02 February 2017 0 10K Report

In the book Bayesian Reasoning and Machine Learning (by David Barber, DRAFT March 9, 2010). There is a definition of Local Markov Property (see pics attached).  It is mentioned that "When conditioned on its neighbours, x is independent of the remaining variables of the graph", which I can understand in a way, but how to reach the Equation (4.2.6) based on such a statement? Can anyone help to give a detailed explanation? Or say, can anyone provide a proof reference rather than the intuitive feeling based on the statement? Thanks!

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