Why Convergence Is More Than a “Converged” Message
A GLM fitting routine searches for coefficient values that maximize a likelihood. Its output may include an estimate, a convergence flag, and standard errors, but none of these alone guarantees that the fitted model is numerically reliable. An algorithm can stop because its steps became small even when it is far from a genuine maximum; it can also keep taking steps because the likelihood has no finite maximum under the specified model.
For ratemaking, this distinction matters. A problematic fit may arise from an unsuitable numerical starting point, poorly scaled predictors, sparse rating cells, or data that effectively separate outcomes. The right response depends on the cause. Repeatedly changing iteration limits does not fix a model whose finite maximum-likelihood estimate does not exist.
Newton or Fisher-scoring methods use the local slope and curvature of the log-likelihood to propose an update. With expected information, a Fisher-scoring step is
The proposal assumes that the local quadratic approximation is useful and that the information matrix can be inverted reliably. Neither assumption is guaranteed for a large step, a nearly redundant set of predictors, or a likelihood that keeps improving toward a boundary.
What to Check During Fitting
No single convergence tolerance is universal. A practical assessment combines several checks and uses scales appropriate to the data and parameterization.
- Likelihood behavior. For a maximization routine, accepted steps should ordinarily increase the log-likelihood. A stalled likelihood is not enough to establish convergence: the likelihood may be nearly flat in one direction or may be approaching a finite supremum at an infinite parameter value.
- Parameter-step size. Inspect the change in each parameter, preferably relative to its scale. A small absolute change can be meaningless for a large-valued parameter, while a parameter changing by a fixed amount on every iteration signals a likely boundary problem.
- Score size. Check whether the score is small after accounting for the scale of the information or data. A raw score can be large simply because the data set is large; a scaled score or score relative to information is often more informative.
- Information matrix. A singular or poorly conditioned information matrix can make updates and standard errors unstable. This may reflect redundant predictors, sparse data, or near-confounding among rating variables.
- Fitted values and probabilities. Check for implausible fitted means, probabilities numerically equal to zero or one, and extreme predictions driven by a small number of observations.
A common safeguard is step-halving. If a full proposed step decreases the observed log-likelihood, the algorithm tries a fraction of that step and continues reducing the fraction until it finds an acceptable move. This limits harmful jumps; it does not create information in empty cells or guarantee that a finite estimate exists.
Check the response distribution, exposure or offset, predictor coding, and aggregated cell totals before interpreting an optimizer warning.
Compare log-likelihood, scaled score, parameter changes, and fitted values across iterations. Look for declining likelihood, repeated large steps, or a coefficient moving steadily in one direction.
Try sensible rescaling or alternative starting values for a numerical issue. Investigate sparse cells, redundancy, or separation-like patterns when estimates continue toward a boundary.
Pooling levels, constraining effects, or using a justified regularization method changes the estimation problem. Document why that change is appropriate rather than treating it as a way to silence a warning.
Worked Examples of Convergence Diagnostics
Worked Example: Checking a Poisson Intercept Fit
A hypothetical rating segment is summarized in three cells. The observed claim counts are 2, 0, and 4, and the corresponding exposures are 1, 2, and 2 policy-years. Fit an intercept-only Poisson frequency model with mean claim count \(\mu_j=e_j\exp(\beta)\) in cell \(j\), where \(e_j\) is exposure. Start Fisher scoring at \(\beta=0\). Show the updates and check the result.
Method and assumptions. Counts are treated as independent Poisson variables with the same claim rate per policy-year, and exposure enters as a known multiplicative amount. The total observed count is \(2+0+4=6\); total exposure is \(1+2+2=5\). The model therefore has total fitted count \(5\exp(\beta)\).
Ignoring terms in the log-likelihood that do not depend on \(\beta\),
At the starting value \(\beta_0=0\), \(\exp(\beta_0)=1\), so \(U(\beta_0)=6-5=1\) and \(I(\beta_0)=5\). The first scoring step is \(1/5=0.2\), giving \(\beta_1=0.2\).
At \(\beta_1=0.2\), \(\exp(0.2)=1.2214\). Thus \(U(\beta_1)=6-5(1.2214)=-0.1070\) and \(I(\beta_1)=5(1.2214)=6.1070\). The next step is approximately \(-0.1070/6.1070=-0.0175\), giving \(\beta_2\approx0.1825\).
