  A model for a subject's beliefs about a phenomenon may exhibit symmetry, in
the sense that it is invariant under certain transformations. On the other
hand, such a belief model may be intended to represent that the subject
believes or knows that the phenomenon under study exhibits symmetry. We defend
the view that these are fundamentally different things, even though the
difference cannot be captured by Bayesian belief models. In fact, the failure
to distinguish between both situations leads to Laplace's so-called Principle
of Insufficient Reason, which has been criticised extensively in the
literature.
  We show that there are belief models (imprecise probability models, coherent
lower previsions) that generalise and include the Bayesian belief models, but
where this fundamental difference can be captured. This leads to two notions of
symmetry for such belief models: weak invariance (representing symmetry of
beliefs) and strong invariance (modelling beliefs of symmetry). We discuss
various mathematical as well as more philosophical aspects of these notions. We
also discuss a few examples to show the relevance of our findings both to
probabilistic modelling and to statistical inference, and to the notion of
exchangeability in particular.
