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.)

Bibliography

• Vladimir Vovk (1986). On the concept of Bernoulli property. ''Russian Mathematical Surveys]], 41:247–248.