# Paper No. 12-01

**D Chakrabarty, M Biswas and S Bhattacharya**

**Bayesian Inverse Learning of Milky Way Model Parameters using new Matrix-Variate Gaussian Process-Based Method**

**Abstract:** The modelling of theMilkyWay galaxy is an integral step in the study of galactic dynamics; this owes not only to our natural inquisitiveness about our residence in the Universe, but also because knowledge of model parameters that define the MilkyWay directly influences our understanding of the evolution of our galaxy. Since the nature of phase space in the neighbourhood of the Sun is affected by distinct Milky Way features, an inverse learning of a highdimensional MilkyWay model parameter vector is in principle possible from using measurements of phase space coordinates of individual stars that live in the vicinity of the Sun. In the past this has been attempted via the “calibration method”, a data and computational cost intensive method that limited by such logistical shortcomings, could attempt the learning of a 2-dimensional Galactic model parameter vector, namely the radial and azimuthal locations of the Sun, with respect to the centre of the Galaxy and a chosen axis. This method used synthetic stellar phase space data that were generated from dynamical simulations of distinct astrophysical models of the Milky Way. We argue in this paper that the calibration method is unsatisfactory because of reasons both methodological and computational. Here we develop a Bayesian inverse problem approach, where we model the measurable, i.e. stellar velocity as an unknown function of the Milky Way model parameters, where this function is inverted using Bayesian techniques to predict the model parameters. This unknown function turns out to be matrix-variate, which we model as a matrix-variate Gaussian Process. We develop the general theory of matrix-variate Gaussian Processes and formulate the general inverse problem. For the inference we use the recently advanced Transformation-based Markov chain Monte Carlo (TMCMC). Application of our method to observed stellar velocity data results in estimates that are consistent with those in astrophysical literature. That we could obtain these results using far smaller data sets compared to those required for the calibration approach, is encouraging in terms of projected applications to external galaxies.