William M. Briggs

Statistician to the Stars!

Page 148 of 546

Subjective Versus Objective Bayes (Versus Frequentism): Part II

John Maynard Keynes, Chance Master

Read Part I.

What is the probability that “The Detroit Tigers win today’s game” (which has not yet been played)? The truth of the proposition (in quotes) is not known and is therefore uncertain. Enter probability.

Some will used words to express their answer (“Pretty likely”, “They don’t have a chance”, “No way they can lose”), others will provide rough quantification (“90%”, “3 to 1 against”), while still more will provide serious quantification (“$50 bucks says they win”). Finally, some will not answer at all (“I have no idea”, “I hate baseball”).

Which of these is the right answer? Assuming nobody is fibbing, they all are. (The frequentist response is given below.) Each reply is subjective because each is conditional of a set of premises supplied by each individual, premises which may or may not be articulated.

For example, “Pretty likely, given that they won their last three and the Astros (their opponent) are dead last.” Another thinks, “They don’t have a chance because I suspect Justin Verlander (Tigers’ starting pitcher) is injured.” But when you ask the man who was willing to bet $50 why, he might say, “I don’t know. It feels like the right amount.” Or he might say, “I always bet $50 on the team I think has the best chance” which again fails to provide a list of premises why he thinks the Tigers have the best chance.

This kind of situation is what people have in mind when they think of subjective probability. Answers can range from no probability at all (“I hate baseball”), to vague but real probabilities (“Pretty likely”), to actual quantifications (“3 to 1 against”, “$50 to win”). All depend on individual premises which we may or may not be able to elicit. This includes those situations where the person doesn’t want to or has no answer. For example, you might be asked, “底特律老虎隊奪冠的概率今天的比賽是什麼”? If you don’t speak Mandarin and haven’t any idea of the context the best answer is, “I have no idea what you’re talking about.” (Real speakers of Mandarin will say the same thing of this translation.) That is, there is no probability for you.

It is never an answer to say, “‘The event will either happen or it won’t’ therefore, the probability is 50%”. That number can never be deduced from a tautology. That is, it is always (as in always) true “The event will either happen or it won’t” for any event, which is what makes it a tautology, and that adding a tautology to a list of premises cannot change the truth or probability of a proposition. Any number of tautologies may be added, not just one. For example, “At today’s game it will either rain or it’ll stay dry, Verlander will either pitch well or he won’t, so the Tigers will lose or they will win.” There is no content in that phrase except that the Tigers will play (and Verlander will be the pitcher).

From these simple examples, we may conclude several things. (1) Probability is not always quantifiable; that is, not every probability is a precise number; (2) Probability is sometimes a range (another example: “They’ll either lose or come close to losing”); (3) Probability can be a fixed number; (4) Not all probabilities can be known; (5) The weight of evidence, how stable the probability of the proposition appears, depends on the list and strength of the premises.

What a person says may be at odds with what he believes, not only because of deception, but because sometimes words take slightly different meanings for different people or because not everybody is attentive to grammar. Our man might list as one of his premises, “Verlander will either pitch well or he won’t”, which is formally a tautology and therefore of no probative value, but subjectively he gives more weight to “pitch well” than to “not pitch well”, and so this tautology-in-form is actually informative. This is why there is confusion on the subject.

Consider two men. One gives the premise, “I don’t know much about the Tigers, but they won their last three.” Another says, “The Tigers’ batting is on fire; here are their stats. And Verlander is the best pitcher in baseball, and here is why” plus many more (a real fan). But suppose both men say the chance the Tigers will win is 80%. Adding or subtracting a premise from the second man will not change his stated probability by a great degree. But adding or subtracting a premise (particularly subtracting!) from the first man changes his by a lot. We would say the weight of evidence of the first is less than that of the second, even though both have the same probability. And this is because of the differences in the premises.

What is the objective probability the Tigers win? There isn’t one, at least, not yet. And the frequentist probability? Same answer: there isn’t one yet.

Now the difference between subjective and objective probability is this: when presented with a list of premises (of unambiguous words) a subjectivist can state any probability for the conclusion (proposition) he wishes, but the objectivist must take the premises “As Is” and from these deduce the probability. The subjectivist is free, while the objectivist is bound. This is why there is no objective probability the Tigers win, because there is no “official” list of premises for the proposition.

The lack of an official list of premises is also why the frequentist must remain mute, because in order to calculate any probability the frequentist must embed the proposition of interest in an infinite (as in infinite) sequence of events which are just like the event on hand, except that the other events are “randomly” different. This constrains the type and kind of premises which are allowable. (I discuss “random” in another post.)

For example, if the “official” list—which merely means those premises we accept for the sake of argument—are one: “The Tigers always win 80% of the time against the Astros”, the objectivist must say (given the plain English definitions of the words) the probability of a win is 80%. The subjectivist may say, if he likes, 4%. He won’t usually, but he is free to to do. The frequentist may be tempted to say 80%, but he has to first add the premise that “Tigers vs. Astros” events are unchanging (except “randomly” different) and will exist in perpetuity. Perpetuity means “in the long run.” But as Keynes reminded us, “In the long run we shall all be dead.” In other words, unless the frequentist “cheats” and adds to the official list of premises suppositions about infinite “trials”, he is stuck. Incidentally, the subjectivist who does say other than 80% is also usually cheating by adding or subtracting from the official premise list, or by (subjectively) changing the meaning of the words.

