Podcast Lecture #3: Understanding Statistics and Probability


Today’s lecture:

Review Logic: If our premises are that we have a six-sided die that we’ll toss, and that only one side will show on that toss, and that just one side of the six has six spots, then the probability that a six spot will show given these premises is 1/6. Probability, then, is a matter of logic. Usually non-deductive logic, and not necessarily inductive logic. Whew! We finally have that out of the way.

Frequentism The largest rival to logical probability is frequentism. This is the belief that the probability of some event is the limiting relative frequency of that event. Take an event, like a die roll, toss is a number of times which approaches infinity. Count the number of times it comes up “6 spot”, then divide by the number of throws. That limiting fraction becomes the probability. But only after we get to infinity. While this is a perfectly fine mathematical definition, it has nothing to do with reality.

What is the probability that “Hillary Clinton wins the 2012 USA presidential election”? This is a contingent event, and given that information, via logical probability, it is greater than a 0% chance and less than a 100% chance. Given other assumed information, like “She will be the Democrat nominee” then we can say, logically, the probability is 1/2 (implicitly, this assumes she has only one opponent). Incidentally, it is not pertinent that these assumptions are false in fact: assuming they are true, we can calculate a logical probability.

In any case, frequentism can never give a probability for any unique event, or any series of events that cannot, in theory, be infinite. And since those two classes encapsulate all real-world events of interest to humans, frequentism cannot deliver useful probabilities. Even if you wanted to embed the “Clinton wins” event into a set of events that can approach infinity, you are stuck with how. All women running for president of the USA? All women running for leader in democratic nations? All women whose husbands have been presidents? All women who take over any large organization? Each of these will arrive at difference answers—a theme that will often come back to haunt frequentist procedures.

Further, given the premise “Briggs was abducted by a green UFO”, the conclusion “Briggs was abducted by a UFO” has logical probability 1. But it can never have a relative frequency because, of course, it’s premise is false in fact. I think.

Subjective Bayesian The other name for logical probability is “objective Bayesian”, so you can imagine there is a lot of overlap with its subjective, willful brother. This is true: the math is, for all intents, identical; and in practice, nearly all subjectivists act like objectivists, so these objections are somewhat trivial.

However, a subjectivist is allowed to take, “If our premises are that we have a six-sided die that we’ll toss, and that only one side will show on that toss, and that just one side of the six has six spots” and announce the probability as any number he likes. He can say 0.01, or 0.987, or anything between 0 and 1—even including 0 and 1! However, as mentioned, hardly anybody does such a thing.

Next time Demystifying randomness. And finally some examples!

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  1. “Count the number of times it comes up “6 spot”, then divide by the number of throws. That limiting fraction becomes the probability. But only after we get to infinity. While this is a perfectly fine mathematical definition, it has nothing to do with reality.”

    Huh? Empiricism has nothing to do with reality, while thought experiments in “logical probability” do?

  2. Mike B,

    We haven’t learned how to map the (logical) arguments—that is, model—real-life stuff yet. But we will.

  3. Yes indeed.

    I would point out, not an original observation, but sometimes discussed, that humans seem to tend to think in frequentist ways. There are many suggestions as to why that might be so, from evolution, if you find lots of fish or game at one place rather than another it might well pay to visit again, to even more arcane ideas.

    No matter. Whatever the reason it is well known empirically to bookmakers, operators of casinos, and of course stage magicians and illusionists. And amongst less reputable others who seek to deceive or defraud the public.

    Oddly I have never seen a detailed study of this: but stand to be corrected.

    Kindest Regards

  4. a jones,

    There’s a whole range of psychological work asking, “Do people think like Bayesians or frequentists.” Whichever answer is correct does not prove that the underlying theory is the correct one, of course. Most work indicates Bayesian thought. See Kahneman, Tversky, Gilovich and others. They think people are Bayesian-ish, meaning Bayes + built-in bias. Other workers, like Gigerenzer et al., say just Bayes and the bias can be trained out or that it is an artifact from the odd way psychologists ask questions. A very rich literature out there!

