\log\mu-\mu-\log(y!)\). Comparing this with the exponential-family form identifies \(\theta=\log\mu\), \(b(\theta)=e^\theta\), and \(a(\phi)=1\). Therefore the canonical link is \(g(\mu)=\log\mu\).

In a frequency model, if the count mean is exposure \(e\) times a rate \(\lambda\), then \(\log\mu=\log e+\log\lambda\). Thus the exposure offset \(\log e\) appears on the canonical link scale. The rate may itself be modeled through covariates, for example \(\log\lambda=\mathbf{x}^{\mathsf T}\boldsymbol{\beta}\). The resulting count mean remains positive because it is the exponential of its predictor.

Why the Gamma Log Link Is Noncanonical

Consider a Gamma response \(Y>0\) with mean \(\mu>0\) and variance \(\phi\mu^2\), where \(\phi>0\). One corresponding density is \( f(y;\mu,\phi)=y^{1/\phi-1}\exp[-y/(\phi\mu)]/\{\Gamma(1/\phi)(\phi\mu)^{1/\phi}\}\). Taking its logarithm, the terms involving \(y\) and \(\mu\) are \(-y/(\phi\mu)-(1/\phi)\log\mu\). In particular, the log-mean term has a negative sign.

Set \(\theta=-1/\mu\), so that \(-\theta=1/\mu>0\), and let \(b(\theta)=-\log(-\theta)\). Then \(\log\mu=-\log(-\theta)\), and \(y\theta-b(\theta)=-y/\mu-\log\mu\). Consequently the terms just identified equal \([y\theta-b(\theta)]/\phi\). The natural parameter is \(\theta=-1/\mu\), and differentiating \(b\) verifies the mean and variance relationship: \(b'(\theta)=-1/\theta=\mu\) and \(b''(\theta)=1/\theta^2=\mu^2\). With dispersion \(\phi\), the variance is \(\phi b''(\theta)=\phi\mu^2\), as specified. Hence the canonical link in this convention is \(g(\mu)=-1/\mu\).

The log link \(g(\mu)=\log\mu\) is not this natural parameter. It is often chosen for Gamma severity because any real-valued predictor maps to a positive mean, and predictor differences correspond to multiplicative mean changes. Some presentations refer to the Gamma canonical link as the inverse link; sign conventions can differ. In the natural-parameter representation above, the canonical parameter is explicitly \(-1/\mu\). A positive inverse link \(1/\mu\) is its negative, and the corresponding coefficients must be sign-reversed to represent the same linear-predictor structure.

Canonical Status Does Not Mean “Best” or “Positive”

Canonical links have a special relationship to the likelihood because they use the distribution’s natural parameter directly. That relationship does not make every noncanonical link unsuitable. A modeler may prefer a noncanonical link for interpretability, multiplicative rating effects, or a convenient mean domain. The model must still use a response distribution and mean structure appropriate to the data, and its fitted means must satisfy the response’s requirements.

For Gamma severity, the log link guarantees positive means for all real predictors. The canonical link also produces positive means only on its valid predictor domain: since \(\mu=-1/\eta\), it requires \(\eta<0\). A canonical Gamma model with an unrestricted linear predictor can therefore produce invalid values unless its structure or fitting procedure keeps predictors in the negative domain. This is a practical reason a log link is often attractive; it is not evidence that the log link is canonical.

A Reliable Way to Classify a Link

When asked whether a link is canonical, do not start by asking whether its fitted means are positive or whether it is commonly used in insurance. Start from the response distribution’s exponential-family representation and identify its natural parameter. Then compare that parameter with the link applied to the mean. The following sequence keeps the two questions separate.

A common exam trap is to reason that the log link must be canonical whenever the mean is positive. That conclusion confuses a range property with a distributional property. Another trap is to lose a negative sign while deriving the Gamma natural parameter. The Gamma terms are \(-y/(\phi\mu)-(1/\phi)\log\mu\), not \(-y/(\phi\mu)+(1/\phi)\log\mu\). With the correct signs, the natural parameter is \(-1/\mu\). For Poisson counts, the corresponding natural parameter is \(\log\mu\), so the log link is canonical there.