Representing preorders with injective monotones

TL;DR

This paper introduces injective monotones in preordered spaces, expanding the classification of preorders and connecting to entropy and decision theory, with implications for statistical inference and machine learning.

Contribution

It defines injective monotones, situates them between strict monotones and multi-utilities, and extends classical results to this new class, linking to entropy and inference.

Findings

Injective monotones exist for a broader class of preorders.

Construction of injective monotones from countable multi-utilities.

Connections between injective monotones, Shannon entropy, and maximum entropy principles.

Abstract

We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter-Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective montones can be used to generalize some appealing properties of Jaynes' maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory.