## Wednesday, October 10, 2007

### More math psychology

Did I really let almost a month elapse without posting? Must be because my life is so wild and exciting. Um, yeah.

Here's a question that's been occupying me for a while -- non-mathematicians' perception of probability. What does a man on the street mean when he says a coin has a 50% chance of landing on heads?

The most likely interpretation is frequentist: if you flip the coin a whole lot of times, you'll see heads about half the time, on average. (How many times is a lot? What does on average mean here? Read my thesis -- at least the section on the Law of Large Numbers.)

But what about one-time events? What does it mean that there's a 30% chance of rain tomorrow? (Tomorrow will only happen once, so any talk of averages is meaningless.) Heck, what does the meteorologist mean by that probability?

Cosma helped me resolve the latter quite satisfactorily in private correspondence (so satisfactorily, in fact, that I feel dumb ever having asked the question). But I turn to the readers:

1. can you make rigorous mathematical sense out of the meteorologist's 30% chance of rain prediction?

2. can you ask your non-mathematician friends what that prediction means to them?

Kenny said...

Is it really clear that the person means a frequentist interpretation when talking about the coin? It seems just as plausible that they mean some sort of bayesian interpretation, like "there's no evidence in favor of one or the other" or "I'd be willing to bet on either at the same odds". The problem with the coin case is that all these interpretations actually give the same value, so we can't tell which the person is talking about.

With one time events, I think the Bayesian interpretation makes more sense, though there's presumably some sort of frequentist gloss available as well (even if it's more complicated in that case).

And of course, all this ignores chance interpretations of probability, on which something about physical law is actually non-deterministic, in a way that it makes sense to associate numbers with (as in quantum mechanics). Then you can get single-case chances from the physical laws, even if frequencies can't apply to the single case. (And I think people will probably claim in the coin case that they're talking about chances, although they'd eventually be forced to admit that they don't really believe physical indeterminacy plays a relevant role here.)

Geet Duggal said...

I appreciate you bringing that up...funny how those two seemingly simple questions defy an easy answer...not to say I have that answer, but it got me looking around a little.

1. On the kick that some form of numerical prediction for atmosphere dynamics had a role in the method that determined this chance, I came across this technique of ensemble forecasting where it appears that errors due to well-behaved noise as well as that due to the nonlinear/chaotic nature of the system are handled in such a way (perhaps by repeated numerical simulation) that the probability of an event (e.g. rain) occuring can be obtained. How good is that model for inference? I wonder how well-behaved this aggregate noise is and is this actually the way that modern more sophisticated weather systems use ensemble forecasting to give us a percentage chance of rain?

2. How people interpret this seems quite varied. There even seems to be a journal article on the topic. The frequentist interpretation does come up more than once in these posts...even for the one-time event of rain (though it's really weird how the posts are trying to explain it).

My hunch is that proper statistical inference methods (Bayesian or frequentist) for such chaotic systems are still in their infancy. But I'm just talking crap now and I'm curious if that's true at all.

Any thoughts welcome...

Aryeh said...

Kenny --

I agree that if a person (any reasonable person) is asked to guess the bias of an unknown coin, 50/50 is the most rational guess; and the rationale is perfectly Bayesian. What I was asking, however, is the following: What does a non-mathematician mean when he claims that a coin is 50/50?

Likewise, chance probabilities make perfect sense to a mathematician. I'm perfectly happy to define "probability" as "a positive Borel measure normalized to 1" and not think about the philosophical implications. However, I'd be curious to know what a physicist means when he says that a photon will go through the left slit with probability 1/2. Multiple universes serve us nicely here, and I don't know of any other "physical" interpretation.

Geet -- thanks for the comments. I'll let some more readers have a stab at it over the weekend, and then ask Cosma's permission to quote the relevant excerpt from the email.

Unknown said...

It seems reasonable to me to suppose that a man in the street will interpret the statement "there is a 30% chance that it will rain tomorrow" by comparison (or analogy) to the situation where there is an urn containing 10 balls, 3 of which are black and the rest are white. Then the "chance" that it rains is the same as the "chance" that you pull out a black ball without looking. Note that there are no repeated experiments here. If I recall correctly, an explanation of this sort was offered by John Hartigan in his excellent book Bayes Theory.

