# Conformal testing

The main property of validity of conformal transducers is that they output a sequence of p-values that are independent and uniformly distributed on ; therefore, the whole sequence is uniformly distributed on . This can be used for testing the assumption of randomness (or a different on-line compression model) on-line. Namely, fix a martingale on the probability space with the uniform distribution (and standard filtration: is the -algebra generated by the first coordinates of ); we will say that is a test martingale if it is non-negative, for all , and starts from 1, . Therefore, only depends on the first numbers of its argument . The base martingale and the conformal transducer then define the process that is an exchangeability martingale, i.e., a martingale with respect to any exchangeable distribution on . Such processes are called conformal exchangeability martingales. They are randomized in that depends not only on the first observations but also on the internal coin tosses of the conformal transducer. We say that a conformal exchangeability martingale is a test conformal martingale if its base martingale is a test martingale. More generally, a test exchangeability martingale is a nonnegative exchangeability martingale with initial value 1.

Conformal exchangeability martingales are a natural tool for anomaly detection.

There is an unrelated notion of a conformal martingale in stochastic calculus: see, e.g., Revuz and Yor (1999), Section V.2. Therefore, it is best to avoid dropping "exchangeability" in "conformal exchangeability martingale" (and there is a hope that the conformal exchangeability martingales are the only exchangeability martingales, in which case the term "conformal exchangeability martingales" will become redundant).

### Open questions

Bibliography

• Valentina Fedorova, Ilia Nouretdinov, Alex Gammerman, and Vladimir Vovk (2012). Plug-in martingales for testing exchangeability on-line. In: Proceedings of the Twenty Ninth International Conference on Machine Learning, pp. 1639-1646. Omnipress.
• Daniel Revuz and Mark Yor (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.
• Vladimir Vovk, Alexander Gammerman, and Glenn Shafer (2005). Algorithmic learning in a random world. Springer, New York.
• Vladimir Vovk, Ilia Nouretdinov, and Alex Gammerman (2003). Testing exchangeability on-line. In: Proceedings of the Twentieth International Conference on Machine Learning, pp. 768-775. AAAI Press, Menlo Park, CA.