Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 3 / ii

For a distribution faithful to a Directed acyclic graph, the smallest Markov blanket of a vertex consists of

Conditioning on this set blocks every path from the vertex to all remaining vertices. Each listed neighbor is necessary under faithfulness of a directed acyclic graph: omitting a parent or child leaves its direct edge active, while omitting a child's other parent leaves the collider path through that conditioned child active. Faithfulness rules out accidental cancellations that could otherwise make a smaller blanket sufficient.