Simulators run but likelihoods do not: an anytime-valid confidence sequence for parameters, with neural scores near exact ratios

For models with intractable likelihoods, parameter uncertainty can now be reported as a confidence sequence that stays valid under continuous monitoring and at data-dependent stopping times, rather than only as a fixed-sample confidence region. Existing frequentist likelihood-free approaches mainly construct and calibrate fixed-sample tests or confidence regions, and the time-uniform guarantee requires a sequentially valid construction; when likelihoods are available, likelihood-ratio test martingales can be inverted, but in implicit models the ratios cannot be evaluated and substituting a neural estimate generally destroys the martingale property. On synthetic benchmarks, neural scores yield confidence sequences close to conformalized likelihood ratios; in a semi-synthetic audit of model routing in an LLM API and in inference of the stellar initial mass function from OGLE-IV microlensing events, neural score sequences perform comparably to application-specific scores.

The construction draws a fresh reference batch from each candidate parameter, compares the observation and its references through the same score chosen from past information only, normalizes the observed score by the batch average into a conditional e-value, and multiplies these factors over time into a test martingale that is inverted into a parameter set. Normalized-score conformal e-values existed before; SLICE combines them with direct null simulation, predictable alternatives, and amortized neural ratio scores for sequential likelihood-free parameter inference. The score approximates the likelihood ratio with a likelihood-to-evidence network trained under the NRE-C objective on independent simulations before monitoring and then held fixed; experiments cover Gaussian and g-and-k location models, the five-parameter SLCP benchmark, the LLM API audit, and microlensing data.

The method targets models that can be simulated but whose likelihood is intractable, evaluates parameters on a finite candidate grid, and its coverage guarantee applies to true values on that grid; each observation requires fresh reference simulations for every candidate. Demonstrated settings include Gaussian and g-and-k location models, the SLCP benchmark, a semi-synthetic audit of model substitution in an LLM API using real model replies, and inference of the low-mass-star and brown-dwarf initial mass function slopes from OGLE-IV microlensing events. In the microlensing implementation references come from shared simulation banks, and the authors assess coverage empirically for that implementation without establishing an exact time-uniform guarantee.

Validity holds for any positive predictable score and any reference batch size, but score quality governs how fast evidence accumulates, and a finite reference batch limits the expected log evidence obtainable per observation. On SLCP, conformalization has a visible efficiency cost and retained fractions level off once few candidates remain; on the g-and-k model, replacing the minimum-evidence alternative with a model-based running estimate widens the sets substantially. In the microlensing application, high values of the brown-dwarf slope are hard to exclude because short events are rarely detected in low-cadence fields.