GSM Filter: A Learning Model of Measurement Noise Distributions for Bayesian Filtering

Abstract

In this paper, a new universal Bayesian filter with a Gaussian scale mixture (GSM) model, called the Gaussian scale mixture (GSM) filter, for adaptively learning any measurement noise distribution of a state space system is presented. It is demonstrated that the discrete measurement noise distribution of the system takes on a Gaussian sum form when described by a GSM model. It implies that, as the Gaussian sum model does, the GSM model can also be used to approximate any measurement distribution as closely as desired, whatever the probability density function (pdf) of the measurement noise is. If its covariance is known as a priori or can be estimated, it is found that the scale parameter and matrix of the GSM model at each time step can be achieved adaptively to make the following Bayesian inference much simpler. Further analyses show that the GSM model's auxiliary random variable and the system's state posterior at each time step can be inferred by a variational Bayesian (VB) approach adaptively. At each time step, the posterior distribution of system states shown is a Gaussian whatever the measurement noise pdf of the system is. As a result, the Bayesian filtering for nonlinear systems with any measurement noise pdf can be done by our GSM filter, almost as simple as the general Gaussian filters for filtering a nonlinear Gaussian system. Both simulations and experiments are conducted to verify the effectiveness and correctness of the analytical results. The results on the performances of our algorithms fully outperforming those of the existing state of the art Bayesian filters are given.