Maximum entropy inference for mixed continuous-discrete variables

We represent knowledge by using probability distributions of mixed continuous and discrete variables. In the case of complete knowledge of the joint distribution of all items, one can compute arbitrary conditional distributions, which may be used for prediction. However, in many cases only some marginal distributions, inverse probabilities, or moments are known. Under these conditions, a principle is needed in order to determine the full joint distribution of all variables. The principle of maximum entropy (Jaynes; 1957, 2003; Haken; 1977; Guiasu and Shenitzer; 1985) ensures an unbiased estimation of the full multivariate relationships using only known facts. In the case of discrete variables, the expert shell SPIRIT implements this approach (cf. Rödder; 2006; Rödder and Meyer; 2006; Rödder et al.; 2006). In this paper the approach is generalized to continuous and mixed continuous-discrete distributions and applied to the problem of credit scoring.