William M. Briggs

Statistician to the Stars!

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Chance Of Heat Wave Only 1 in 1.6 Million? Or, Probability Gone Wrong

My dad took a swing with his nine-iron and the wiffle simulacrum of a golf ball took flight, arched upwards, spun left and, without bouncing, landed atop my favorite blade of grass! Yes, this really happened.

That a nasty little white plastic ball with holes drilled through it would land on my favorite blade of grass could not be a coincidence. I mean this in the full quantitative sense. For if these calculations are correct, and there is no reason to suggest they are not, my father’s back yard has in it about 1,204,346,880 individual grass blades (his yard is just over three-fifths of an acre).

That makes the chance that my very favorite blade would have been viciously assaulted to be just over one in a billion!—a number so incredibly small that it should be written in bold font. Now whenever you see a probability so low, it can only mean that some kind of directing force, some guiding principle, some entity must have had a hand in causing the event. There’s just no other way to get a low probability!

And since I can think of nothing else, the cause must be global warming, a.k.a. climate change, a.k.a. climate disruption, a.k.a. climate tipping points, a.k.endless.a. etc. Take that skeptics!

Or so reason the people who lately claimed that the “U.S. heat over the past 13 months” was only a “one in 1.6 million event.” After making this dubious calculation, it was argued that because the result was so incredibly tiny, something-that-could-only-be-global-warming caused the temperature to take the values it did.

The 1-in-1.6-million came from the NCDC via reasoning like this: a month’s temperature can occur (they claim) in one of three buckets, below normal, normal, above normal. The chance it “falls” into one of these buckets is 1/3. Therefore, seeing 13 months in a row of monthly temperatures in the above-normal bucket has a probability of (1/3)13.

Ignore the simplification about the buckets and the assumption of a 1/3 chance that a month’s temperature lands in any bucket. Rather, accept them both as true. Then, given we believe in these premises, it is indeed true that the probability of 13 out of 13 “above normals” is 1 in 1.6 million. Ok then. Now what? It was also true that the probability my dad’s wiffle ball would crush my favorite blade of grass was 1 in 1.2 billion.

Now it is also true that the probability of any sequence of 13 monthly temperatures is the same: e.g. below-below-below-normal-above-above…above has the same chance as above-below-above-below-above…below; you get the idea. This means if I see one of these sequences—and I must by definition see one of them—the event I witness will be “rare.” Just as the ball-blade meeting was rare.

The argument is then supposed to be that since this probability is so low it could not have happened “by chance.” But chance doesn’t cause anything. There is no wily devil-in-the-machine rolling cosmic dice to determine outcomes of temperature or wiffle balls or anything. Instead, something physical caused the temperature sequence to take the values it did, just as something physical caused the ball to land where it did. If it is this “something physical” in which my interest lies, I do myself no good at all by calculating dubious probabilities and then worrying over them. I should better spend my time investigating real causes then in inventing probabilistic bogeymen.

Calculating probabilities the way we did in these two examples is to purposely, willfully turn a blind eye to all the evidence we have about actual monthly temperatures and actual clubs hitting real balls, and then to say to ourselves, “These probabilities are so low that there must be physics that we purposely, willfully ignored.” When we were kids we had a comeback about Sherlock when hearing observations of this kind (but since this is a family blog, I won’t repeat it).

For the golf ball, I’m ignoring where my dad routinely stands relative to my favorite blade, the distances balls fly when hit with a nine-iron, and on and on. For the temperature, I’m ignoring just about everything there is to know about temperature, which is a lot. Such as how on 30 June the temperature does not “reset” itself so that on 1 July it begins anew, and so forth. It really is a sad business to pretend we don’t know all of this and then to intimate that that some mysterious cause, like global warming, is the real culprit for actual events.

Low probabilities are not proof of anything—except that certain propositions relative to certain premises are rare. If those certain premises are true, then so are the probabilities accurate. Whatever the probabilities work out to be is what they work out to be end of story. If the chance a ball hits my favorite blade of grass is tiny, this does not mean that therefore global warming is real. Who in the world would claim that it is? Yet why if relative to unrealistic premises about temperature buckets the probability of 13 out of 13 above-normal monthly temperature is tiny would anyone believe that therefore global warning is real? You might just as well say that the same rarity of 13 out of 13 meant therefore my dad was a master golfer. The two pieces of evidence are just as unrelated as were the rarity of the grass being hit and global warming true.

