A Note on the Geometry of Kullback-Leibler Information Numbers.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The Kullback-Leibler information number is a well-known measure of statistical distance between probability distributions. Previous authors have shown that when endowed with this distance measure, the space of probability distributions possesses geometrical properties analogous to Euclidean geometry. This paper proves a new geometrical property by showing that one can in fact define the shortest line between two probability distributions as well as its mid-point. It turns out that the probability distributions comprising this line have long ago been used as a tool in the important problem of testing statistical hypotheses involving nuisance parameters. Apart from pure mathematical convenience, there has been little justification for its use. The results in this paper are the first attempt at such explanation.
- Statistics and Probability