Conditioning
G.L. Wise, Eric B Hall · 1993
Abstract The topic of u-algebras is basic to the subject of conditioning since conditioning is conventionally taken with respect to au-algebra. In many cases the u-algebra of interest is that generated by some random variables representing data. Hence, in applications, it is common to treat u-algebras as somehow representing knowledge or information associated with data. The following example from Billingsley [1986] shows that associating u algebras with knowledge or information, as commonly understood, may lead to incorrect conclusions. Consider the probability space ([0, 1], B([0, 1]), .X), where A denotes Lebesgue measure on B([0, 1]) and consider the u-sub algebra g given by the family of all subsets of [0, 1] that are either countable or co-countable. Now, for BE B([0, 1]), consider the conditional probability P(B IQ). Since g contains all singletons {w}, and hence might be seen as being completely informative, one might suppose that P(B IQ) is equal to 18 . In other words, one might rationalize that to know the sets in g implies that one knows w itself and hence knows whether or not w is contained in B, leading to the conclusion that P(B IQ) should be one when w is contained in Band zero otherwise. It follows quickly, however, from the definition of conditional probability that P(B IQ)= P(B), except possibly off a countable subset of [0, 1]. D