  Motivated by indirect measurements and applications from nanometrology with a
mixed noise model, we develop a novel algorithm for jointly estimating the
posterior and the noise parameters in Bayesian inverse problems. We propose to
solve the problem by an expectation maximization (EM) algorithm. Based on the
current noise parameters, we learn in the E-step a conditional normalizing flow
that approximates the posterior. In the M-step, we propose to find the noise
parameter updates again by an EM algorithm, which has analytical formulas. We
compare the training of the conditional normalizing flow with the forward and
reverse KL, and show that our model is able to incorporate information from
many measurements, unlike previous approaches.
