Tuesday, July 11, 2017

Probability is about our knowledge...

and not a fixed feature of the world "out there."

A couple members of the commentariat I have complained that in this model, it is necessary to have "inside knowledge" to beat someone who thinks the odds are 50-50 on any given shot. Now, I don't care whether you want to call what "Gene" knows in that model "inside knowledge" or not. Either way, that is missing the more important point: "the odds" change with our knowledge of a situation.

To illustrate: imagine I ask you to predict the odds that an American, male, 40-year-old will live to be 78? Well, if that is all the information you have, you should answer "Even odds." (I looked that up, but from here on out my odds are all just plausible-sounding guesses.)

But now I tell you, "Oh, and he's a heavy smoker."

Oops, better revise that forecast: say, 2-1 against.

But then I add, "And so were all of his deceased male relatives that we can identify, and they all lived to be at least 90."

Aargh, now the odds are 2-1 in favor.

However, I finally add "By the way, he has terminal pancreatic cancer, and the doctors only give him a month to live."

Now you had better revise your odds to 1000-1 against.

Supposing that my guesses after the first odds I gave are accurate, your answer each step of the way was "correct," given the knowledge you had at hand. When we know more about a situation the odds change. And it doesn't matter at all whether this is "inside knowledge" or not.

This applies even to something as seemingly straightforward as a claim that, in a flip of a fair coin, the odds are 50-50 of getting heads. If we could somehow see all the forces at work in a particular flip, we would be able to state with certainty, "This toss is going to be heads (or tails)." And, in fact, it turns out that with practice, a person can learn to flip a coin so that it almost always comes up in its original orientation, or vice versa. If all we know is that we have "a person" flipping a fair coin, it is correct to say the odds are 50-50 for getting heads. But if we learned we were dealing with one of these skilled coin flippers, and we had a reason to think he was trying to produce heads, we would instead be correct to say that the coin would come up heads with near certainty.

An application: the above considerations are why a simple mastery of the odds of drawing various card hands are not enough to make one a top poker player. The top players have of course internalized that knowledge, but they have gone much further: they have learned to read the "tells" of less skilled players, so that they can see from the reaction of an amateur whether he has just completed his full house or not. Once they can do that, the formal odds of his having drawn the card he needed become irrelevant: they know whether or not he got it. This is not "inside knowledge": the tell was right out in the open, for anyone to see. But only someone practiced at looking for it will recognize it as information to be used in betting.


12 comments:

  1. The odds are what they are, regardless of what we know. Our knowledge of them doesn't affect them, it just affects whether or not we're right about them.

    ReplyDelete
    Replies
    1. "The odds are what they are, regardless of what we know."

      Well, I just wrote a whole post proving this is wrong. I don't think "Nope" is really a very good counter-argument.

      If we were omniscient, we would just know what will happen. The whole concept of "odds" only exists because we are not.

      Delete
    2. Well, you wrote a whole post conflating "the odds" with "a person's estimate of the odds."

      Or at least that's the way it reads to me; but maybe you're subtly referencing some concept that's just way over my head.

      Delete
    3. One caveat: In SOME circumstances, our knowledge of the odds can affect the odds. For example, a 1 in 20 chance of living through some disease becoming a 19 in 20 chance because our knowledge led us to administer medication.

      But apart from that, generally speaking: If there is a 50% chance that the coin is going to come up heads, there is a 50% chance of that coin coming up heads _whether I know there is a 50% choice of that coin coming up heads or not).

      Delete
    4. Wrong Tom: the odds of a coin coming up heads being "50%" is simply a statement of our ignorance concerning exactly how it was flipped. In fact, given the exact nature of the flip, there are no "odds" at all: it either WILL come up heads or it WILL come up tails.

      Delete
    5. "The odds" just ARE a person's best estimate of the odds. There is nothing else there at all.

      You have hypostatized a tool of human thinking and shoved it out into the world as if there were some magical leprachaun forcing half of all coin flips to come up heads.

      Delete
    6. "Well, you wrote a whole post conflating "the odds" with "a person's estimate of the odds."

      As you may note in my recent post, some of the greatest pioneers of probability theory said exactly what I am saying here.

      Delete
  2. But its possible to imagine a non-deterministic world where probability is a feature of the world "out there" , right ?

    Take someone for whom the world "out there" is a version of your basketball hot-hands model and the randomizer is truly random (uses quantum indeterminacy or something). Even if they developed a perfect understanding of their world they would have to conclude that that odds on a basket were either 30% or 70% but could never call it for sure.










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  3. OK, _now_ I understand what you are saying and agree that you are correct.

    ReplyDelete
  4. Gene, sometimes probability is just a subjective measure of belief, but sometimes it is not. The clearest places where it is not is in quantum mechanics. It seems that the odds in quantum theory really are objective.

    ReplyDelete
    Replies
    1. 1) I hold that purely subjective beliefs are impossible: all belief is in fact aiming at objectivity. (See Oakeshott, Experience and Its Modes, and Polanyi, Personal Knowledge.)
      2) I know about QM, Ken. The jury is out, but your interpretation is certainly popular with physicists. (In other words, it is a sensible belief, although not everyone accepts it, e.g., Bohm.)

      Delete

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