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Profile Likelihood

Profile-likelihood confidence intervals are constructed by examining how the objective function changes as each parameter is varied away from its optimum, while all other parameters are re-optimized at each step. Unlike Wald intervals, which assume a locally quadratic log-likelihood, profile intervals can capture asymmetric uncertainty and are better suited to parameters near boundary constraints or in models with nonlinear reparameterizations. They are considered the gold standard for frequentist confidence intervals in nonlinear models.

In NoLimits.jl, profile-likelihood UQ is accessed through:

compute_uq(res; method=:profile, ...)

The underlying interval computation is performed via LikelihoodProfiler.jl.

Applicability

Profile UQ is available for fitted results from the following estimation methods:

  • MLE
  • MAP
  • Laplace
  • LaplaceMAP

Minimal Usage

using NoLimits
using Random

uq_profile = compute_uq(
    res;
    method=:profile,
    level=0.95,
    profile_method=:LIN_EXTRAPOL,
    profile_scan_width=1.0,
    profile_scan_tol=1e-2,
    profile_loss_tol=1e-2,
    profile_max_iter=300,
    rng=Random.Xoshiro(1),
)

Core Controls

The following parameters govern the profile-likelihood algorithm and are exposed through compute_uq:

  • profile_method (default :LIN_EXTRAPOL): the profiling algorithm used by LikelihoodProfiler.jl.
  • profile_scan_width (must be positive): search window around the point estimate, specified in transformed-coordinate units and subject to parameter bounds.
  • profile_scan_tol: tolerance for the scanning phase of the profiler.
  • profile_loss_tol: tolerance on the objective function difference used to determine the interval boundary.
  • profile_local_alg (default :LN_NELDERMEAD): local optimization algorithm applied near interval endpoints.
  • profile_max_iter: maximum number of iterations for the local optimizer.
  • profile_ftol_abs: absolute function tolerance for the local optimizer.
  • profile_kwargs: additional keyword arguments forwarded directly to the underlying profiler call.

In practice, profile_scan_width determines how far from the estimate the profiler searches. If intervals appear truncated, increasing this value or raising profile_max_iter may help the profiler locate the true boundary.

Fit-Context Overrides

The profile backend accepts the same fit-context overrides available in other UQ backends:

  • constants
  • constants_re
  • penalty
  • ode_args, ode_kwargs
  • serialization
  • rng

When not provided, stored values from the original fit are reused.

Parameter Inclusion Rules

Profile UQ is evaluated only on free fixed-effect coordinates that are eligible for uncertainty calculation.

A coordinate is excluded when:

  • its fixed-effect block is held constant via constants, or
  • its block has calculate_se=false.

If no eligible coordinates remain, profile UQ raises an error.

fe = @fixedEffects begin
    a = RealNumber(0.2, calculate_se=true)     # included
    b = RealNumber(0.1, calculate_se=false)    # excluded
end

Returned Quantities

The result is a UQResult with backend :profile. The available accessors are shown below.

backend = get_uq_backend(uq_profile)                # :profile
source = get_uq_source_method(uq_profile)
names = get_uq_parameter_names(uq_profile)

est_nat = get_uq_estimates(uq_profile; scale=:natural)
est_tr = get_uq_estimates(uq_profile; scale=:transformed)

ints_nat = get_uq_intervals(uq_profile; scale=:natural)
ints_tr = get_uq_intervals(uq_profile; scale=:transformed)

V_nat = get_uq_vcov(uq_profile; scale=:natural)     # nothing
draws_nat = get_uq_draws(uq_profile; scale=:natural) # nothing

diag = get_uq_diagnostics(uq_profile)

Because profile likelihood characterizes uncertainty by tracing the objective function surface rather than by sampling, covariance matrices and draw matrices are not available for this backend. Only interval estimates are returned.

Diagnostics and Boundary Behavior

get_uq_diagnostics(uq_profile) returns profiler metadata that is useful for assessing the quality of the computed intervals:

  • Algorithm settings: profile_method, tolerances, and local algorithm used.
  • Objective values: loss_at_estimate and loss_critical (the threshold corresponding to the requested confidence level).
  • Per-parameter endpoint status: left_status and right_status indicate whether each boundary was successfully located.
  • Per-parameter endpoint counters: left_counter and right_counter report the number of profiler evaluations at each boundary.
  • Endpoint availability: endpoint_found flags whether both interval endpoints were determined.
  • Per-parameter errors: errors captures any profiler-level issues encountered during computation.

These diagnostics are essential for identifying incomplete or numerically unstable intervals. If an interval endpoint was not found, common remedies include widening profile_scan_width, increasing profile_max_iter, or relaxing profile_loss_tol.