Why AIC Balances Fit and Complexity
A more flexible statistical model can often fit observed data better simply because it has more parameters available to adapt to the sample. Comparing maximized likelihoods alone can therefore reward unnecessary complexity. The Akaike Information Criterion (AIC) adjusts the maximized log likelihood by adding a penalty for the number of estimated parameters.
For an actuarial candidate model, AIC is a relative selection measure: among models fitted to the same observations under a consistent likelihood convention, the smaller AIC indicates the preferred balance of fit and complexity. AIC does not establish that the selected model is an adequate description of the data, nor does it give the probability that the model is true.
The log likelihood must be maximized over the model’s free parameters before calculating AIC. The parameter count is not the sample size: it counts fitted parameters, such as a severity distribution’s shape and scale or a frequency model’s mean and dispersion. A parameter fixed in advance rather than estimated from the data does not contribute to \(k\). The likelihood value and parameter count should refer to the same model fit.
How to Calculate and Compare AIC
The calculation has two parts. First, convert the maximized log likelihood to a deviance-like fit term, \(-2\hat{\ell}\). Second, add \(2\) for each estimated parameter. Because a log likelihood is commonly negative, a larger (less negative) value of \(\hat{\ell}\) gives a smaller fit term and improves AIC. More parameters increase the penalty and, all else equal, increase AIC.
Identify the candidate distributions, the observations used, and each maximized log likelihood. AIC values are meaningfully compared when the candidates are fitted to the same data using a consistent likelihood basis.
Determine \(k\) for each model, including its fitted distribution parameters. Do not count each observation as a parameter, and do not count a value fixed in advance as an estimated parameter.
Evaluate \(-2\hat{\ell}+2k\) separately for every candidate, retaining the negative sign on the log likelihood.
Choose the smallest AIC among the candidates under consideration, then state that this is a relative ranking rather than a proof of absolute model adequacy.
The reason for the two opposing terms is important. Maximizing likelihood rewards a model for describing the observed sample well, so \(-2\hat{\ell}\) decreases as the maximized log likelihood improves. But using more fitted parameters can improve in-sample fit without guaranteeing better performance beyond that sample. AIC therefore adds a penalty that grows with the number of parameters. The formula does not simply select the model with the best in-sample fit; it trades fit against complexity.
Worked Example: Comparing Two Fitted Severity Models
A hypothetical insurer fits an exponential severity model and a gamma severity model to the same 80 fully observed claim amounts, measured in thousands of dollars. A maximum-likelihood fitting report gives an exponential rate estimate of \(0.0042\) per thousand dollars with maximized log likelihood \(\hat{\ell}=-246.8\). The gamma model has estimated shape \(1.7\) and scale \(165\) thousand dollars, with \(\hat{\ell}=-238.9\). Calculate and compare AIC.
Method. Use \(\operatorname{AIC}=-2\hat{\ell}+2k\). The exponential model estimates one parameter, its rate, so \(k=1\). The gamma model estimates two parameters, shape and scale, so \(k=2\). The sample size of 80 is not the parameter count.
Exponential model. Its fit term is
Its penalty is \(2(1)=2\), so
Gamma model. Its fit term is
Its penalty is \(2(2)=4\), giving
The gamma model has the lower AIC: \(481.8<495.6\), a difference of \(13.8\). Its improved likelihood more than offsets the extra parameter’s penalty. The conclusion is that gamma is preferred by AIC among these two fitted candidates on this data set; it is not a claim that gamma is the true severity distribution.
Common mistake avoided: Comparing \(-238.9\) directly with \(-246.8\) would select gamma based only on fit. AIC still selects gamma here, but only after the two-parameter penalty is included.
Reading the Fit and Penalty Together
AIC can be thought of as a sum of a fit term and a complexity term. The fit term is determined by the maximized log likelihood; the penalty is determined by the parameter count. When a model adds parameters, its log likelihood cannot be lower than that of a nested version fitted to the same observations, because the larger model can reproduce the smaller model’s parameter choices. However, AIC does not automatically prefer the more flexible model: it must improve the log likelihood enough to compensate for the added penalty.
For example, adding one free parameter increases the penalty by \(2\). If doing so raises the maximized log likelihood by less than \(1\), then the fit-term reduction is less than \(2\), and AIC increases. If it raises the maximized log likelihood by more than \(1\), the AIC decreases. This is an arithmetic comparison, not a general test of whether the added parameter is statistically significant.
Worked Example: A Fixed Assumption Versus an Estimated Rate
A hypothetical portfolio has three observed claim severities, \(1\), \(2\), and \(3\) thousand dollars. Consider two exponential models for these positive losses. Model A fixes the rate at \(\lambda=0.4\) per thousand dollars in advance. Model B estimates the rate by maximum likelihood. For an exponential density \(f(x)=\lambda e^{-\lambda x}\), calculate both AIC values and interpret the ranking. Use the ordinary likelihood for these three observations.
