2012 UEFA European Football Championship And What Probability Is

A pair of statisticians have calculated that, as of 10 June, the probability that Germany takes the 2012 Euro Cup is 26.8%. Russia, they say, comes in with the next highest probability at 17.5%. Poor Ireland was given only a 0.1% chance to win. (I am rooting for Germany and Poland.)

Magne Aldrin and Anders Løland say that

The probabilities are found by simulating (“playing” on a computer) all remaining matches numerous times. The remaining part of the tournament is “played” 10,000 times and from these simulations we calculate all the teams’ chances of winning the Championship, their group, to reach the second round etc.

In other words, they have some model which takes in information selected by the pair and out spits the probabilities. The natural question many want to ask is: are these the right probabilities? And the answer is: yes, as long as they haven’t made any errors in the calculations or in typing out the results for press.

Now compare the MAAL (authors’ initials) probabilities with those produced by the patented Briggs Soccer Model-O-Matic. This model says that Germany has a 6.3% chance of winning. But it also says that Russia has a 6.3% chance; and so does Ireland. So do each of the 16 teams; indeed, they all have equal probability of winning. The natural question now to ask is: are these the right probabilities? And the answer again is yes.

This sounds nuts. How can the MAAL and the BSMOM probabilities both be right? Shouldn’t there exist, somewhere out there, one final set of probabilities which we can discover, a true set which we can all know? Well, no and yes.

The MAAL probabilities are true assuming the model which produced them is true. The BSMOM probabilities are true assuming the model which produced them is true instead. Which model is truly true? Well, if we knew that there would be no use deriving different models. And the MAAL and the BSMOM aren’t the only models in contention: each bookie has a different one, and probably so do other statisticians.

Consider what is happening here. All probabilities (indeed, all statements of knowledge) are conditional on propositions which are known or assumed to be true. These propositions are what the models are. I mean, any logically different set of propositions, which includes observational propositions (e.g. “Germany won so many games last year”), make up what a model is.

For example, the BSMOM model, i.e. its list of propositions, are this: “There are 16 teams in contention (and here is the list), just one of which can win.” From assuming this model is true we deduce the chance that Germany wins is 6.3%, etc.

The MAAL model’s list of propositions is longer and more complex, but in the end it is just a list of propositions which we assume is, or really is, true and thus let us deduce the chance each team will win. All statistical models are the same: mere lists of propositions we assume are true, or really are true, from which we deduce probabilities of events.

There is still a sense that the MAAL is a better model than the BSMOM, however. We have the feeling that, for any situation, there exists what we can call the Omniscient Model. This is a list of propositions which are true and which lead us to deduce that the chance that (in this situation) Germany wins is either 0% or 100%. Since the propositions which make up the OM are true, the chance that Germany wins is 0% or 100%.

There always exists an Omniscient Model, even for quantum mechanical events; the trouble is that, especially for events on the smallest scale, we rarely know what this model is (and for QM it seems we cannot know). But here is where the sense that the MAAL is better than the BSMOM arises: the closer any model’s propositions are to the propositions in the Omniscient Model’s list, the better that model will be. Surely, our gut tells us, the BSMOM is father from the truth.

Our gut is probably right, but it’s relying on its own model which says roughly, “In my experience, models like the BSMOM rarely produce useful information; while models like the MAAL are better.” We can only confirm our gut after the fact, but measuring how close the probabilities of each model are from the actual outcomes. Those measures become yet another proposition which feeds back to our gut (or into a meta-model in a formal sense) so that when we meet MAAL-like and BSMOM-like models in the future we’ll have an idea which to prefer.

Update Fixed asinine mistake and typo. Thanks Uncle Mike and Stephen D!

18 Comments

  1. There always exists an Omnipotence Model, even for quantum mechanical events; the trouble is that, especially for events on the smallest scale, we rarely know what this model is (and for QM it seems we cannot know).

    Exactly wrong. In QM there is no “Omnipotence Model”, and that’s its uniqueness. From that characteristic, things like “spooky actions at a distance” and other queer quantum phenomena mystifies our classical line of thinking.

    It’s not that there’s some “hidden” truth that the Heisenberg equation prevents our “eyes” of catching. The amazing feat of QM is that there are no hidden variables of this kind at all (this is what drove Einstein nuts). Subsequent surreal “impossible” experiments have confirmed this hypothesis numerous times.

  2. Oh and yeah, the final will be Italy vs Spain, and it will be Italy who wins. I’d love for Portugal to win it, of course, but I’m afraid that we will probably tie the game today and fail to even qualify for the quarters, let alone frighten anyone with our lacking might.

  3. Luis,

    Re: QM. You confuse our inability to know what’s causing QM events with there being no cause. Bell’s theorem does not say “no cause” or “causality suspended for QM.”