Now this is not nearly yet a complete proof which shows that frequentism or subjectivism are doomed; merely a taste of things to come. What is clear is that probability can seem subjective, but only because, as was showed in Part I, the list of agreed upon premises for a proposition can be difficult or impossible to discover. Next time: simpler examples. Maybe where “priors” come from.

Read Part III.


Mexican Hat Fallacy

Hola!

Hola!

Reader Kip Hansen asks, “Can you please run a brief explanation of what the Mexican Hat fallacy is statistically?”

I can. The Mexican Hat Fallacy, or Falacia Sombrero, is when a man moves from sunny to cloudy climes, such as when a hombre shifts from Veracruz to Seattle, and thus believes he no longer need wear a hat. This is false. A gentleman always wears a hat—not a baseball cap—not just because it regulates heat and keeps you dry, but because it completes any outfit.

Well, that’s the best joke I could come up with.

The term was coined by Herren von Storch and Zwiers in the book Statistical Analysis in Climate Research and it came about like this: Some fellows were wandering in the Arizona dessert sin sombreros and came upon the curious rock formation pictured above (image source).

One fellow said to the other, “Something caused those rocks to resemble a sombrero.” The other fellow, more sun-stroked then the first, disagreed, “No, no thing was its cause. That’s my null.” Quoting from a paper by Herr Gerd Bürger (because I had never heard of this fallacy before):

By collecting enough natural stones from the area and comparing them to the Mexican Hat [formation], one would surely find that the null hypothesis ‘the stone is natural’ is quite unlikely, and it must be rejected in favor of human influence. In view of this obvious absurdity [von Storch and Zwiers] conclude: ‘The problem with these null hypotheses is that they were derived from the same data used to conduct the test. We already know from previous exploration that the Mexican Hat is unique, and its rarity leads us to conjecture that it is unnatural.’ A statistical test of this kind ‘can not [sic] be viewed as an objective and unbiased judge of the null hypothesis.’

Which leads me to (hilarious) joke number two. There are two kinds of people: those who find null hypotheses a useful philosophical concept, and those who don’t. This description is confusing—but then so ultimately are most stories about “null” hypotheses.

If the “fallacy” merely means that the closeness of model fit to observed data is not necessarily demonstrative of model truth, then I am with them (this is why p-values stink). You can always (as in always) find a model which fits data arbitrarily well—as psychoanalytic theory does human behavior, for example—but that does not mean the model/theory is true. Good fit is a necessary but not sufficient condition for model/theory truth. A (nearly) sufficient condition is if the model/theory predicts data not yet known, or not yet used (never used in any way) to fit or construct or posit the model/theory—as psychoanalytic theory does not well predict new behavior.

The parenthetical “nearly” is there to acknowledge that, in most cases, we are never (as in never) 100% certain an empirical model/theory is true. But we can be pretty sure. Thus we do not say “It is 100% certain evolutionary theory is true,” but we can say, “It is nearly certain evolutionary theory is true.”

So much is Stats 101. Yet I’m still perplexed by Bürger-von Storch-Zwiers’s example. If we “already know from previous exploration” that the Mexican Hat formation was caused by (say) weathering, then collecting rocks from nearby isn’t useful—unless one wants to play King of the Hill. And what does “comparing” these rocks to the formation mean? Should the individual stones resemble the formation in some way for the formation to be “natural”? The rocks nearest will be made of the same material as the formation, so this is no help.

Regarding the possible causes or hypotheses of formation, they are infinite. It is we who pick which to consider. It could be, for example, that we’ll soon see a History Channel “documentary” which claims ancient Egyptians were flown to Arizona by aliens under the guidance of angels to build the Sombrero so that the Hopi could use it in a religious ceremony that was eventually secretly used by Hitler in his bid to conquer the USSR.

Let’s call this the “null” hypothesis. Why not? The “null” is ours to choose, so it might as well be something juicy. I bet if we link this around that, give the ingenuity of internet denizens, within a week we would have enough corroborative evidence for it to satisfy any NPR listener.

Speaking of hats, if you’re looking for a genuine Panama to cool your pate in the summer months, may I recommend Panama Hats Direct? I get nothing for this endorsement, except the satisfaction of helping this fine company stay in business. (If this is your first, go for the $95 sub fino. It is a fantastic deal.)


Woman To Marry Fairground Ride. A New Sexual ‘Orientation’?

Weeeeeeeeeeeee!

Weeeeeeeeeeeee!

I want you to tell me exactly why Amy Wolfe, a 33-year-old “US church organist”, can’t marry her favorite roller coaster. You heard me. Roller coaster—an “80ft gondola ride.” Exactly, now.

No fair saying, “It’s illegal”. Same-sex “marriage” is illegal in many spots too, but lots of people are still for it. Why shouldn’t she be allowed to do as she pleases?

