William M. Briggs

Statistician to the Stars!

Category: Links (page 3 of 78)

The statistics of both climatology and meteorology.

Mexican Hat Fallacy



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.)

Good Ways Of Speaking About Truth

Quid est veritas?” Pilate asked. Famously, his interlocutor did not answer, perhaps because Pilate didn’t give Him the chance. Then Pilate may have been (understandably) addled because the Answer was standing there.

Anyway, Aristotle, under less pressure, had a go at a definition (one that Pilate almost certainly would have known). He said, “To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.”

That is lovely, understandable, and complete. It—the definition—is called realism, a pleasant and accurate label. Actually, it is called Aristotelian or ‘moderate’ realism to distinguish it between the hyper and over-literal realism of his pal, Plato. That difference makes no difference to us today.

There are other ideas of truth, all of them wrong, which follow two main roads: idealism (everything exists in your head, therefore your head doesn’t exist) and nominalism (what’s in your head doesn’t exist, therefore there is nobody there to think up and fret over nominalism). But we’ll pass these by today, too.

Yesterday, we agreed it was true that ‘all men are mortal’ and that ‘2 + 3 = 5′. It is the nature of men to die and for integers to behave undeviatingly according to certain rules. These are universal truths. There also exist contingent truths, which are propositions that accord with Aristotle’s definition conditionally. Unfortunately, there is only word in English for both, which means universal and contingent truths are often confused—which leads to hurt feelings.

To explain. Universal truths are those which begin with indisputable axioms and lead inexorably and necessarily to certain truths. For example, once we accept, without proof and based on no evidence save introspection, that “For all natural numbers x and y, if x = y, then y = x” and a couple of other similar sounding axioms, it is necessarily true that ‘2 + 3 = 5′. Because we don’t know why or how the axioms can be true—we just know that they are—we don’t know why or how ‘2 + 3 = 5′ is true, except in the weak sense that we say the equation is true because the axioms and intermediate theorems are. But we cannot say why it didn’t turn out that ‘2 + 3 = 7′ (don’t even think of arguing over the symbols).

Contingent truths are those propositions which follow from premises that might themselves not be universally true. For example, if we accept “All cats speak French & Whiskers is a cat” then it follows, i.e. it is contingently true, that “Whiskers speaks French”. Yet nobody but a cat lady would run into the street and claim Whiskers’s linguistic ability were universally true. That’s because the first premise is, according to other well known premises, false. Therefore, on that evidence, the conclusion, while contingently true, is universally false. True and false simultaneously, at least speaking loosely, and therefore something to fight over.

The “Laws” of science are all contingently true. Any one or even all of these Laws may be universally true, but we don’t (possibly yet) know it. If they were universally true, then they would all be in the same epistemic boat as mathematics and logic. We would start with introspection, decide what follows from beliefs we just know are true, and then build theorem upon theorem until we reached the Law of Gravity.

That’s almost how it works, but not quite. Inside the Law of Gravity are several fudge factors, “constants” of the universe which are derived via observation, i.e. which are not deduced from first principles. And (we read) there are one or two other dicey premises which are not entirely convincing. The Law of Gravity, which nobody doubts in practice, cannot be said to be universally true (no pun; nay, not even from me), even though it contingently is.

Because the Laws of science are only, or at least, contingently true the premises which accompany them may be argued over. It is not unscientific to do so. It is prudent. When physicists argue over gravitation, it is clear to everybody that the conclusion is accepted because it is observed to hold in most places, and so discussion centers on the premises which would make the Law hold in all places.

The situation is different in climate science, for example, where the conclusion itself is in doubt (rampant global warming will kill half the population by 2009—oops, I meant 2017), and where the premises are so beloved that they are Not To Be Questioned. The (suitably modified to keep current) doom conclusion is contingently true, but that does not make it truly true, i.e. universally true. Failing to understand that distinction is what leads the weak to shout “Denier!”

Climategate 3.0—Update: Hacker A Coder?

