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COMPUTING SCIENCE

First Links in the Markov Chain

Probability and poetry were unlikely partners in the creation of a computational tool

Brian Hayes

Peterburgian Math

In trying to understand how Markov came to formulate these ideas, we run straight into one of those long chains of contingent events extending deep into the past. One place to start the story is with Peter the Great (1682–1725), the ambitious Romanov tsar who founded the Academy of Sciences in St. Petersburg and fostered the development of scientific culture in Russia. (Other aspects of his reign were less admirable, such as the torture and murder of dissidents, including his son Alexei.)

At roughly the same time, elsewhere in Europe, the theory of probability was emerging from gambling halls and insurance brokerages to become a coherent branch of mathematics. The foundational event was the publication in 1713 of Jacob Bernoulli’s treatise Ars Conjectandi (The Art of Conjecturing).

Back in St. Petersburg, the mathematics program prospered, although initially most of the accomplishments were by imported savants. Visitors to the Academy included two younger members of the Bernoulli family, Nicholas and Daniel. And the superstar was Leonhard Euler, the preeminent mathematician of the era, who spent more than 30 years in St. Petersburg.

By the 19th century indigenous Russian mathematicians were beginning to make their mark. Nikolai Lobachevsky (1792–1856) was one of the inventors of non-Euclidean geometry. A few decades later, Pafnuty Chebyshev (1821–1894) made contributions in number theory, in methods of approximation (now called Chebyshev polynomials) and in probability theory. Chebyshev’s students formed the nucleus of the next generation of Russian mathematicians; Markov was prominent among them.

2013-03HayesFB.jpgClick to Enlarge ImageAndrei Andreevich Markov was born in 1856. His father was a government employee in the forestry service and later the manager of an aristocrat’s estate. As a schoolboy Markov showed enthusiasm for mathematics. He went on to study at St. Petersburg University (with Chebyshev and others) and remained there for his entire career, progressing through the ranks and becoming a full professor in 1893. He was also elected to the Academy.

In 1906, when Markov began developing his ideas about chains of linked probabilities, he was 50 years old and had already retired, although he still taught occasional courses. His retirement was active in another way as well. In 1883 Markov had married Maria Valvatieva, the daughter of the owner of the estate his father had once managed. In 1903 they had their first and only child, a son who was also named Andrei Andreevich. The son became a distinguished mathematician of the Soviet era, head of the department of mathematical logic at Moscow State University. (To the consternation of librarians, both father and son signed their works “A. A. Markov.”)

An intellectual thread extends all the way from Jacob Bernoulli through Chebyshev to Markov. In Ars Conjectandi Bernoulli stated the law of large numbers, which says that if you keep flipping an unbiased coin, the proportion of heads will approach 1/2 as the number of flips goes to infinity. This notion seems intuitively obvious, but it gets slippery when you try to state it precisely and supply a rigorous proof. Bernoulli proved one version; Chebyshev published a broader proof; Markov offered further refinements.

Markov’s later studies of chains of dependent events can be seen as a natural continuation and generalization of this long line of work. But that’s not the whole story.








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