Normalizing Flow Regression for Bayesian Inference with Offline Likelihood Evaluations

Bayesian inference provides a principled framework for quantifying uncertainty in both parameters and models by computing full posterior distributions and model evidence (Gelman et al., 2013). However, Bayesian inference is often analytically intractable, requiring the use of approximate methods like Markov chain Monte Carlo (MCMC; Brooks, 2011) or variational inference (VI; Blei et al., 2017). These methods typically necessitate repeated evaluations of the target density, and many require differentiability of the model (Neal, 2011; Kucukelbir et al., 2017). When model evaluations are computationally expensive - for instance, involving extensive numerical methods - these requirements make standard Bayesian approaches impractical. Due to these computational demands, practitioners often resort to simpler alternatives such as maximum a posteriori (MAP) estimation or maximum likelihood estimation (MLE); 1 see for example Wilson and Collins (2019); Ma et al. (2023). While these point estimates can provide useful insights, they fail to capture parameter uncertainty, potentially leading to overconfident or biased conclusions (Gelman et al., 2013). This limitation highlights the need for efficient posterior approximation methods that avoid the computational costs of standard inference techniques.1.