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Under ideal conditions, the probability density function (PDF) of a randomvariable, such as a sensor measurement, would be well known and amenable tocomputation and communication tasks. However, this is often not the case, sothe user looks for some other PDF that approximates the true but intractablePDF. Conservativeness is a commonly sought property of this approximating PDF,especially in distributed or unstructured data systems where the data beingfused may contain un-known correlations. Roughly, a conservative approximationis one that overestimates the uncertainty of a system. While prior work hasintroduced some definitions of conservativeness, these definitions either applyonly to normal distributions or violate some of the intuitive appeal of(Gaussian) conservative definitions. This work provides a general and intuitivedefinition of conservativeness that is applicable to any probabilitydistribution that is a measure over $\mathbb{R}^m$ or an infinite subsetthereof, including multi-modal and uniform distributions. Unfortunately, weshow that this strong definition of conservative cannot be used to evaluatedata fusion techniques. Therefore, we also describe a weaker definition ofconservative and show it is preserved through common data fusion methods suchas the linear and log-linear opinion pool, and homogeneous functionals,assuming the input distributions can be factored into independent and commondistributions. In addition, we show that after fusion, weak conservativeness ispreserved by Bayesian updates. These strong and weak definitions ofconservativeness can help design and evaluate potential correlation-agnosticdata fusion techniques.

Formal Definitions of Conservative Probability Distribution Functions (PDFs)