Relative Divergence and Maximum Relative Divergence Principle for Grading Functions on Partially Ordered Sets

TL;DR

This work generalizes the Insufficient Reason Principle to the Maximum Relative Divergence Principle (MRDP) for grading functions on partially ordered sets, extending Relative Divergence (RD) from chains to complex poset structures. By linking RD to Shannon entropy and leveraging chain-based reductions, the paper derives classic probabilistic results (e.g., conditional probability and independence) as MRDP solutions, and develops structural tools (block-chains, block-splits, and l-g embeddings) to handle general posets. It applies MRDP to powersets, chain bundles, and partition-induced grading functions, yielding explicit least-presuming updates for group-testing costs and queueing-service costs, including height-dependent and additively separable cases. The approach provides a principled, least-assuming framework for updating models under prior information, with practical implications for decision-making under uncertainty in networked systems and population testing.

Abstract

Relative Divergence (RD) and Maximum Relative Divergence Principle (MRDP) for grading (order-comonotonic) functions (GF) on posets are used as an expression of Insufficient Reason Principle under the given prior information (IRP+). Classic Probability Theory formulas are presented as IRP+ solutions of MRDP problems on conjoined posets. RD definition principles are analyzed in relation to the poset structure. MRDP techniques are presented for standard posets: power sets, direct products of chains, etc. "Population group-testing" and "Single server of multiple queues" applications are stated and analyzed as "IRP+ by MRDP" problems on conjoined base posets.