Learning in the absence of training data—A galactic application

Abstract

There are multiple real-world problems in which training data is unavailable, and still, the ambition is to learn values of the system parameters, at which test data on an observable is realised, subsequent to the learning of the functional relationship between these variables. We present a novel Bayesian method to deal with such a problem, in which we learn the system function of a stationary dynamical system, for which only test data on a vector-valued observable is available, though the distribution of this observable is unknown. Thus, we are motivated to learn the state space probability density function (pdf), where the state space vector is wholly or partially observed. As there is no training data available for either this pdf or the system function, we cannot learn their respective correlation structures. Instead, we perform inference (using Metropolis-within-Gibbs), on the discretised forms of the sought functions, where the pdf is constructed such that the unknown system parameters are embedded within its support. The likelihood of the unknowns given the available data is defined in terms of such a pdf. We make an application of this methodology, to learn the density of all gravitating matter in a real galaxy.