MinNorm approximation of MaxEnt/MinDiv problems for probability tables

Categorical data are found in a wide variety of important applications in environmental sciences and dealing with multivariate analyses is a challenging topic. Rebuilding a multivariate probability table becomes an issue and is expected to lead to poor probability estimates when a very limited number of samples are at hand. In order to take into account the lack of data, the information can be rewritten as inequality constraints instead of using the few sampled values as direct probability estimates. There is thus a need for an efficient method that allows us to rebuild a multivariate probability table from equalities and inequalities constraints. Rebuilding a probability function from equalities constraints can be done through a classical maximum entropy (MaxEnt) methodology. MaxEnt problem can be implemented by using iterated minimum norm (MinNorm) approximations. Minimum divergence (MinDiv) methodology extends the problem to the case of inequalities constraints and, again, MinNorm approximations can be applied and iterated. Thus, iterated MinNorm approximations are a fast and efficient way to combine equalities and inequalities constraints to rebuild a multivariate probability table. MinNorm methodology for solving problems involving both equalities and inequalities constraints can be applied in a wide variety of applications. MinNorm approximations become useful, for instance, when only few data are available or when taking into account experts opinion rewritten as equalities and inequalities constraints is of prime interest in probability estimates. An example in environmental sciences is presented in order to illustrate the benefits of the methodology