Invariant metric under deformed Markov embeddings with overlapped supports

TL;DR

This paper explores the deformation of Markov embeddings while preserving sufficiency, establishing the existence and uniqueness of invariant metric families on probability measure spaces.

Contribution

It introduces a deformation of Markov embeddings that maintains invariance and proves the existence and uniqueness of such invariant metrics.

Findings

Existence of invariant metric families under deformed Markov embeddings.

Uniqueness of these invariant families.

Extension of Cencov's theorem to deformed embeddings.

Abstract

Due to \v{C}encov's theorem, there exists a unique family of invariant symmetric $(0,2)$-tensor fields on the space of positive probability measures on a set of $n$-points indexed by $n\in \mathbb{N}$ under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant families under the embeddings.