Approximation of powers of Gaussian mixtures

Gaussian mixtures are used to approximate general probability density functions. In fusion problems, the weighted geometric mean of densities exploits density powers. Since the operation of exponentiation of Gaussian mixtures is not closed, the problem of approximating the mixture powers arises. This paper proposes and inspects several such approximations. The first is based on a limit case, the second is based on an optimisation, the third is given by a numerical solution to the optimisation problem and the last one is given by the exact power of an approximation of the Gaussian mixture. The numerical examples compare component weights and the means and covariance matrices.