If you study economics, or statistics, or chemistry, or mathematical biology, or thermodynamics, you’re sure to encounter the notion of a Markov chain — a random process whose future depends probabilistically on the present, but not on the past. If you travel through New York City, randomly turning left or right at each corner, then you’re following a Markov process, because the probability that you’ll end up at Carnegie Hall depends on where you are now, not on how you got there.
But even if you work with Markov processes every day, you’re probably unaware of their origins in a dispute about free will, Christianity, and the Law of Large Numbers.
The Law of Large Numbers says (roughly of course) that if you flip a very large number of fair coins, and if no coin exerts any influence on any other coin, the fraction that come up heads will be very close to one-half. The Law could fail, however, if the coins somehow affect each other’s “choices”. For example, if the first coin lands heads and yells “follow me”, and all the other coins attempt to conform, then the fraction of heads is likely to be much more than a half.
Coins, of course, don’t behave that way, but other random processes do. If all vacationers are a priori equally likely to choose the beach or the mountains, but if they all like to go where the crowds are, then one vacation spot is going to draw all the tourists.
- In point of fact, we know by observation that the Law of Large Numbers often holds.
- Therefore, we can conclude that events often do not influence each other.
- The existence of events that do not influence each other is an expression of free will.
- The existence of free will is a confirmation of the truth of Christianity.
- Therefore the empirical validity of the Law of Large Numbers is evidence for the truth of Christianity.
A reasonable person might take issue with pretty much any step in this argument, but Markov was sufficiently annoyed by it (and, apparently, by pretty much everything else about Nekrasov) that he wanted to make sure every step was refuted. He was therefore fond of pointing out that, just because the non-influence condition implies the Law of Large Numbers, it does not follow that the Law of Large Numbers implies the non-influence condition. When Nekrasov failed to get the point, Markov was motivated to write down an explicit example of a random process in which events do influence each other, but the Law of Large Numbers nevertheless holds. And so the theory of Markov chains was born.
More details of this odd and little-known story are here. I can’t wait for Bob Murphy to weigh in on this.