Prior- and likelihood-free probabilistic inference with finite-sample calibration guarantees

Motivated by parametric models for which the likelihood is analytically unavailable, numerically unstable, or prohibitively expensive to compute or optimize, we develop a prior- and likelihood-free framework for fully probabilistic (Bayesian-like) uncertainty quantification with finite-sample calibration guarantees. Our method, a type of inferential model, produces data-dependent degrees of belief about claims concerning the unknown parameter while controlling the frequency with which high belief is assigned to false claims, even in finite-sample settings. Our procedure is general in that it requires only the ability to simulate from the model. We first rank candidate parameter values according to how well data simulated from the model agree with the observed data, and then rescale these rankings in a way that yields the desired finite-sample calibration guarantees. The key idea is to employ a permutation-invariant function, such as a depth function, to rank parameter values. We show that such a choice yields closed-form calibration rescaling calculations, making the procedure computationally simple. We illustrate our method's broad appeal with four examples, including differential privacy and Ising models. An analysis of the spatial configuration of 2025 measles outbreaks in the U.S. showcases our method's practical advantages.