Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference
This work addresses the challenge of likelihood-free conditional sampling for researchers in computational statistics and machine learning, offering a new method with applications in Bayesian inference and image processing.
The paper tackles the problem of conditional sampling of probability measures by introducing a novel framework using block triangular transport maps and monotone generative adversarial networks (M-GANs), achieving accurate sampling in synthetic examples, Bayesian inverse problems, and image in-painting.
We present a novel framework for conditional sampling of probability measures, using block triangular transport maps. We develop the theoretical foundations of block triangular transport in a Banach space setting, establishing general conditions under which conditional sampling can be achieved and drawing connections between monotone block triangular maps and optimal transport. Based on this theory, we then introduce a computational approach, called monotone generative adversarial networks (M-GANs), to learn suitable block triangular maps. Our algorithm uses only samples from the underlying joint probability measure and is hence likelihood-free. Numerical experiments with M-GAN demonstrate accurate sampling of conditional measures in synthetic examples, Bayesian inverse problems involving ordinary and partial differential equations, and probabilistic image in-painting.