Whoever it was that snatched the cache of emails from prominent climatologists and created Climategate 1.0, then 2.0 has come forward, in a sort of way, to begin Climategate 3.0. He—sounds like a he, perhaps a Russian he?—sent an email to several climate scientists explaining why he did what he did, and including a password to open 200,000 files that have been previously hidden. (I wasn’t one of these; I don’t have the files nor the password.)

The hero/thief/activist/concerned citizen-of-the-world, call him what you will, I prefer jokester for this individual has a fine sense of humor, is clearly a computer geek and is careful covering his tracks. His missive can be read in several places, such as at Anthony Watt’s place.

I have not seen the file nor the password, but others have started burrowing through. Early results suggest, as Tom Nelson discovered, much boredom awaits. There has been a tidbit or two, such as one email from a serious, working, peer-reviewed and -reviewer climatologist that called Mann’s hockey stick “crap.” This curiously is the precise statistical word to describe Mann’s work, so perhaps it was a statistician and not climatologist who wrote those words.

This means the, the, THE, THE Consensus isn’t. Ah well.

The other (so far) slice of fun came from my pal Gav Schmidt, who in reaction to the refreshed controversy tweeted this:

This is the interest over time in Climategate. I must admit this curves tracks my attentiveness, too. But here’s why this is funny. A new email from Tom Wigley admits the pseudo-science (actually cheap journalism) of counting papers as proof of consensus or truth. After trying his own hand at counting citations, he said:

Analyses like these by people who don’t know the field are useless.
A good example is Naomi Oreskes work.

The press naturally loves Oreskes’s work, because journalists nearly always fall prey to and cherish the fallacy that interest equals truth. But Oreskes has always been engaged in persiflage.

So we now understand that plots of interest are a standard newsman’s dodge and reveal nothing but political hotness. This includes Gav’s plot, which is misleading even as a political thermometer, since it was taken before Climategate 3.0 hit.

Ah well, so much of science is theatre these days, yet another avenue for agitation. I’m guessing 3.0 doesn’t reach the peak that 1.0 or 2.0 did, since, though activist scientists haven’t yet ceased discovering new ways to announce the sky is falling, people have tired of hearing them.

But see this page for updates which I find of note.

Update May as well engage in amateur forensics. I think the hero-hacker is a coder. The facility with all things computer makes this easy to guess, but so does his language, which doesn’t sound like a scientist but with somebody who works with them. He also appears to be somebody who uses English for his day job, but whose native language is something else.

Probably a coder tasked with implementing parts of climate models (data assimilation, connectivity between modules, output generation, etc.) and who sees these creations resemble smelly sausage rather than prime rib. Somebody who is aware that the certainty and confidence publicly stated in the models is far more than is actually warranted.

Doomed Planet: Changing Sun, Changing Climate

Today’s post is at Quadrant. Have you been there before? If not, you have a treat in store.

The editors supplied this quotation:

“Today’s debate about global warming is essentially a debate about freedom. The environmentalists would like to mastermind each and every possible (and impossible) aspect of our lives.”

Vaclav Klaus
Blue Planet in Green Shackles

Bob Carter, Willie Soon, and I provided this start:

Scientists have been studying solar influences on the climate for more than 5000 years.Chinese imperial astronomers kept detailed sunspot records, and noticed that more sunspots meant warmer weather. In 1801, celebrated astronomer William Herschel, the first to observe Uranus, noted that when there were fewer spots the price of wheat soared. He surmised that less “light and heat” from the sun resulted in reduced harvests.

Head over, then come back here to leave comments.

But while you’re there, stay and read Keith Windschuttle’s piece “Inventing massacre stories“, tales the Left uses to induce pleasurable guilt and sense of purpose.

See also the (not-too-often updated) Quandrant TV page, and watch Jim Franklin’s video, “What Science Knows.”

EPA’s Climate Change Adaptation Plan

Motto: “It’s for your own good.”

The EPA has asked for my advice.1 They will not listen to, or harken, or act on this advice, but asking for it gives them a sense that they have participated in our great Democratic Experiment. Since one of the goals of this column is to make people feel good about themselves, even bureaucrats in our ever-increasing government, I’ll provide constructive counsel.

