Robust Universal Inference For Misspecified Models

Summary In statistical inference, it is rarely realistic that the hypothesized statistical model is well-specified, and consequently, it is important to understand the effects of misspecification on inferential procedures. When the hypothesized statistical model is misspecified, the natural target of inference is a projection of the data generating distribution onto the model. We present a general method for constructing valid confidence sets for such projections, under weak regularity conditions, despite possible model misspecification. Our method builds upon the universal inference method and is based on inverting a family of split-sample tests of relative fit. We study settings in which our methods yield either exact or approximate, finite-sample valid confidence sets for various projection distributions. We study rates at which the resulting confidence sets shrink around their target of inference and complement these results with a simulation study and a study of causal discovery using a linear causal model with the CausalEffectPairs dataset.