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We discuss some properties of conditional expectation operators, and usethese facts to prove an interesting counterexample regarding sufficientstatistics. In particular, we show that there exists sufficient randomvariables X and Y, such that (X, Y) are jointly not sufficient. We follow thework of Burkholder and Chow, presenting complete proofs and supplying missingsteps.

Properties of Conditional Expectation Operators and Sufficient Subfields