Abstract
In this paper we develop an inverse Bayesian approach to find the value of the unknown model parameter vector that supports the real (or test) data, where the data comprises measurements of a matrix-variate variable. The method is illustrated via the estimation of the unknown Milky Way feature parameter vector, using available test and simulated (training) stellar velocity data matrices. The data is represented as an unknown function of the model parameters, where this high-dimensional function is modelled using a high-dimensional Gaussian Process ($\mathcal{GP}$). The model for this function is trained using available training data and inverted by Bayesian means, to estimate the sought value of the model parameter vector at which the test data is realised. We achieve a closed-form expression for the posterior of the unknown parameter vector and the parameters of the invoked $\mathcal{GP}$, given test and training data. We perform model fitting by comparing the observed data with predictions made at different summaries of the posterior probability of the model parameter vector. As a supplement, we undertake a leave-one-out cross validation of our method.
Highlights
Curiosity about the nature of the parameter space of the Milky Way that we earthlings live in, is only natural
We discuss the learning of the parameters characterising those Milky Way features that bear influence upon the motion of individual stars that lie in the neighbourhood of the Sun
Using the methodology discussed above we attempt an estimate of the unknown Milky Way feature parameter vector S ∈ Rd using the available stellar velocity data
Summary
Curiosity about the nature of the parameter space of the Milky Way that we earthlings live in, is only natural. The inversion of such measured motions will in principle, allow for the learning of the unknown feature parameters This approach has been adopted in the modelling of our galaxy, to result in the estimation of the angular separation of the Sun from a chosen axis of the bar, and the distance of the Sun from the Galactic centre (Minchev et al, 2010; Fux, 2001; Dehnen, 2000; Simone et al, 2004). We discuss the generic methodology that we use to learn the unknown location vector of the observer in the Milky Way disk, given the matrix-variate test and training stellar velocity data. We note that Equation 2.1 is the same as saying that the likelihood is matrix normal
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