Mixture density network estimation of continuous variable maximum likelihood using discrete training samples

Mixture density networks (MDNs) can be used to generate posterior density functions of model parameters $$\varvec{\theta }$$ given a set of observables $${\mathbf {x}}$$ . In some applications, training data are available only for discrete values of a continuous parameter $$\varvec{\theta }$$ . In such situations, a number of performance-limiting issues arise which can result in biased estimates. We demonstrate the usage of MDNs for parameter estimation, discuss the origins of the biases, and propose a corrective method for each issue.