# Conformal Predictive System

Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017). Essentially, these are conformal transducers that, for each training sequence and each test object, output p-values that are increasing as a function of the label , assumed to be a real number. The function is then called a *predictive distribution*.

A wide class of conformity measures that often lead to conformal predictive systems is

where is the prediction for the label of based on the training sequence . The width of such conformal predictive distributions is typically equal to , where is the length of the training sequence, except for at most values of .

The formal definition of conformal predictive systems takes account of the fact that, in the case of smoothed conformal predictors, also depends on the random number , and a fuller notation is . It is also required that as and as .

Notice that in the context of conformal predictive systems the p-values acquire properties of probabilities. Besides, they have some weak properties of object conditionality: e.g., the *central prediction regions* are not empty, except in very pathological cases.

**Bibliography**

- Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017). Nonparametric predictive distributions based on conformal prediction. On-line Compression Modelling Project (New Series), Working Paper 17, April 2017.