Chain Reaction


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.

Now: The mathematician Andrey Markov had a largely forgotten contemporary named Pavel Nekrasov, who was fond of making the following argument:

  • 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.

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38 Responses to “Chain Reaction”


  1. 1 1 David Wallin

    It is always funny when people try to build “mathematical models” to prove their religious beliefs (e.g., specified complexity). But, maybe they serve their purpose: give comfort in to those who will believe anyway. I would think they know they’ll never convince the rest. But, that is the fun; sometimes I think they think they may convince the nonbelievers.

  2. 2 2 Harold

    As stated a half-wit might take exception to most steps.

  3. 3 3 Andy

    I like the Pavel Nekrasov link, at least I now know what he looked like!

  4. 4 4 Andy

    Ah, should have guessed he was the weird-looking one in your two pictures…

  5. 5 5 thomasblair

    Color me a half-wit, then – I don’t see the problem with statement 1.

  6. 6 6 Ken B

    Wait no longer! The fact that Markov, seeking to refute the claims about christianity, was lead unknowingly to create beautiful and useful mathematics that makes the world and our lives better is a miracle.

  7. 7 7 Harold

    Statement 1 is not a step. Also, I think IF all the other things were true, the last conclusion is valid.

    I think Statement 1 says that if we toss coins we get about 1/2 and 1/2. This essentially is the law of large numbers, and therefore “fact”.

    The false inference is that this proves that events often do not influence eachother.

    As I understand it, Markov showed that the law of large numbers can apply even if events do influence eachother. “Thus independance of quantities does not constitute a necessary requirement for the existence of the law of large numbers” This makes the initial inference invalid. QED.

    It seems to me that Nekrasov could have said “we know for a fact that events sometimes do not influence eachother” This is as much a “fact” as the other. We do not know the law of large numbers works forever, or over extremely long timescales, or at extreme detail. Maybe the coin takes 10,000 years to “hear” the first one, or the odds of the next toss are shifted by a miniscule amount. Presumably there are more compelling reasons why he framed it in this way.

    The next statement “The existence of events…” How is this an expression of free will? Event 1 – star goes nova in galaxy A. Event 2 – Another star goes nova in galaxy B at about the same time. No influence as it is impossible without faster than light travel, but why is this an expression of free will? I am sure there are subtleties in the argument not expressed in the version here.

    And why is expression of free will confirmation of the truth of christianity? I can’t even begin to figure that one.

  8. 8 8 Manfred

    It says in Wikipedia that the Russian Orthodox Church did ex-communicate Markov.
    “In 1912, Markov, protesting Leo Tolstoy’s excommunication from the Russian Orthodox Church, requested that he himself be excommunicated. In response, the Church formally excommunicated him.”

  9. 9 9 Steve Landsburg

    Harold:

    As I understand it, Markov showed that the law of large numbers can apply even if events do influence eachother. “Thus independance of quantities does not constitute a necessary requirement for the existence of the law of large numbers” This makes the initial inference invalid. QED.

    Yes, exactly right. And the remainder of your comment is, in my opinion, exactly dead on correct in every particular.

  10. 10 10 Ken B

    Harold: “And why is expression of free will confirmation of the truth of christianity? I can’t even begin to figure that one.”

    I think I can, and annoy the believers too. Such a twofer I cannot resist.

    If you look at the arguments christian apologists put forward they often turn on a false dichotomy: christianity vs everything else, with the everything else being conflated into one thing. (Just browse some of the discussions of the argument from multiplicity, or the anthropic principle on Bob Murphy’s blog for examples.) Now in this mindset we have on the one hand a cold Laplacian clockwork universe with no room for free will, and on the other christianity. So if free will exists and cannot exist in Laplace’s clockwork then (because all alternatives to christianity are one) christianity is vindicated.

  11. 11 11 Roger

    Christianity has everything to do with free will. The prominent scientific atheists, such as Sam Harris, spend much of their time arguing against free will. Harris just wrote a book on the subject. Just today, Jerry Coyne wrote free will is pushed largely by the faithful.

  12. 12 12 nobody.really

    Three thoughts:

    1. It’s not obvious to me that we need a concept such as a Markov Chain to demonstrate that the Law of Large Numbers does not imply that events are uncorrelated. Yeah, if a flipped coin could send a “follow me!” signal, I would expect to find that the outcome of multiple coin flips would tend to favor the outcome of the first coin flip. But what if a flipped coin sent the opposite signal: “DON’T follow me!”? Between any two flips I would tend to find a negative correlation (rather than no correlation), but in aggregate I would still expect to find a 50/50 distribution of heads and tails. Thus, auto-correlation would still be consistent with the Law of Large Numbers.

    2. Karl Pearson introduced the concept of a Random Walk in 1905; Markov introduced his chain in 1906. What’s the difference?

