  This paper is devoted to the prediction problem in extreme value theory. Our
main result is an explicit expression of the regular conditional distribution
of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$
given observations $\{\eta(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is
the point process representation of max-infinitely divisible processes by
Gin\'e, Hahn and Vatan (1990). We carefully analyze the structure of the
underlying point process, introduce the notions of extremal function,
sub-extremal function and hitting scenario associated to the constraints and
derive the associated distributions. This allows us to explicit the conditional
distribution as a mixture over all hitting scenarios compatible with the
conditioning constraints. This formula extends a recent related result by Wang
and Stoev (2011) dealing with the case of spectrally discrete max-stable random
fields. We believe this work offers new tools and perspective for prediction in
extreme value theory together with numerous potential applications.