Aryeh said...

Right -- this corresponds to an implicit "multiple universes" interpretation of the physical world. Otherwise, what are we to make of these "urns"?

This may well be how humans intuitively interpret physical probabilities.

Geet Duggal said...

The example above with 3 balls out of 10 being black and the rest being white brings up a distinction that is small but has some affect on how I conceptualize things:

The connection to 30% probability comes as a result of the ratio of the black balls to the total number. It's usually given that we draw samples from the urn uniformily. I think this explanation makes sense.

Another way to think about it is grabbing the topmost ball from the urn and between every grab the urn shuffles the balls randomly.

In the case that we draw samples from the urn uniformily, we abstract ourselves away from the process that may conceptually "order" these balls, and it is quite a jump to say "drawing balls from an urn uniformily or with no bias".

Perhaps it's more likely that to some people, a 30% chance of rain is more like the god of nature shuffling a deck of cards and picking the top one...not necessarily implying the 3/10 fraction of white balls.

(By the way, I neglected to mention that that based on the journal article linked to above, that the percentage rain actually reported by common weather services is usually the fraction of days in the past with the same conditions where rain occured...very frequentist.)

Anonymous said...

I see this as just another nonsensical question brought about by the metaphysical obsession about "existence", where does the hypothetical "non actually occuring cases" live?
They have to "exist" somewhere in some "alternate universe", LOL...
There is no need to answer such questions in order to proceed with a rational action like placing a bet.
There is no "meaning" in a rational bet, yet it can still be qualified as rational, can't it?
For those lingering in anguish about what "is" and "is not" there are still people working on alternative foundations of logic and maths : Nik Weaver's Mathematical conceptualism.
But I doubt such research will soothe the fears of the metaphysic oriented nor have much import on actual mathematical practice.

Aryeh said...

Kevembuangga,

you seem to be saying that all "physical" probabilities only have a psychological interpretation, and no inherent physical one. I take particular issue with your claim that "There is no 'meaning' in a rational bet" -- certainly some bets are more rational than others!..

Let me post Cosma's crystal clear explanation of probabilities assigned by models:
<<
The model employed by the forecaster involves only certain observables, thereby defining a filtration which is smaller than the natural filtration of the weather-process.

Condition, and you have probabilities, saying "given observations like this, we will see rain x% of the time". If this is correct, one expects that there will be rain on 30% of the "30% chance of rain" days, etc., which is pretty close to the case.)
>>

Anonymous said...

leo
I take particular issue with your claim that "There is no 'meaning' in a rational bet" -- certainly some bets are more rational than others!..

Likely a matter of terminology.
I do agree that "some bets are more rational than others" in that they allow you to win a larger reward.
What I am saying is that it does not matter if you made your bet by "believing" in a bayesian or frequentist interpretation and therefore winning by being rational doesn't settle the dispute between the two schools of thought which I still see as a vain metaphysical quibble.

In the same vein I see most foundational questions in maths and logic (platonism v/s nominalism, v/s realism, etc...) widely off the mark with respect to actual practice.
I think that actual practice, in maths as well as in many other intellectual or artistic endeavours, entails some inscrutable evidence that we are absolutely at loss to explain.
What is an "elegant proof", Eh?

Unknown said...

Leo --

I would like to add to Cosma's very crisp explanation by saying that the "30% chance of rain" statement made by the weather forecaster should be interpreted in the context of the so-called "prequential principle" (see "Prequential probability: principles and properties" by A.P. Dawid and V.G. Vovk). The prequential principle says, roughly, that a good forecasting strategy should behave empirically as one would expect from the "true" probabilistic model. For example, if the underlying statistical law satisfies, say, the law of large numbers or the law of iterated logarithm, then the forecaster's strategy (which is a function of past observations only, just like Cosma pointed out) should also satisfy these laws.

The second paragraph in your quotation of Cosma's explanation states exactly that. Moreover, when Geet says that "the percentage rain actually reported by common weather services is usually the fraction of days in the past with the same conditions where rain occurred," he is invoking the prequential principle as well.