On impossibility of sequential algorithmic forecasting

Abstract

The problem of prediction future event given an individual 
sequence of past events is considered. Predictions are given 
in form of real numbers $p_n$ which are computed by some algorithm 
$varphi$ using initial fragments $omega_1,dots, omega_{n-1}$ 
of an individual binary sequence $omega=omega_1,omega_2,dots$ 
and can be interpreted as probabilities of the event $omega_n=1$
given this fragment. 
According to Dawid's {it prequential framework} 
%we do not consider
%numbers $p_n$ as conditional probabilities generating by some 
%overall probability distribution on the set of all possible events.
we consider partial forecasting algorithms $varphi$ which are 
defined on all initial fragments of $omega$ and can 
be undefined outside the given sequence of outcomes. 
We show that even for this large class of forecasting algorithms
combining outcomes of coin-tossing and transducer algorithm
it is possible to efficiently generate with probability close 
to one sequences 
for which any partial forecasting algorithm is failed by the
method of verifying called {it calibration}.