Problems of Information Transmission, Vol. 39, No. 1, 2003, pp. 21–31. Translated from Problemy Peredachi Informatsii, No. 1, 2003, pp. 24–35.
Original Russian Text Copyright
2003 by Vovk, Shafer.
to the Foundations of Probability
V. G. Vovk
and G. R. Shafer
Department of Computer Science, Royal Holloway, University of London
Rutgers School of Business – Newark and New Brunswick
Abstract—Andrei Nikolaevich Kolmogorov was the foremost contributor to the mathematical
and philosophical foundations of probability in the twentieth century, and his thinking on the
topic is still potent today. In this article we ﬁrst review the three stages of Kolmogorov’s work
on the foundations of probability: (1) his formulation of measure-theoretic probability, 1933; (2)
his frequentist theory of probability, 1963; and (3) his algorithmic theory of randomness, 1965–
1987. We also discuss another approach to the foundations of probability, based on martingales,
which Kolmogorov did not consider.
The exposition of this paper is based on the ﬁgure (see ﬁgure).
The center of the ﬁgure is Kolmogorov’s earliest formalization of the intuitive notion of proba-
bility in his famous book  in 1933. This formalization, measure-theoretic probability,hasserved
and still serves as the standard foundation of probability theory; virtually all current mathematical
work on probability uses the measure-theoretic approach. To connect measure-theoretic probability
with empirical reality, Kolmogorov used two principles, which he labeled A and B. Principle A is a
version of von Mises’s requirement that probabilities should be observed frequencies. Principle B is
a ﬁnitary version of “Cournot’s principle,” which goes back to Jacob Bernoulli  (1713) and was
popularized in the 19th century by Antoine Cournot.
The goal of Kolmogorov’s later attempts to formalize probability was to provide a better math-
ematical foundation for applications. In Section 3 we discuss his frequentist theory of probability
and in Section 4, his algorithmic theory of randomness.
Another strand of work also derived from von Mises’s ideas. In his 1939 book , Jean Ville
proposed an improvement on von Mises’s approach that used game-theoretic ideas going back to his
famous compatriot Blaise Pascal. The resulting notion of martingale was never used by Kolmogorov
in his studies of the foundations of probability; he developed von Mises’s deﬁnition in directions
quite diﬀerent from Ville’s. In Section 5, we discuss this strand of research, including our recent
suggestion  to base the mathematical theory and interpretation of probability directly on the
notion of martingale.
We conclude by discussing the usefulness of Kolmogorov’s algorithmic randomness. We argue
that, even though its usefulness as a framework for stating new results about probability might
be limited, it has a great potential as a tool for discovering new facts. We describe in detail an
example from our own research.
Supported in part by the EPSRC, grant no. GR/R46670/01; BBSRC, grant no. 111/BIO14428; EU, grant
no. IST-1999-10226; and NSF, grant no. SES-9819116.
2003 MAIK “Nauka/Interperiodica”