Formal definitions of conservative probability distribution functions (PDFs)

Abstract

Under ideal conditions, the probability density function (PDF) of a random variable, such as a sensor measurement, would be well known and amenable to computation and communication tasks. However, this is often not the case, so the user looks for some other PDF that approximates the true but intractable PDF. Conservativeness is a commonly sought property of this approximating PDF, especially in distributed or unstructured data systems where the data being fused may contain un-known correlations. Roughly, a conservative approximation is one that overestimates the uncertainty of a system. While prior work has introduced some definitions of conservativeness, these definitions either apply only to normal distributions or violate some of the intuitive appeal of (Gaussian) conservative definitions. This work provides a general and intuitive definition of conservativeness that is applicable to any probability distribution that is a measure over Rm or an infinite subset thereof, including multi-modal and uniform distributions. Unfortunately, we show that this strong definition of conservative does not hold with any of the commonly used data fusion techniques. Therefore, we also describe a weaker definition of conservative and show it is preserved through common data fusion methods, assuming the input distributions can be factored into independent and common PDFs that can be normalized over Rm. By illustrating what is possible and not possible in terms of conservativeness during data fusion, an improved understanding of data fusion methods for general PDFs can be obtained.