Conformal Exchangeability Martingale

Conformal exchangeability martingales are defined in the article conformal testing. The notion of a conformal exchangeability martingale has several pitfalls:

• Conformal exchangeability martingales are randomized: they depend not only on the observations ⚠ $z_1,z_2,\ldots$ but also on independent random numbers taking values in 0,1$.
• Each conformal exchangeability martingale ⚠ $S_n$ is a martingale only in the sense of satisfying ⚠ $E(S_{n+1}\mid S_1,\ldots,S_n)=S_n$. It does not satisfy ⚠ $E(S_{n+1}\mid \mathcal{F}_n)=S_n$ (where ⚠ $\mathcal{F}_n$ is the past including the observations ⚠ $z_1,\ldots,z_n$) except in trivial cases.
• Therefore, the sum of two conformal exchangeability martingales does not have to be a conformal exchangeability martingale.

The main open question about conformal exchangeability martingales is whether they exhaust the class of exchangeability martingales.