A Gamma Filter for Positive Parameter Estimation

In many data fusion applications, the parameter of interest only takes positive values. For example, it might be the goal to estimate a distance or to count instances of certain items. Optimal data fusion then should model the system state as a positive random variable, which has a probability density function that is restricted to the positive real axis. However, classical approaches based on normal densities fail here, in particular whenever the variance of the likelihood is rather large compared to the mean. In this paper, it is considered to model such random parameters with a Gamma distribution, since its support is positive and it is the maximum entropy distribution for such variables. For a Bayesian recursion, an approximative moment matching approach is proposed. An example within the framework of an autonomous simulation and further numerical considerations demonstrate the feasibility of the approach.