Discrete approximation of a mixture distribution via restricted divergence

Mixture distributions arise in many application areas, for example as marginal distributions or convolutions of distributions. We present a method of constructing an easily tractable discrete mixture distribution as an approximation to a mixture distribution with a large to infinite number, discrete or continuous, of components. The proposed DIRECT (Divergence Restricting Conditional Tesselation) algorithm is set up such that a pre-specified precision, defined in terms of Kullback-Leibler divergence between true distribution and approximation, is guaranteed. Application of the algorithm is demonstrated in two examples.