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I am reading up on D-separation and Markov Blankets. The definition for D-separation is clear enough.

De finition 0.2 (Blocked Path). We say that a path from a node X to a node Y is blocked by Z if it contains a variable such that either

  1. this variable is in Z and it is head-to-tail or tail- to-tail or
  2. the node is head-to-head and neither this node nor any of its descendants are in Z .

My questions are.

  1. Explanations on markov blanket always use a figure where the arrows go 'downwards'. Any particular reason why ?
  2. For the figure on the right, is B independent of A given the markov blanket ? Note that the arrow direction has been reversed.

enter image description here

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    $\begingroup$ Questions 1 and 2 seem to be separate questions, could you edit the question to focus on one $\endgroup$ Jan 13, 2018 at 19:51

1 Answer 1

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EDIT: I just noticed the fact that, in the right image, B is actually part of the Markov Blanket of A as it is a parental node of a child of A. A should, by definition, be independent of all nodes outside it's Markov Blanket. The answer now talks independence of A and B given the blank node in between.

B is not independent from A given the blank node in between. Knowing about the node between A and B will enable B to get information about A. Assume B is a bomb that explodes, A is an alarm and the blank node represents whether you react to some noise.

When we know you react to some noise (conditioned on blank node), the probability that a bomb exploded and thus caused a reaction will depend on whether or not the alarm went off. The probability that the bomb exploded will be smaller if the alarm went off then when it didn't. This is because only one is necessary to get a reaction and them happening both will in this case be small.

This picture provides an overview of different connections and the independence they encode:

enter image description here

The above comes from David Barber's Bayesian Reasoning and Machine Learning (Chapter 3)

About the other question, I think it's just more convenient to illustrate as there is you can flip the illustration horizontally and it will still represent the same conditional independency assumptions.

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  • $\begingroup$ hi thanks ! so just to make it clear, for the figure on the left, B is independent of A given the markov blanket (picture f) but on the right, B is not independent of A given the markov blanket (picture h) ? $\endgroup$
    – Kong
    Jan 13, 2018 at 20:06
  • $\begingroup$ Yes, although I would say picture g instead of f but thats just me :) $\endgroup$
    – Lejafar
    Jan 13, 2018 at 20:09
  • $\begingroup$ Are nodes outside of the Markov blanket d-separated from each other? $\endgroup$ Jan 23 at 4:09

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