Wolfram in Chains

The 100 year anniversary of Andrei Markov's presentation on Eugene Onegin to the Imperial Academy of Sciences in St. Petersburg has come and gone. (January 23 was the date.) It was during this presentation that he first proposed the mathematical model we now call, after him, a Markov chain. Markov chains represent a first go at introducing structure into a stochastic process. (The zeroth order go being a sequence of independent random variables.) Basically, if you have a sequence of random variables \(X_{1}, X_{2}, X_{3}, \ldots\), they form a Markov chain (of first order) if \(P(X_{t+1} | X_{t}, X_{t-1}, X_{t-2}, \ldots, X_{1}) = P(X_{t+1} | X_{t})\) for all \(t\). In words, the future \(X_{t+1}\) is independent of the past \(X_{t-1}, X_{t-2}, \ldots\) given the present \(X_{t}\) . Or, as a graphical model1,

The other site I saw mention this was the Wolfram Blog. Which made me wonder:

If Wolfram had invented Markov chains, would he have patented them?

What do I mean? Stephen Wolfram patented one of the cellular automata he invented. And then sued a researcher that used that cellular automata in an academic paper.

Which begs the question: how much money could Markov have made off of his chains?

They show up in

  • speech recognition
  • weather prediction
  • gene annotation
  • Google's PageRank algorithm
  • the generation of fake papers

amongst many other places2.

He'd certainly be rich.

Thankfully, he had better things to worry about:

By subscribing to them I can only weaken that which for me is particularly dear: the rigor of judgements I permit.

For example, it is very important to me to observe that Poisson did not prove his theorem; moreover, I cannot consider a statement a theorem unless it is established precisely.

-- from a letter by Andrei Markov (probabilist) to Alexander Chuprov (mathematical statistician), quoted in The Life and Work of A. A. Markov

  1. Now you can see why these are called Markov chains.

  2. Including, of interest to me, my own research.