At \(\beta_2\approx0.1825\), \(\exp(\beta_2)\approx1.2002\). The score is approximately \(6-5(1.2002)=-0.0010\), and the information is approximately \(6.0010\). The next step is about \(-0.0002\), giving \(\beta_3\approx0.1823\). The log-likelihood (with the same omitted constants) rises from \(-5.0000\) at the start to about \(-4.9070\) at \(\beta_1\), then to about \(-4.9061\) near the maximum.
The score equation gives \(6-5\exp(\hat\beta)=0\), so \(\exp(\hat\beta)=6/5=1.2\) and \(\hat\beta=\log(1.2)\approx0.1823\). The fitted total is \(5(1.2)=6\), matching the observed total. This check supports the numerical result.
Common mistake avoided: Calling the first small improvement in log-likelihood convergence. Here the parameter step and score also become small, and the fitted total agrees with the score-equation solution.
Worked Example: Recognizing a Separation-Like Logistic Fit
In a hypothetical binary underwriting outcome study, four records have a fixed baseline predictor of zero and all four outcomes equal 1. A single coefficient \(\gamma\) is added to the logit for this group, so \(P(Y=1)=p=\exp(\gamma)/(1+\exp(\gamma))\). Starting from \(\gamma=0\), use Newton's method to diagnose whether a finite maximum-likelihood estimate exists.
Method and assumptions. Outcomes are independent Bernoulli variables, and the baseline predictor is fixed rather than estimated in this one-parameter illustration. The log-likelihood is \(\ell(\gamma)=4\log(p)\). Its score is \(U(\gamma)=4(1-p)\), and its information is \(I(\gamma)=4p(1-p)\). Therefore, when \(p<1\), the Newton step is \(U/I=1/p\).
At \(\gamma_0=0\), \(p=0.5\), so \(U=2\), \(I=1\), and the step is \(2\). Thus \(\gamma_1=2\), where \(p\approx0.8808\) and the log-likelihood is \(4\log(0.8808)\approx-0.5077\).
At \(\gamma_1=2\), the step is \(1/0.8808\approx1.1353\), giving \(\gamma_2\approx3.1353\). Here \(p\approx0.9583\), and the log-likelihood is approximately \(4\log(0.9583)\approx-0.170\). The next step is approximately \(1/0.9583=1.0435\), so \(\gamma_3\approx4.1788\). The log-likelihood continues to increase as \(p\) approaches 1.
For every finite \(\gamma\), \(p<1\), so the likelihood is below its supremum. As \(\gamma\) tends to \(+\infty\), \(p\) tends to 1 and the log-likelihood tends to 0. There is no finite maximum-likelihood estimate. In an ordinary logistic model, a predictor pattern that perfectly distinguishes outcomes can produce this separation behavior. A routine may eventually stop because probabilities round to 1 or information becomes numerically tiny, but that is not evidence of a finite estimate.
Common mistake avoided: Treating the last reported coefficient as a stable estimate just because the software stopped. Here the steps remain near 1 and the likelihood keeps improving; investigate the data pattern and model rather than interpreting that coefficient as an ordinary finite fit.
Worked Example: A Zero-Count Cell With Its Own Effect
A hypothetical rating cell has 10 policy-years of exposure and zero claims. Suppose it has an unrestricted cell-specific Poisson log-mean \(\eta\), so its mean count is \(\mu=10\exp(\eta)\). Starting at \(\eta=0\), determine what the likelihood and scoring steps imply.
Method and assumptions. The cell count is Poisson with known exposure 10 and no constraint linking its rate to another cell. Omitting constants, the log-likelihood is \(\ell(\eta)=-10\exp(\eta)\). Its score is \(-10\exp(\eta)\), and its information is \(10\exp(\eta)\). Each Fisher-scoring step is therefore \(-1\).
At \(\eta=0\), the fitted mean is \(10\) and the log-likelihood contribution is \(-10\). The first step gives \(\eta=-1\), with mean \(10e^{-1}\approx3.6788\) and log-likelihood \(-3.6788\). The next step gives \(\eta=-2\), with mean \(10e^{-2}\approx1.3534\) and log-likelihood \(-1.3534\). At \(\eta=-3\), the mean is \(10e^{-3}\approx0.4979\), and the log-likelihood is \(-0.4979\). Each step improves the likelihood, but the parameter does not settle.