There is no reason to ask for advice on what to do about the climate because nobody knows what the ideal climate is. By “nobody” I mean no body, no person, no living or dead soul, no individual no matter his, or even her, “degree” or credential or experience. The very concerned, the activists, the nerve-racked environmentalists, the emotionally and financially committed, the mindful, the most deeply vexed greens; none of these have any idea what the climate should be.

If you don’t have a target it is futile to take aim. It is worse to shoot. It is insane to ask strangers to pay for your arrows.

The climate has never been static. It has always changed. From the day a wanton pile of mass coalesced into a molten ball and called itself Earth (Gene “von Frahnkensteen” Wilder was in my mind here), to when it cooled sufficiently to form a surface crusty enough to host oceans, to the point yesterday afternoon when the weather was clement enough for the Tigers to host the Pirates in Lakeland, Florida, the only constant has been change.

Strike that: there was another constant. Mankind’s ability to adapt. He has thus far lived through all environmental vicissitudes. He has not flourished everywhere and always—there have been times when the change of seasons forced a change of address—but he surely prospered, usually without the “benefit” of central planning. Evidence for this can be had by comparing head-counts from then to now.

When mankind was far less technologically sophisticated, he managed, even thrived. The consequence is that life is better now on average than it has ever been, speaking purely in terms of comforts. Paradoxically, however, the biggest danger to mankind is comfort. Everywhere it is introduced, fewer children are produced. Too much of a good thing is deadly. Comfort is therefore far scarier than a few tenths of a degree increase in average temperature that might, or again might not, happen.

Next consider the layers of uncertainty of climate change. First is measurement. We don’t have a perfect idea of what climate was everywhere; uncertainty increases backwards through time; we only know (with reasonable certainty) that it was different and that it changed. The EPA, in its request for comments, says the climate is changing “more rapidly than society has experienced in the past.” This is false, or at least not at all certain. To prove this article of faith requires having certain or near certain historical observation, which we lack. We also need proof that recency bias has been accounted for, which it has not (this is when periods of frequent measurement are compared against times with sparse measurement: the frequent measures can often falsely cause one to conclude that changes are happening more rapidly).

So we lack conclusive evidence of what the past was. It is clear, however, that what happened previously is a good predictor of what will happen (climatologists call this “persistence forecasting”). Yet EPA claims “the past is no longer a good predictor of the future.” This is false. Persistence forecasts for climate have remarkable accuracy, far better than climate models.

If the past cannot be used, what can? EPA says expert opinion (codified into computer circuitry). Yet to gain our trust experts must first demonstrate superiority over the past (skill over persistence). Have they? No, sir, they have not. For years experts said it was going “to be worse than we thought.” Has it been? No, sure, it has not.

We don’t know for sure how it was, we have a reasonable idea of what weather and climate conditions are now, but we do poorly predicting skillfully the future. What about things that are affected by climate? We must necessarily be less sure of what will happen to them. This is a provable, logical statement. Thus: if there is probability X that polar bears will die off if the climate changes, and there is probability Y the climate will change, then there is only X x Y probability—noting X x Y < X or Y—that the climate will change and polar bears hand in their dinner pails as a species.

This isn’t nearly the end. You must multiply the uncertainty of every other thing you say will be “impacted” (like a tooth?). If there is probability X x Y of climate change and polar bears dying, there is probability X x Y x Z of climate change and polar bears dying and corn crop decreases, where X x Y x Z < X or Y or Z. Etc. ad nauseum.

That simple, and true, math married to the other arguments given above implies the best thing to do is nothing until such time we see items which actively need attention. None do currently.

This won’t satisfy because government bureaucracies are created to do something, anything. Therefore they will do something, even if it something as small as calling for “more study”. To admit nothing need be done is to concede one does not need to exist. This the EPA will never do. See? It really is worse than we thought.


1The link provides another link to EPA’s Climate Change Adaptation Plan, the document from which I draw my quotes.

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