Wolfe sure looks happy in a picture of her and her intended, surnamed “1001 Nachts”. The Daily Telegraph reports:

“I love him as much as women love their husbands and know we’ll be together forever,” she said.

Miss Wolfe first fell for the ride when she was 13: “I was instantly attracted to him sexually and mentally.

“I wasn’t freaked out, as it just felt so natural, but I didn’t tell anyone about it because I knew it wasn’t ‘normal’ to have feelings for a fairground ride.”

Point one: she said, “I’m not hurting anyone and I can’t help it…It’s a part of who I am.” So Wolfe ‘oriented’ towards roller coasters. If you say no, why?

It’s not an argument to laugh or scoff and say, “That’s absurd!” Maybe it is absurd: your job is to tell us why. It’s also not an argument to say why you’re in favor (if you are) of same-sex “marriage.” I don’t care—not here, anyway—unless the reason you’re in favor of SSM is the same as why Wolfe should be bound by matrimony to her ride.

No good using the word “obvious” in your proof. It’s obvious to me, for example, that SSM shouldn’t happen and that Wolfe needs a good psychiatrist; perhaps it’s obvious to you that SSM is A-okay and Wolfe is free to do what she wants. We have to go deeper than obvious.

Point two: though she didn’t use the word, in effect she said she’s “oriented” towards playground rides. Very well. Perhaps this “orientation” is biological; I mean, caused (in some way, we know not how) by her genes, raging hormones, or perhaps a swollen amygdala.

Coincidentally, I was reminded of a study by Bearrman and Bruckner: “Opposite-sex Twins and Adolescent Same-Sex Attraction (in American Journal of Sociology, 2002, pp 1179-1205).

Oddly, despite the popularity of the idea, the evidence for genetic and/or hormonal effects on same-sex orientation is inconclusive at best. The most publicized genetic findings, for example, the discovery of a marker for homosexuality in men (Hamer et al. 1993) has not been replicated, and studies purporting to establish a genetic or hormonal foundation to human sexual orientation to have serious methodological flaws.

The reason genetic markers are looked for is that it has been noticed that, sometimes, homosexual behavior runs in families. But then so does trout fishing. Meaning sometimes behavior is not genetically predetermined. Meaning that’s it’s possible, and even likely, that sexual behavior is at least sometimes chosen, a politically unwelcome idea. Anyway, with human behavior as complex as it is, it would be a wonder if there was just one explanation for our amorous proclivities.

Science News also reports “no major gene for homosexuality has been found despite numerous studies searching for a genetic connection”; instead some people are looking for epigenetic “shocks”, so to speak. But it’s all theoretical at this point, meaning the epigenetic idea “isn’t based on actual experiments,” quotes one scientist. (This is all necessary because evolutionary theory predicts genes for non-heterosexuality would quickly disappear because non-heterosexuals acting non-heterosexually don’t pass on their “selfish” genes.)

Even if the epigenetic shock theory were true, it wouldn’t explain people like Wolfe, those who are ‘oriented’ towards our four-legged friends, those oriented towards the infertile (children and the deceased; not much evolutionary advantage here), or folks with other non-standard desires.

The old-fashioned natural, scientific answer to today’s question is that marriage is between one man, one woman, mated for life and for the purpose of making and raising children. Used to be, up until about five, six years ago it was customary to acknowledge this, at least tacitly, especially the bit about kids.

People (even “researchers”) used to say that kids raised in homes with biological mom and dad did best. Evolutionary psychologists were even on board with this idea, saying (for example) adopted kids in man-woman families were under increased “risk” of death (step dads want to eliminate rival genes, you see). “Researchers” are now having to present data which “shows” that kids raised in any state whatsoever are equal to kids raised in scientific families. Given the loose requirements of statistical evidence, they’ll find it, too.


More On Flying

SFO to JFK. A Russian couple, both double-plus sized, the man at the aisle the woman the window. Me in the middle. I offered many times to let them sit together. No interest. They weren’t displeased with each other, judging by the matériel passed between them (and over me). They just didn’t want to move seats. They were both nice people.

The guy grabbed a magazine and flipped it open randomly. Full page line drawing of the kind of thing an OB-GYN would find familiar. Lots of precise anatomical detail, lovingly depicted. Because of my wide and diverse life experiences I knew just what I was seeing. I wondered if the people seated behind the man knew too.

Turns out the man was an OB-GYN. The article he was perusing, in a professional journal, was demonstrating all the different ways one might do an episiotomy. Where to place the fingers, best viewing angle, recommended knots, that sort of thing. I’m no expert, of course, but I thought the cross-patterned stitch was the most artistic.

This was Jet Blue and they unfortunately have televisions in the seat backs. It’s unfortunate because people seated at the windows find the televisions so fascinating that they almost always close the shades. Here we are, soaring through clear skies over what has to be the most beautiful mountain range in the world, blind to it. And wouldn’t it have been nice to see the lights of New York City (we came in at night)?

On the flight before last all but five windows were shuttered. The plane was illuminated to an artificial murk. Depressing.


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