Conditional and Unconditional Statistical Independence
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Conditional independence almost everywhere in the space of the conditioning variates does not imply unconditional independence, although it may well imply unconditional independence of certain functions of the variables. An example that is important in linear regression theory is discussed in detail. This involves orthogonal projections on random linear manifolds, which are conditionally independent but not unconditionally independent under normality. Necessary and suﬀicient conditions are obtained under which conditional independence does imply unconditional independence.
Phillips, Peter C.B., "Conditional and Unconditional Statistical Independence" (1987). Cowles Foundation Discussion Papers. 1067.