  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 $\mathbb{R}^m$ or an infinite subset
thereof, including multi-modal and uniform distributions. Unfortunately, we
show that this strong definition of conservative cannot be used to evaluate
data fusion techniques. Therefore, we also describe a weaker definition of
conservative and show it is preserved through common data fusion methods such
as the linear and log-linear opinion pool, and homogeneous functionals,
assuming the input distributions can be factored into independent and common
distributions. In addition, we show that after fusion, weak conservativeness is
preserved by Bayesian updates. These strong and weak definitions of
conservativeness can help design and evaluate potential correlation-agnostic
data fusion techniques.