    3. Random walking far afield, which series of events are NOT part of a Markov Chain?

    If I’m on 2d base with no outs, and no other runners, then my next move is arguably part of a Markov Chain. It doesn’t matter if I got to 2d base by hitting a double, or by hitting a single and stealing 2d, or by getting a walk and stealing, or by getting hit by a wild pitch and stealing, or by hitting a single and having the 1st basement fumble the ball, or by hitting a triple but then stopping at 2d inadvertently, or whatever: My future options and probabilities of success are pretty much identical under any of these scenarios.

    In contrast, when is my next move influenced by my past moves in a manner that is NOT reflected in my current state?

  13. 13 13 Ken B

    Et voila. Barely has the ink dried on comment 10 when comment 11 comes rushing in!

  14. 14 14 Ken B

    Are posters here superstitious?? I seem to be able to bag comment 13 nearly every time …

  15. 15 15 Roger

    Ken B, if you always get 13, that raises the question of whether the Law of Large Numbers applies, whether you are influenced by comment 12, whether you had the free will to post, and whether others had free will to compete with you for that comment. Since you reference comment 11, I am guessing a Markov chain with each comment depending on the previous 2 comments.

  16. 16 16 Martin-2

    What makes you think Bob Murphy has a second opinion? Just because someone believes in God doesn’t mean they want to defend every proof ever attempted.

    By the way, I’m not sure I have the right definition of free will. From what I can tell a “free willed” decision must be-

    1. Unpredictable no matter how much information is given
    2. Not random

    That’s silly. Even divine intervention wouldn’t help you out of that hole.

  17. 17 17 Drew

    Step One in proving a statement is true, maybe even Step Zero, is having some idea what the statement actually says.

    This step has never, that I’ve come across, been successfully taken in the case of Free Will. Every explanation of what it is I’ve ever heard boils down to something so trivial and obvious that it can’t really be what claimants are claiming (i.e., because no one put a gun to my head to force me to do something, I had free will in choosing to do it), or completely incoherent/self-contradictory (I am able to make choices that are not determined by anything… which taken literally includes EVERYTHING SPECIFIC TO WHO I AM… which makes it kind of hard to see how it was _my_ choice in the first place).

    The whole purpose of Free Will in terms of theology appears to be to try and cut the chain of causality that would ordinarily run from a Creator God and his creations, absolving him of any responsibility for them acting ways that are consistent with how he made them.

    But this gets really silly fast. If God thinks extramarital sex is bad, and we truly have Free Will, then why give us very specific biological mechanisms that translate out into huge biological urges? Maybe it’s because we are cursed with temptation… but wait a minute: if we have “Free Will” why should temptation MATTER as to whether we’re horndogs or not? Aren’t we choosing everything freely? Why would that or any natural inclination have a direct effect on anything? And yet, it clearly does: there people with naturally very very low libidos (often due to unusual hormone conditions) who in fact don’t have lots of extramarital sex. Why aren’t they choosing to exercise their Free Will to have extramarital sex despite their low libidos? Why do treatments that radically reduce these hormones in normally horny people then also affect not only their libidos, but also how much sex they have or want?

    And so on…

  18. 18 18 iceman

    It seems to me a presumption of free will is required to judge behavior by *any* sense of morality. But it’s not at all clear how the existence of free will would prove the truth of any particular moral code.

    Drew – good questions briefly considered, but maybe for present purposes it’s sufficient to just consider “free will” broadly as the alternative to the “clockmaker” view of God, i.e. we are capable of providing some self-restraint (so yes no one put a gun to your head) around certain biological urges that might not be ‘good’ for us (physically or otherwise). Some might even say it’s primarily *our choices that MAKE us who we are*? (I think Albus Dumbledore had a nice line about this.) E.g. setting God aside, your wife might not approve of your extramarital sex, e.g. because her particular biological urges led her to seek a mate that would be faithful and supportive as you represented you would be? I would think most moral codes, including those that endorse polyamory, tend to take a negative view of dishonesty.

  19. 19 19 Bob Murphy

    This is really interesting Steve. I’m glad the Holy Spirit led you to post this.

    Actually, Markov chains were one of my favorite things that I learned in the evil mathematical econ program at NYU.

    I would have to delve into the specifics before rendering a final opinion (“I hate to judge on something like this before all the facts are in,” as General Turgidson would say), but if his argument is what you say, it seems pretty weak. However, if he didn’t use the word “confirmation” but instead said “evidence of” then it might not be so bad.

  20. 20 20 Bob Murphy

    To be clearer, I bet most readers here think the “Intelligent Design” movement is utterly absurd. In contrast, I think some of its leaders have made very brilliant points. Part of the reason for our difference is that the people who think ID is silly often just rely on secondhand accounts of what their position actually is, rather than slogging through the technical expositions of its leaders.

