
Given a sample space X, the traditional likelihood function is equal to the conditional probability
of the data given a parameter theta in Theta, i.e., a family of
probability distribution functions (PDFs) over X parameterised
by theta:
As originally proposed by Shafer and Wasserman [28, 32, 33], belief
functions can indeed be built from traditional likelihood functions. However, as we argue here, one can directly
define a belief likelihood function, mapping a sample observation
x in X to a real number, as a natural setvalued generalisation of the conventional likelihood. It is natural
to define such a belief likelihood function as family of belief functions on X, BelX(.theta), parameterised by
theta in Theta. As the latter take values on sets of outcomes, A X, of which singleton outcomes are mere special
cases, they provide a natural setting for computing likelihoods of setvalued observations, in accordance with the
random set philosophy.
