A Cautionary Note on Model Choice and the Kullback-Leibler Information

The Kullback-Leibler information has found application in many areas of statistical science. It typically arises in model choice and model dimension questions in a way which suggests its use as a distance. Indeed, it has been widely described as a distance although it comprehensively fails to be a metric. Some pitfalls in interpreting it as a distance are discussed and, in particular, its application to discriminate between prospective risky asset returns distributions.