Approximate Bayesian Neural Doppler Imaging

The non-uniform surface temperature distribution of rotating active stars is routinely mapped with the Doppler Imaging technique. Inhomogeneities in the surface produce features in high-resolution spectroscopic observations that shift in wavelength depending on their position on the visible hemisphere. The inversion problem has been systematically solved using maximum a-posteriori regularized methods assuming smoothness or maximum entropy. Our aim in this work is to solve the full Bayesian inference problem, by providing access to the posterior distribution of the surface temperature in the star. We use amortized neural posterior estimation to produce a model that approximates the high-dimensional posterior distribution for spectroscopic observations of selected spectral ranges sampled at arbitrary rotation phases. The posterior distribution is approximated with conditional normalizing flows, which are flexible, tractable and easy to sample approximations to arbitrary distributions. When conditioned on the spectroscopic observations, they provide a very efficient way of obtaining samples from the posterior distribution. The conditioning on observations is obtained through the use of Transformer encoders, which can deal with arbitrary wavelength sampling and rotation phases. Our model can produce thousands of posterior samples per second. Our validation of the model for very high signal-to-noise observations shows that it correctly approximates the posterior, although with some overestimation of the broadening. We apply the model to the moderately fast rotator II Peg, producing the first Bayesian map of its temperature inhomogenities. We conclude that conditional normalizing flows are a very promising tool to carry out approximate Bayesian inference in more complex problems in stellar physics, like constraining the magnetic properties.