Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers and generalized minimizers are explicitly constructed from solutions of modified dual problems, not assuming the primal constraint qualification. A generalized Pythagorean identity is presented using Bregman distance and a correction term for lack of essential smoothness in integrands. Results are applied to minimization of Bregman distances. Existence of a generalized dual solution is established whenever the dual value is finite, assuming the dual constraint qualification. Examples of `irregular' situations are included, pointing to the limitations of generality of certain key results.
A solution to the marginal problem is obtained in a form of parametric exponential (Gibbs-Markov) distribution, where the unknown parameters are obtained by an optimization procedure that agrees with the maximum likelihood (ML) estimate. With respect to a difficult performance of the method we propose also an alternative approach, providing the original basis of marginals can be appropriately extended. Then the (numerically feasible) solution can be obtained either by the maximum pseudo-likelihood (MPL) estimate, or directly by Möbius formula.
Within the framework of discrete probabilistic uncertain reasoning a large literature exists justifying the maximum entropy inference process, \ME, as being optimal in the context of a single agent whose subjective probabilistic knowledge base is consistent. In particular Paris and Vencovská completely characterised the \ME inference process by means of an attractive set of axioms which an inference process should satisfy. More recently the second author extended the Paris-Vencovská axiomatic approach to inference processes in the context of several agents whose subjective probabilistic knowledge bases, while individually consistent, may be collectively inconsistent. In particular he defined a natural multi-agent extension of the inference process \ME called the social entropy process, \SEP. However, while \SEP has been shown to possess many attractive properties, those which are known are almost certainly insufficient to uniquely characterise it. It is therefore of particular interest to study those Paris-Vencovská principles valid for \ME whose immediate generalisations to the multi-agent case are not satisfied by \SEP. One of these principles is the Irrelevant Information Principle, a powerful and appealing principle which very few inference processes satisfy even in the single agent context. In this paper we will investigate whether \SEP can satisfy an interesting modified generalisation of this principle.
Several algorithms have been developed for time series forecasting. In this paper, we develop a type of algorithm that makes use of the numerical methods for optimizing on objective function that is the Kullbak-Leibler divergence between the joint probability density function of a time series xi, X2, Xn and the product of their marginal distributions. The Grani-charlier expansion is ušed for estimating these distributions.
Using the weights that have been obtained by the neural network, and adding to them the Kullback-Leibler divergence of these weights, we obtain new weights that are ušed for forecasting the new value of Xn+k.