Uncertainty Quantification in Simulation-based Inference

Complex phenomena in engineering and the sciences are often modeled by feed-forward simulators that implicitly encode the likelihood function. Classical methods for constructing confidence sets and hypothesis tests are poorly suited for such settings. While the field of simulation-based inference has undergone a revolution in terms of the complexity of problems that can be tackled, the development on the statistical methodology front has fallen behind. Indeed, many techniques have been developed for learning a surrogate likelihood using forward-simulated data, but these methods do not guarantee frequentist confidence sets and tests with nominal coverage and Type I error control, respectively, outside the asymptotic and low-dimensional regimes. In this thesis we introduce a series of statistical tools for uncertainty quantification in a simulation-based inference setting. In the first part of the thesis, we provide inferential tools with frequentist guarantees for a high-dimensional simulator-based setting. We introduce a statistical framework that unifies classical statistics with modern machine learning algorithms to achieve the following

goals: (i) confidence sets and hypothesis tests with finite-sample guarantees of coverage and power, (ii) diagnostics for checking empirical coverage over the entire parameter space and (iii) scalable and modular procedure which can also be used with other simulation-based approaches. We showcase the applicability of this framework across a diverse range of data and parameter settings. In the second part of the thesis, we consider the problem of assessing the quality of fit of approximate likelihood models. Approximate likelihood models are used in settings with high-resolution and computationally intensive simulators for faster inference. We propose a statistically consistent validation method that can pinpoint the locations in the parameter space where the fit is inadequate as well as provide insights as to how the high-resolution simulator and approximate likelihood model differ In the third and final part of the thesis, we propose a new flexible method for nonparametric conditional density estimation that can transform any neural network

regression architecture into a conditional density estimator. We demonstrate the versatility of our approach for two applications with convolutional and recurrent neural networks, respectively.