Notes on imprecise probabilities from ISIPTA summer school

The lower prevision of categorical information "x in A" is P(f) = min f(x), x in A.

The lower and upper prevision of precise probabilistic beliefs is the expectation of f.

For real variables, we also have lower and upper cumulative distribution L(x) = P( (-∞ , x] )

There are always three points of view: Lower expectations, Credal sets and Desirable gambles.

P(f|B) is defined as the willingness to pay x for a gamble that rewards f(), canceled with money back if B happens. Its definition implies a way to compute it, the Generalized Bayes Rule:
P(f|B) is the x solution of P(1_B (f - x) ) ) = 0.

If P(B)>0, then P(f|B) is also the lower enveloppe of the P(f|B), for all P in the credal set, conditioned using Bayes rules : p(x,y) = p(x) p(y|x)

Full conditional measures are Probabilities coming in layers:
in layer n+1, you define p(x|y) for the y such that p(y)=0 in layer n...
and nothing blows up !

There are no probabilities, only conditional probabilities. See Krauss-Dubins representation. See Isaac Levi's (and Peter Walley's) book.

If you speak only about desirable gambles, you are limited to convex credal sets because they define half-spaces. But see this 2007 paper by Seidenfeld, Kadane and Schervish working with general partial preferences ordering, it gives meaning to non-convex credal set. In practice convexity is not so important for computation, one is always at the vertices.