Multiple Local Maxima in Selection-Model Likelihoods

Why Multiple Maxima Need Special Attention

Selection models describe both an outcome distribution and a response mechanism. Their observed-data likelihood can therefore have complicated curvature: different combinations of outcome and response parameters may explain the observed records similarly well. Weak identification and likelihood ridges can make optimization difficult, but a ridge and multiple local maxima are not the same thing. A ridge is a relatively flat path of similar likelihood; distinct local maxima are separate points, each locally at least as high as nearby feasible points.

A routine optimizer usually returns a solution from the neighborhood of its starting value. If the likelihood is not globally concave, convergence from one start does not establish that the returned solution is the best one. The practical task is to search for distinct candidates, compare them on the same likelihood scale, and report precisely what the search does—and does not—establish.

For a selection model with outcome \(Y\) and response indicator \(R\), a typical observed-data likelihood includes a density contribution when \(Y\) is observed and an integrated contribution when it is missing. For example, if \(Y\mid x\) has density \(f(y\mid x;\beta)\) and \(\pi(y,x;\gamma)=P(R=1\mid Y=y,x;\gamma)\), the individual contributions are:

Here \(\theta\) collects the outcome and response parameters. A comparison of fitted candidates is meaningful only when they use the same observed-data likelihood, data, parameter constraints, and numerical accuracy. In particular, a likelihood evaluated from an approximation that is poor in one region can make a spurious candidate look superior or conceal a real one.

What Counts as Evidence of Competing Maxima?

The most direct warning is that dispersed initial values produce converged solutions with both different parameter estimates and materially different attained log likelihoods. Distinct estimates alone are not enough: a flat ridge can yield noticeably different estimates with almost identical likelihoods. Conversely, similar estimates and similar likelihoods across runs are reassuring but do not prove that no other maximum exists.

Treat a run as a candidate maximum only after checking its status and the model domain. At an interior differentiable maximum, the score should be close to zero, and the Hessian of the log likelihood should be negative semidefinite; for a well-behaved strict local maximum, it is typically negative definite. At a boundary, the interior score condition does not apply in the same way: feasible one-sided changes must not increase the objective. A failed convergence code, a large score, a non-finite objective, or an infeasible estimate is not evidence of a local maximum.

Cluster runs by both parameter values and likelihood. Parameter distances should be considered on sensible scales, because a raw Euclidean distance can be dominated by a parameter measured in large units. For each distinct cluster, retain the estimate, log likelihood, convergence status, score norm, Hessian diagnostics, and whether any constraint is active. Then compare the candidates' likelihoods at adequate numerical precision. Rounded values can conceal a small but consequential difference.

A Careful Search and Local Verification

Dispersed starts are a search strategy, not a theorem about global optimization. A disciplined analysis preserves every run, identifies distinct solutions, and checks the strongest candidates independently. This extends the practical work of choosing starting values: the purpose here is to establish whether the fitted likelihood appears to have competing maxima and to qualify the resulting claim.

A useful local result follows from concavity, but its domain matters. Let \(U\) be a convex region contained in the parameter space, and suppose the Hessian \(H(\theta)\) of \(\ell\) is negative definite throughout \(U\). For any two distinct points in \(U\), the line segment between them lies in \(U\). Along that segment the second derivative of the log likelihood is strictly negative, so the log likelihood is strictly concave on \(U\). Consequently, an interior stationary point in \(U\) is the unique maximum in \(U\). This conclusion is local to \(U\); it says nothing about points outside it.

Negative definiteness throughout a nonconvex neighborhood is not, by itself, enough for this line-segment argument: the segment joining two points may leave that neighborhood. A convex neighborhood is a straightforward sufficient condition. In computation, checking the Hessian only at the candidate supports a local diagnostic, not a claim that it is negative definite throughout a region.

What Can Establish a Global Maximum?

No finite collection of ordinary starts proves global optimality for a general nonconcave likelihood on a continuous parameter space. A stronger conclusion requires additional structure or a more exhaustive argument. For example, if the log likelihood is proved concave over the entire convex feasible parameter space, then any feasible stationary maximum is global; strict concavity makes it unique. Alternatively, a rigorous global-search procedure can sometimes bound the objective over a compact parameter domain, subdividing it until every region is either excluded or shown unable to beat the candidate. Such a certificate must cover the whole domain, including boundaries.

If the parameter space is unbounded, a global argument must also address the tails or demonstrate that a maximum is attained within a bounded region. Reparameterizing a constrained parameter does not remove the need to check the original feasible boundary: a boundary may become an infinite limit in the transformed scale. In many applications, the available evidence is numerical rather than a formal certificate. Then the accurate description is “the largest log likelihood found across the reported starts,” not “the global maximum.”

Multiple maxima matter beyond optimization. If distinct candidates have meaningfully different selection effects or outcome parameters, inference based on just one candidate can give a misleading impression of stability. In contrast, nearly equal likelihoods with different parameter estimates may indicate a weakly identified direction rather than distinct, sharply separated peaks. Hessian diagnostics, profiles, and likelihood comparisons help distinguish these situations; none should be substituted for the others.

Common Pitfalls and Alternatives

• Equating convergence with global optimality. A convergence flag concerns the optimizer's behavior near its endpoint. It does not compare that endpoint with all feasible parameter values.
• Calling different estimates different maxima. Similar likelihoods along a flat ridge can produce different estimates. Compare both parameter separation and attained likelihood, and inspect curvature.
• Ignoring constraints. A boundary candidate may be best, and an unconstrained score test is not the right stationarity check at that boundary.
• Using a rounded objective to rank candidates. Retain sufficient precision and recompute all candidate values consistently.
• Calling the best-found candidate global. Multistart increases the chance of finding a better solution; absent a global argument, it does not certify one.

Compared with a single-start fit, multistart optimization is a practical way to expose competing solutions, but its coverage depends on the starting points and optimizer. A profile likelihood can help explore how the objective changes as a parameter varies, although profiling nuisance parameters also requires reliable optimization and does not automatically certify a global maximum. A Hessian check describes local curvature, while a rigorous global bound addresses the whole domain. These methods answer related but different questions.

The main limitation is that a numerical search can fail silently: a narrow basin may be missed, an integral approximation may distort the surface, or convergence tolerances may be too loose to distinguish candidates. Keep a reproducible record of starts, estimates, objective values, constraints, and diagnostics. Treat substantial disagreement between candidates as a finding to investigate, not as a reason to select whichever estimate is easiest to interpret.