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The objective of the paper is that of constructing finite Gaussian mixture
approximations to analytically intractable density kernels. The proposed method is
adaptive in that terms are added one at the time and the mixture is fully re-optimized
at each step using a distance measure that approximates the corresponding importance
sampling variance. All functions of interest are evaluated under Gaussian product
rules. Since product rules suffer from an obvious curse of dimensionality, the proposed
algorithm as presented is only applicable to models whose non-linear and/or nonGaussian
subspace is of dimension up to three. Extensions to higher-dimensional
applications would require the use of sparse grids, as discussed in the paper. Examples
include a sequential (filtering) evaluation of the likelihood function of a stochastic
volatility model where all relevant densities (filtering, predictive and likelihood) are
closely approximated by mixtures.