The cut metric for probability distributions

Abstract

Guided by the theory of graph limits, we investigate a variant of the cut metric for limit objects of sequences of discrete probability distributions. Apart from establishing basic results, we introduce a natural operation called pinning on the space of limit objects and show how this operation yields a canonical cut metric approximation to a given probability distribution akin to the weak regularity lemma for graphons. We also establish the cut metric continuity of basic operations such as taking product measures.

Inference and mutual information on random factor graphs

“…Coming with this embedding of ({−1, 1} n into the space of functions f : [0, 1] → P ({−1, 1}), there is an embedding of the corresponding probability measures µ ∈ P (({−1, 1} n ) into the space of functions μ : [0, 1] 2 → P (({−1, 1}) by taking well-defined limits. A detailed discussion and formal justification of the procedure is provided by [18].…”