Beyond Prediction: Conformal Inference for Latent Distributional Parameters

Many prediction problems seek to infer an unobserved, instance-specific parameter governing the distribution of an observable response, even when latent parameters are unavailable for both historical and future instances. We develop \latentcp, a prior-free conformal framework that constructs uncertainty sets for latent distributional parameters. The method first builds a conformal prediction set in the response space and then retains latent candidates whose induced response distributions assign sufficient probability to that set. We further introduce a multilevel procedure that aggregates uncertainty sets across response-space miscoverage levels to improve set efficiency. We theoretically show that the sets admit a finite-sample validity guarantee for the latent parameter under an exchangeability condition, and empirically demonstrate that \latentcp maintains nominal latent coverage under weak identification, nonidentifiability, heterogeneity, and multimodality, where competing methods can substantially under-cover.