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Uncertainty, Entropy, Variance and the Effect of Partial Information

James V. Zidek and Constance van Eeden
Lecture Notes-Monograph Series
Vol. 42, Mathematical Statistics and Applications: Festschrift for Constance van Eeden (2003), pp. 155-167
Stable URL: http://www.jstor.org/stable/4356236
Page Count: 13
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Uncertainty, Entropy, Variance and the Effect of Partial Information
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Abstract

Uncertainty about the value of an unmeasured real random variable Y is commonly represented by either the entropy or variance of its distribution. If it becomes known that Y lies in a subset A of the support of Y's distribution, one might expect uncertainty about Y to decrease. In other words, one might expect the entropy and variance of Y's conditional distribution given Y ∈ A to be less than their counterparts for the unconditional distribution. Going further it might be conjectured that the uncertainty about Y would be greater given the knowledge that Y ∈ B as compared with Y ∈ A ⊂ B. We do not know whether these conjectures are correct. However, we give sufficient conditions in certain cases where they are true. In particular, when Y is normally distributed we can make considerable progress. For example, we show in the case that A = [a, b] and Y normally distributed with mean η and variance 1, that the variance of the conditional distribution of Y given that a ≤ Y ≤ b is less than that of the unconditional distribution, thereby confirming our intuitive reasoning in this case. This last example also shows that for this exponential family the variance is less than 1 for all a < b and all η--a result that is not known among the experts on exponential families we consulted.

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