  21. 21 21 Bob Murphy

    Oops sorry one final clarification: The above might make it sound like I’m saying “Steve Landsburg is lazy and hasn’t delved into the specifics.” That’s not what I’m saying. Rather, *I’m* too lazy to delve into the actual guy’s arguments, so maybe if I read him in the original I would see something that Steve is overlooking.

  22. 22 22 Steve Landsburg

    Bob: I’m also too lazy. All I know is what’s in the paper I linked to.

  23. 23 23 Bob Murphy

    Oh OK Steve. I guess that’s why we’re armchair economists.

  24. 24 24 Mike H
  25. 25 25 Mike H

    @nobody.really “In contrast, when is my next move influenced by my past moves in a manner that is NOT reflected in my current state?”

    This, perhaps, depends on how you define “state”. If you use the word “state” in the same way a physicist does, the answer might be “never”. On the other hand, it’s often useful to simplify a problem to make it tractable. So if “state” is “where I went on holidays last summer”, the probabilities for next summer might depend not just on the current state, but also on earlier states : “We haven’t gone to the mountains for ages! Can we please go this time???”

  26. 26 26 Darzil

    “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.”

    I’d suggest that this isn’t a Markov process, as where you go next depends on your current facing, which depends on how you got there. However, if you changed turning left or right to taking a random junction, I might agree.

  27. 27 27 Harold

    John Calvin back in the reformation preached predestination. Calvinism seems to reject free will, but is Christian. The idea that free will be definative about Christianity seems to be flawed.

  28. 28 28 Dave
  29. 29 29 Dave

    Bob Murphy: “I bet most readers here think the “Intelligent Design” movement is utterly absurd”

    I do but would be interested in any links that you might provide that could sway me.

  30. 30 30 Steve Landsburg

    Darzil:

    I’d suggest that this isn’t a Markov process, as where you go next depends on your current facing, which depends on how you got there. However, if you changed turning left or right to taking a random junction, I might agree.

    Point taken. Thank you.

  31. 31 31 Al V.

    I like the pictures at the top of the post. Is Markov the handsome one with with piercing eyes, or the crazy looking one with the wild eyes? I can guess, but that might demonstrate my prejudices.

  32. 32 32 Winston Marley

    What is the probability that a random variable X satisfies the Law of Large Numbers?

  33. 33 33 Steve Landsburg

    Al V:

    Is Markov the handsome one with with piercing eyes, or the crazy looking one with the wild eyes? I can guess, but that might demonstrate my prejudices.

    You ought to be able to get the answer by mousing over them.

  34. 34 34 Roger

    Harold, Calvinism is the exception that proves the rule. Calvinism is considered exceptional because it differs from mainstream Christianity.

  35. 35 35 David Wallin

    Roger, I’m sorry but it is a pet peeve. Exceptions do not prove rules. In the former use of “to prove” to mean “to test” (like in proving grounds and to proof the yeast), an exception proved (tested) the rule. And, in the case of absolute rules (all swans are white), the rule failed with one exception. An exception never “helps” the rule’s veracity.

  36. 36 36 Ken B

    @David Wallin (& Roger): That expression goes back in English a long way, and the ‘proves’ there is in its older sense of ‘probes, tests, weighs’. So the saying means really, “an exception helps clarify the extent and appplicability of the rule.”

    I belong to the pedantry of the day club and this fulfills my quota!

  37. 37 37 James Knight

    In his book The Big Questions Steve Landsburg says we do have free will. Well I usually submit the following challenge to those who think we do have free will, and I’ve never heard a satisfactory answer yet. I say don’t believe anyone who says we have ‘free will’ until they have first told you what they mean by free will, and what being ‘free’ means in a universe that drives organisms thermodynamically without the aid of human thought. I think you’ll be surprised how infrequent it is that anyone can even explain free will in terms of what I just asked for.

    I’d be pleased if Steve or any of his pro-free will commentators could defend free will with regard to the above.

  38. 38 38 Ken B

    @James Knight: What if they choose not to?

    ‘Free will’ I think means something about your mental apparatus reacting to logic and stimuli, so that your actions can be explained only in terms of mental states explicable and comprehensible to others, and amenable to change based on information. If you have Tourrette’s Syndrome your cursing is not the result of ‘free will’ in this view, but if you read John Roberts’s opinion your cursing probably is.

  1. 1 Free to Choose at Steven Landsburg | The Big Questions: Tackling the Problems of Philosophy with Ideas from Mathematics, Economics, and Physics
  2. 2 Dominic Yeo
  3. 3 Invariant Distributions of Markov Chains | Eventually Almost Everywhere
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