IID Vs Exchangeability

It is shown in Vovk (1986), in the binomial case, that the IID deficiency of a data sequence $x$ can be split into two components:

  • the exchangeability deficiency of $x$
  • the randomness deficiency of the multiset of the observations in $x$ (i.e., $x$ with the order of its elements erased).

The problem is to extend this to general observation spaces.

Can we characterize, in a simple way, the randomness deficiency of a multiset (as in the second component above)? (In the binary case, this is done in Vovk 1986.)