Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy

Summary:Within the framework of discrete probabilistic uncertain reasoning a large literature exists justifying the maximum entropy inference process, $\operatorname{\mathbf{ME}}$, 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 $\operatorname{\mathbf{ME}}$ 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 $\operatorname{\mathbf{ME}}$ called the social entropy process, $\operatorname{\mathbf{SEP}}$. However, while $\operatorname{\mathbf{SEP}}$ 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 $\operatorname{\mathbf{ME}}$ whose immediate generalisations to the multi-agent case are not satisfied by $\operatorname{\mathbf{SEP}}$. 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 $\operatorname{\mathbf{SEP}}$ can satisfy an interesting modified generalisation of this principle