Information Criteria for Risk Models¶

Definition¶

For a fitted likelihood model with maximized log-likelihood $\hat\ell$ , $k$ estimated parameters and sample size $n$ , Akaike's information criterion is $\operatorname{AIC}=-2\hat\ell+2k$ $2 k$ , and Bayesian information criterion is $\operatorname{BIC}=-2\hat\ell+k\log n$ $k lo g n$ . Smaller values are preferred when models use comparable likelihoods on the same observations.

Interpretation¶

The fit improvement from extra parameters must compensate for their penalty. Stepwise selection searches neighbouring models and need not find the best model among all possible subsets.

Stacked bars for models A and B: the fit term −2ℓ̂ (100 and 96) plus the penalty 2k or k log n, giving AIC 104 and 102 and BIC 109.21 and 109.82, the smaller of each pair in green.

The figure draws the Examples case with $n=100$ $100$ ; the vertical axis starts at 90. Model B fits better, so its fit term $-2\hat\ell$ is 4 lower, but it has one more parameter. AIC charges 2 for that parameter, so B wins with 102 against 104; BIC charges $\log 100\approx4.61$ $4.61$ , more than the fit gain of 4, so A wins with 109.21 against 109.82 (B's BIC penalty is $3\log 100\approx13.82$ $13.82$ ).

Change the maximized log-likelihoods $\hat\ell$ , the parameter counts and $n$ : AIC prefers model B here, while BIC, which charges $\log n$ per parameter, prefers model A once $n$ exceeds $e^4\approx54.6$ $54.6$ .

Maximum likelihood estimation supplies the maximized log-likelihood $\hat\ell$ . When comparing fitted generalized linear model specifications with different rating predictors on the same observations, an extra coefficient can improve fit but also increases the parameter-count penalty.

Key Results¶

Count all estimated likelihood parameters, including an estimated dispersion parameter where applicable.
For $n>\mathrm{e}^2$ $e^{2}$ , BIC penalizes each added parameter more than AIC.
AIC and BIC are selection aids, not evidence of causal interpretation or predictive validation.
Ordinary likelihood-based AIC is not directly available from a quasi-Poisson mean–variance specification alone.

Examples¶

With $n=100$ $100$ , model A has $\hat\ell=-50$ $- 50$ , $k=2$ $2$ , while B has $\hat\ell=-48$ $- 48$ , $k=3$ $3$ . AIC values are 104 and 102, favouring B; BIC values are about 109.21 and 109.82, favouring A.

Prerequisites¶

Sources¶

Risk Insurance · Lecture 6

Aliases: AIC, BIC, Schwarz criterion, Akaike information criterion, Bayesian information criterion, information criterion, Akaike's information criterion, Schwarz Bayesian criterion, Schwarz information criterion Keywords: stepwise model selection, AIC = -2ℓ̂ + 2k, BIC = -2ℓ̂ + k log n, model selection, parameter penalty

Last updated: 17 October 2026 21:53