As \(\eta\) tends to \(-\infty\), the fitted mean tends to zero and the log-likelihood tends to 0. No finite \(\eta\) attains that value. The unrestricted cell effect therefore has no finite maximum-likelihood estimate. Pooling the cell with others, imposing a justified constraint, or using a justified regularization approach can change the estimation problem; simply increasing the iteration limit cannot supply missing data.
Common mistake avoided: Reporting a zero rate as though it were the finite estimate of this log-link model. A zero observed count is not itself a finite log-mean parameter; in this unrestricted cell model it sends the estimate toward the boundary.
Why the Update Formula Works, and When It Can Mislead
The scoring update follows from a local approximation to the log-likelihood. Near a current parameter vector, approximate \(\ell\) by a quadratic with slope \(\mathbf{U}\) and curvature \(-\mathbf{I}\). The gradient of that quadratic at an increment \(\mathbf{h}\) is approximately \(\mathbf{U}-\mathbf{I}\mathbf{h}\). Setting the approximate gradient to zero gives \(\mathbf{h}=\mathbf{I}^{-1}\mathbf{U}\), which yields the scoring update.
This derivation also explains its limitations. The approximation is local. A large step may land in a region where the quadratic is a poor description of the actual likelihood, so step-halving or a trust-region method may be needed. If \(\mathbf{I}\) is nearly singular, small changes in the score can produce large changes in the proposed step. If the likelihood has no finite maximum, local optimization cannot make one exist.
Common Numerical Traps in Rating Data
- Using only a likelihood tolerance. A tiny likelihood change can occur while a coefficient continues toward an infinite boundary, as in separation or an all-zero unrestricted cell.
- Assuming a zero cell always prevents fitting. A zero count need not cause a problem when its rate is estimated jointly with informative cells under shared effects. It is especially problematic when a free parameter can drive that cell's fitted mean to zero without a countervailing observation.
- Ignoring scale and coding. Predictors with very different numerical magnitudes can worsen conditioning. Rescaling a continuous predictor can help numerical computation, although the parameterization and interpretation change accordingly.
- Confusing poor starting values with nonexistence. Alternative reasonable starts can reveal a numerical sensitivity. If all fits show a coefficient moving toward the same boundary with continuing likelihood improvement, investigate separation or a boundary maximum instead.
- Forcing a fit by suppressing warnings. More iterations may help a genuinely slow but finite fit. They do not resolve nonidentifiability, empty information directions, or an absent finite maximum.
The practical sequence is to verify the response, exposure, and design; inspect likelihood and parameter paths; check the score and information matrix; then decide whether rescaling, improved starting values, or a justified change to the model is warranted. This disciplined diagnosis is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus because it connects estimation mechanics to the reliability of indicated rating results.
Check Your Understanding
For each problem, identify the relevant numerical or statistical issue and check the final result.
- An intercept-only Poisson model has 9 claims over 12 policy-years. What are the fitted frequency per policy-year and the intercept estimate? (Answer: fitted frequency \(0.75\); intercept \(\log(0.75)\approx-0.2877\).)
- A Poisson cell has exposure 4, zero claims, and its own unrestricted log-mean parameter \(\eta\). Starting at \(\eta=0\), what are the first two scoring updates and fitted means? (Answer: each update is \(-1\), giving \(\eta=-1,-2\) and means \(4e^{-1}\approx1.4715\), \(4e^{-2}\approx0.5413\); no finite maximum exists.)
- In a one-parameter logistic model with three successes and no failures, the initial probability is \(0.5\). What is the first Newton step, and what does the data imply about the estimate? (Answer: score \(=1.5\), information \(=0.75\), step \(=2\); the likelihood increases toward probability 1, so there is no finite intercept estimate.)
- A GLM routine reports convergence because the log-likelihood changed by less than its tolerance, but one coefficient increases by about 1 at each iteration and fitted probabilities approach 1. What should be checked before accepting the fit? (Answer: inspect for separation-like behavior and a boundary/no-finite-estimate problem; a small likelihood change alone does not establish a finite maximum.)
- A fitted information matrix is nearly singular, and two rating indicators have almost identical patterns across observed records. What is the likely concern, and name one model-level remedy to consider. (Answer: weak identifiability or near-redundancy causing unstable updates; consider a justified pooling or a revised, identifiable predictor specification.)