# Conservative Validity

The main article on conformal prediction deduces the asymptotic conservative validity of conformal predictors from the corresponding result for smoothed conformal predictors. The following theorem asserts the conservative validity (in a non-asymptotic sense) of conformal predictors.

Theorem All conformal predictors are conservatively valid, i.e. for any exchangeable probability distribution  on  there exists a probability space with two families  of {0, 1}- valued random variables such that:

• for a fixed  ,  is a sequence of independent Bernoulli random variables with the parameter  ;
• for all  and  ,  ;
• the joint distribution of  coincides with the joint distribution of  , where  is the random variable  if  ;  otherwise.

Of course, this theorem also implies the asymptotic conservative validity of conformal predictors.

Corollary All conformal predictors are asymptotically conservative, i.e., for any exchangeable probability distribution  on  and any significance level  ,  with probability one.