  5. Here’s a stat problem for you.

    For Yosemite bears, dinner arrives in a minivan


    YOSEMITE NATIONAL PARK, Calif. – What’s bigger than a picnic basket and even better than one in the eyes of black bears that live in Yosemite National Park? A study published this month in the Journal of Mammalogy says it’s minivans driven by families with children who leave behind a trail of spilled juice boxes, Cheerios and coolers carrying other snacks.

    Park scientists have found that the bears tore up minivans more frequently than other types of vehicles. It found that minivans represented 29 percent of the 908 vehicles torn into by bears between 2001 and 2007, even though they made up just 7 percent of the cars that visited Yosemite.

    The researchers investigated the relationship after noticing that bears seemed to target that particular vehicle type.

    What’s the Bayesian likelihood of that?

  6. Briggs,

    You mean we haven’t gotten that far in your lectures yet? Or we, as humans, haven’t got that far yet?

    I’m still trying to figure out how your “logical probability” approach will outperform a frequentist approach for a loaded die.

    Or if you really want to have fun, how would you determine if a die is “fair”? How would this different from your experiments on Korean psychics?

  7. Mike D,

    Bears have sensitive noses. They know a high sugar content car when they smell one. It surely wouldn’t take long for a bear that’s no smarter than average to learn which shaped car contained the goodies last time. They are fascinating creatures, anyway. I was pleased to read your comment.

  8. UMM yes.

    I am aware of the psychologists’ investigations but for most of their questions and indeed experiments you might do better to consult a skillful stage magician who understands these things rather better and furthermore can ensure whatever outcome you please. Especially in front of educated, as they imagine themselves to be, audiences such as scientists and the like. Remember Oliver Lodge?

    No my mind was turned another way.

    It seems to me that we actually do have large scale records of how people view probabllity: but nobody has ever collated or analysed these. Partly that may be that they are commercially confidential, partly that they did not and were not designed to collect information in a suitable form, and so on.

    Gambling and how people select their choices is an excellent example and whilst casinos etc. do monitor these patterns to detect what they call cheating they keep their records confidential.

    The best hope would be the big UK bookmaking chains because they not only keep accurate records but also employ good statisticians part of whose job is to devise new complex bets which depend, to make profits, on the way people bet. That is if the punters laid their money randomly the bookmaker would make a loss.

    Into that they calculate both the likely loss but also the difficulty of laying sufficient small bets to affect their profits, large ones are capped by maximum payout, and the level of significance which might suggest there was some kind of organised action against them.

    Unfortunately I am persona non grata with them.

    Kindest Regards

  9. But are they smarter than your average bears? (Note the statistical allusion.)

    The researchers did not test that particular hypothesis, but were impressed enough by the data to the extent that they attributed to bears a capacity for making max/min cost/benefit analyses and exhibiting of behaviors (by the bears) that would make marketeers at our newly formed Government Motors weep for joy.

    Yosemite bears can, for instance, distinguish between minvans and SUV’s, stationwagons, and subcompacts, a feat my wife can’t do — and she had some formal schooling in her youth, something few bears can claim (with any honesty).

    This is one of those cases where the data are or should be highly suspect, IMHO. There is a tendency in our trusting if not gullible species to believe what we are told to believe, if the teller is an authority figure of some kind (has a badge, or wears a white lab coat, or is rather tall, or is giving away free tickets for haircut and neck massage at the newly open barber franchise at the mall, or was accidently elected President, or is a Jewish mother, or otherwise gives off the aroma of authority, which can be purchased at the Sharper Image website).

    Because people, being people, will act in strange ways that are as surprising to bears as they are to us, only more so.

  10. Mike D.

    An odd thing about that paper is the sampling method they used. Perhaps it was too expensive to count all vehicles.

  11. Minivans!!!!!

    It’s just errata from the Cultural Cesspool, but does possess numbers that could be statistically crunched just to illustrate a statistical point, rather than a philosophical point about the absurdity of our PoPoMo world (which looks increasingly like the Dark Ages).

    btw, speaking of statistical purposes, the podcasts are delightful. A real treat. Thank you.

  12. Mike D,

    We can do the bears as an example (eventually).

    Podcasts are going downhill. I’m making some changes.

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