Concept splintering in Imprecise Probability: Aleatoric and Epistemic Uncertainty. 

There is a general phenomena in mathematics [and outside maths as well!] where in a certain context/ theory $T_1$ we have two equivalent definitions ${\varphi }_1,{\varphi }_2$ of a concept $C$ that become inequivalent when we move to a more general context/theory $T_2$. In our case we are moving from the concept of probability distributions to the concept of an imprecise distribution (i.e. a convex set of probability distributions, which in particular could be just one probability distribution). In this case the concepts of 'independence' and 'invariant under group action' will splinter into inequivalent concepts. 

Example (splintering of Indepence) In classical probability theory there are three equivalent ways to state that a distribution is independent 

1. $p(x,y)=p\left(x\right)p\left(y\right)$

 2. $p\left(x\right)=p\left(x|y\right)$

3. $p\left(y\right)=p\left(y|x\right)$

In imprecise probability these notions split into three inequivalent notions. The first is 'strong independence' or 'aleatoric independence'. The second and third are called 'irrelevance', i.e. knowing $y$ does not tell us anything about $x$ [or for 3 knowing $x$ does not tell us anything about $y$].

Example (splintering of invariance). There are often debates in foundations of probability, especially subjective Bayesian accounts about the 'right' prior. An ultra-Jaynesian point of view would argue that we are compelled to adopt a prior invariant under some symmetry if we do not posses subjective knowledge that breaks that symmetry ['epistemic invariance'], while a more frequentist or physicalist point of view would retort that we would need evidence that the system in question is in fact invariant under said symmetry ['aleatoric invariance']. In imprecise probability the notion of invariance under a symmetry splits into a weak 'epistemic' invariance and a strong 'aleatoric' invariance. Roughly spreaking, latter means that each individual distribution in the convex set $p_i$, $i\in I$is invariant under the group action while the former just means that the convex set is closed under the action