Markov Random Fields

Yu. A. Rozanov
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Yu. A. Rozanov: Steklov Mathematics Institute

Chapter Chapter 2 in Markov Random Fields, 1982, pp 55-102 from Springer

Abstract: Abstract Let A1, ℬ, A2 be σ-algebras of events having the following relationship: if the outcomes of all events in ℬ are known, events A2 ∈A2 are independent of events A1 ∈ A1. More precisely, the σ-algebras A1 and A2 are conditionally independent with respect to ℬ; this gives the equation for conditional probabilities: 1.1 $$ P({A_1} \cdot {A_2}|B) = P({A_1}|B) \cdot P({A_2}|B) $$ for any A1 ∈ A1, A2 ∈A2. We say that the σ-algebra ℬ splits A1 and A2 (or is splitting) if (1.1) holds for A1, ℬ, A2.

Keywords: Random Field; Markov Property; Open Domain; Nonempty Intersection; Compact Domain (search for similar items in EconPapers)
Date: 1982
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DOI: 10.1007/978-1-4613-8190-7_2

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