  4. Sorry, that’s just not right. Your “faith” was joined with a very long line of either geniuses that resisted the quantum revolution or more recently just plain crooks who are just unwilling to accept empirical reality. Sorry, but that’s the typical classical line of thought that QM has thrown into the trash (there must be happening *something* under the quantum bubblegum that behaves in a more “classical” way – the classical thougth goes). You have to abandon it. And come on, it’s not like this has been found yesterweek or yesteryear. This stuff almost has 100 years now, it’s older than our previous generation. So you cannot really pretend to be somewhat ignorant of these astonishing results without blushing.

  5. Luis,

    Faith forsooth. You are misunderstanding Bell’s theorem. It says no hidden variables. It does not say that causality ceases at some point. Think of what you’re saying. You imply that (say) a wave function “collapses” for no reason. Now how can that be unless there is no cause or no causality?

    The alternative is to say something, we know now what, we cannot know what, caused this event. All we can say is that whatever this something is cannot be in the form disallowed by Bell’s theorem (which says we cannot know).

    Update With Bell, we are giving up on locality, not causality. See e.g. M. Lange, An Introduction to the Philosophy of Physics, Blackwell, 2002, pp. 268-286.

  6. Wait, by “crooks” I never meant “you”. Sorry if it came that way. I was thinking more of the likes of Smolin (and others alike).

    I did not mention causality btw. I countered the notion that there is some “hidden truth” that the Eisenberg equation is “preventing us” to see (when you said that we cannot know what the omnipotent model of QM is, as if there should or could be one). QM predicts that a wave function “collapses” when it is observed by another particle or wave (like a photon used by an observer), so there’s your “cause” right there.

    With Bell, we are giving up on locality, not causality.

    Yeah.

  7. Luis,

    Gotta disagree with you on the QM thing. If there are no hidden variables then ultimately things happen at the QM level for no reason at all. In a sense that makes physics a futile endeavor since it implies all events have no cause unless the claim is QM events are completely irrelevant on a larger scale. If not, then what’s the point? I think that’s what bothered Einstein.

    QM predicts that a wave function “collapses” when it is observed by another particle or wave

    If you think about it, that’s just a peculiar way of saying the results are unknown until they are known.

  8. Well DAV, we’ll have to agree to disagree then. Things happen in a “dice roll” kinda way in subatomic level, and there seems not to be any “cause” for this to be so. Perhaps the reason is “it couldn’t have been any other way” or “the classical way would be boring”. We’ll have to ask God why he chose to play dices.

  9. Great post that underscores our need to constantly keep an eye on the pea under the thimble with models that proclaim this or that result (whether advertised as a “solution,” a “prediction,” a “scenario” or otherwise).

    I particularly like the point that if we really knew what the true propositions were then we would not need various models. This does not necessarily mean that the existence of various models is bad. After all, we’ve just conceded that we don’t know all the propositions, and there is merit in doing our best to find an answer. But we need to keep front and center before us the elemental fact (perhaps often obscured by its very obviousness?) that we indeed don’t know all the propositions and that the models are — thusly and inevitably — rough guesses at best, completely off base at worst.

  10. That was an artful switch. I take it you did not want to call the Briggs Soccer Model-O-Matic the BSMOM since it is redolent of teenage rebellion.

    What does the ‘C’ in BCMOM stand for?

  11. Stephen, yeah, I noticed that as well and thought “BS” would have sounded better than “BC”. I just chalked it up to a typo, but now that you mention it perhaps Briggs can enlighten us. Inquiring minds want to know. 🙂

  12. I will start out with an absolute truth. If you take your copy of “Speakable and Unspeakable in Quantum Mechanics” (Bell 2004) into a courtroom in the county of Los Angeles Ca – YOU WILL NOT BE SELECTED FOR JURY DUTY!
    Then – on to the debate.
    I prefer John Gribbon’s “Schrodinger’s Kittens and the Search for Reality” for light reading.
    If I get serious, I study Sakurai’s “Modern Quantum Mechanics”. It is on my book shelf
    right next to Von Mises “Probability Statistics and Truth”. (Let’s see if Briggs reacts to that one).
    What I cannot seem to find is a book I know I had by David Bohm which to this day cannot be falsified. Something about a pilot wave providing causality for the collapse of a wave function.
    {Aside. Folks – it has been 20 years since I could follow Bohm or Dirac. Cut me some slack!}
    Also.
    My memory is that Bell does not deny causality. He simply does not confirm special relativity. Spooky “action at a distance” happens. Which my poor Luis – allows something called God to exist outside of time.

  13. What you meant was the Omniscient Model. Omniscient means having complete or unlimited knowledge, awareness, or understanding; perceiving all things. Omnipotent is all powerful. There’s a difference. Not all infinities are alike.

  14. “Russia, they say, comes in with the next highest probability at 17.5%”

    Gales of riotous laughter. Russia didn’t make it out of group behind the Czech and Greek republics.

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