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Due to Cencov's theorem, there exists a unique family of invariant $(0,2)$-tensors on the space of positive probability measures on a set of $n$-points indexed by $n\in \mathbb{N}$ under Markov embeddings. We prove that this invariance does not depend on the geometry on the space, but only on the feature of Markov embeddings related to sufficient statistics. We deform Markov embeddings with sufficiency in mind and construct a family of invariant $(0,2)$-tensors under the embeddings.

Invariant Metric Under Deformed Markov Embeddings with Overlapped Supports