Model A: fixed rate. With \(\lambda=0.4\), the log likelihood is the sum of \(\ln(\lambda)-\lambda x_j\). Since the observations sum to \(1+2+3=6\),
The rate was fixed in advance, so \(k_A=0\). Thus
Model B: estimated rate. For an exponential sample, the maximum-likelihood estimate is the number of observations divided by their total. Therefore
At this estimate, the maximized log likelihood is
Model B estimates one parameter, so \(k_B=1\). Its AIC is
Rounded to four decimal places, the AIC values are \(10.2977\) for Model A and \(12.1589\) for Model B. AIC prefers the fixed-rate model, even though Model B has the better maximized log likelihood: its improvement in log likelihood is only \(0.0694\), too small to compensate for the penalty of \(2\). This comparison is meaningful only if the fixed rate was specified independently of these observations; choosing it after inspecting the sample would not make it a genuinely fixed assumption.
Common mistake avoided: Giving Model A one parameter because its rate appears in the density is incorrect here. The rate is fixed, not estimated from the sample, so it does not add to \(k\).
AIC for a Frequency Model
The same calculation applies to frequency models. The observations might be claim counts rather than severities; what matters is that the maximized log likelihood and parameter count belong to each fitted probability model. A more flexible frequency distribution may capture variation in the counts better, but AIC charges for the additional parameter.
Worked Example: Comparing Poisson and Negative Binomial Frequency Models
A hypothetical actuary fits two distributions to the same 120 policy-year claim counts. The Poisson model has fitted mean \(\hat{\lambda}=1.25\), maximized log likelihood \(\hat{\ell}_{P}=-182.4\), and one estimated parameter. The negative binomial model is parameterized by mean \(\mu\) and size \(r\), with variance \(\mu+\mu^2/r\); the fitted values are \(\hat{\mu}=1.18\), \(\hat{r}=5.6\), and its maximized log likelihood is \(\hat{\ell}_{NB}=-176.9\). The reported log likelihoods are rounded to one decimal place. Calculate the AIC values and choose the lower-AIC model.
Poisson model. It estimates only its mean parameter, so \(k_P=1\). The fit term and penalty are
Therefore,
Negative binomial model. It estimates both \(\mu\) and \(r\), so \(k_{NB}=2\). Its fit term and penalty are
Thus,
The negative binomial model has the smaller AIC: \(357.8<366.8\), by \(9.0\). Its better fit, represented by a maximized log likelihood \(5.5\) higher than the Poisson model’s, outweighs the \(2\)-unit extra penalty. AIC therefore ranks the negative binomial model ahead of the Poisson model for these observations.
Common mistake avoided: Counting 120 policy-years as 120 parameters would produce a meaningless penalty. The models estimate one and two distribution parameters, respectively; the number of observations is not \(k\).
Common AIC Traps and Responsible Use
- Dropping the minus sign. If \(\hat{\ell}=-40\), then \(-2\hat{\ell}=80\), not \(-80\). A more negative log likelihood must produce a larger fit term.
- Forgetting the penalty. The criterion is not just \(-2\hat{\ell}\); it also includes \(2k\).
- Using the number of observations as \(k\). The penalty counts estimated model parameters, not claims, policies, or time periods.
- Comparing unlike likelihoods. AIC rankings require comparable likelihood calculations on the same observations and response definition. If one fit omits observations or uses a different likelihood basis, its AIC should not be ranked directly against the other without addressing that difference.
- Treating the winner as proven true. AIC selects a relative winner among the candidates supplied. It does not establish absolute fit, certainty, or a causal explanation.
Check Your Understanding
For each problem, calculate or interpret AIC using the stated likelihood information. Treat the reported log likelihoods as maximized values.
- A fitted severity model has \(\hat{\ell}=-42.6\) and estimates three parameters. Calculate its AIC. (Answer: \(91.2\))
- Two models fitted to the same data have these results: Model A, \(\hat{\ell}=-120.0\), \(k=2\); Model B, \(\hat{\ell}=-116.5\), \(k=4\). Calculate both AIC values and identify the lower-AIC model. (Answer: Model A AIC \(=244.0\); Model B AIC \(=241.0\); Model B is preferred.)
- A model’s maximized log likelihood improves from \(-75.0\) to \(-74.1\) when one additional parameter is estimated. By how much does AIC change, and in which direction? (Answer: AIC increases by \(0.2\), so the more complex model has the higher AIC.)
- A Poisson frequency model has \(\hat{\ell}=-91.3\) and estimates one mean parameter. A negative binomial model fitted to the same counts has \(\hat{\ell}=-88.7\) and estimates a mean and size parameter. Calculate both AIC values and choose the lower-AIC model. (Answer: Poisson AIC \(=184.6\); negative binomial AIC \(=181.4\); negative binomial is preferred.)
- A candidate model estimates two parameters and has \(\hat{\ell}=-50.0\). A second model has the same maximized log likelihood but estimates five parameters. Which has smaller AIC, and by how much? (Answer: The two-parameter model; its AIC is \(6\) lower.)