If our interest is in different premises—such as the list of premises which specify “global warming”—then we should be calculating the probability of events relative to these premises, and relative to premises which rival the “global warming” theory. And we should stop speaking nonsense about probability.

How Good Are You At Making Sports Predictions?

I’m traveling today so I thought I’d remind us of ShortBurst sports, Edgehogs as was.

Make picks in all major sports, even the God-help-us Olympics, and track your prognosticatory prowess. You’ll be scored with the “X-Rating” a one-of-a-kind, never heretofore seen, unique, sui generis, statistical formula-based number, which runs from 0 (“expert”) to 100 (“in-real-life good”). For you New York Times readers, this means higher is better.

Yes, I am part owner of this fine institution, and that means that, with a fair wind blowing, and everything just so, the stars aligned with the house of Grog infused in the tea of Rum, the Higgs boson creating mass in just the right locality, or, even better ,a generous benefactor finally seeing What Is Investible, I might—I say might—just make some money off this.

My breath is not held. And therefore I am doing the world some good, the carbon dioxide which I ordinarily would have expelled is trapped deep inside, warming my innards through positive feedbacks operating on some tree rings I once ate when Euell Gibbons once thrived and roamed the folding hillsides. I haven’t eaten any cattails, but I once lit one on fire and used it as a punk to startle some fireworks into life. I might have also thrown one at Chuck Coonrod intending to cause him heated distress. It was all in good fun.

See too the ShortBurst Leaderboard, a listing of folks who would have made Cole Porter’s list had they been alive when he was writing about how we should do it to make it to the Coliseum. And by “do it” he meant logging on, via Facebook if you are one of those types, or not if you are not, and making your own prescient picks.

Every time we say goodbye, I wonder why a little. So here are some otters which look like Benedict Cumberbatch.

Observational Bayes > Parametric Bayes > Hypothesis Testing < Looking

This is a completion of the post I started two weeks ago which shows that “predictive” or “observational” Bayes is better than classical, parametric Bayes, which is far superior to frequentist hypothesis testing which may be worse than just looking at your data. Actually, in many circumstances, just looking at your data is all you need.

Here’s the example for the advertising.csv data found on this page.

Twenty weeks of sales data for two marketing Campaigns, A and B. Our interest is in weekly sales. Here’s a boxplot of the data.

It looks like we might be able to use normal distributions to quantify our uncertainty in weekly sales. But we must not say that “Sales are normally distributed.” Nothing in the world is “normally distributed.” Repeat that and make it part of you: nothing in the world is normally distributed. It is only our uncertainty that is given by a normal distribution.

Notice that Campaign B looks more squeezed than A. Like nearly all people that analyze data like this, we’ll ignore this non-ignorable twist—at first, until we get to observational Bayes.

Now let’s run our hypothesis test, here in the form of a linear regression (which is the same as a t-test, and is more easily made general).

Estimate Std. Error t value Pr(>|t|)
(Intercept) 420 10 42 2.7e-33
CampaignB 19 14 1.3 0.19

Regression is this and nothing more: the modeling of the central parameter for the uncertainty in some observable, where the uncertainty is quantified by a normal distribution. Repeat that: the modeling of the central parameter for the uncertainty in some observable, where the uncertainty is quantified by a normal distribution.

There are two columns. The “(Intercept)” must (see the book for why) represent the central parameter for the normal distribution of weekly sales when in Campaign A. This is all this is, and is exactly what it is. The estimate for this central parameter, in frequentist theory, is 420. That is, given we knew we are in Campaign A, our uncertainty in weekly sales would be modeled by a normal distribution with best-guess central parameter 420 (and some spread parameter which, again like everybody else, we’ll ignore for now).

Nobody believes that the exact, precise value of this central parameter is 420. We could form the frequentist confidence interval in this parameter, which is 401 to 441. But then we remember that the only thing we can say about this interval is that either the true value of the parameter lies in this interval or it does not. We may not say that “There is a 95% chance the real value of the parameter lies in this interval.” The interval is, and is designed to be in frequentist theory, useless on its own. It only becomes meaningful if we can repeat our “experiment” an infinite number of times.

The test statistic we spoke of is here a version of the t-statistic (and here equals 42). The probability that if we were to repeat the experiment an infinite number of times, that in these repetitions we see a larger value of this statistic, given the premise that this central parameter equals 0, and given the data we saw and given our premise of using normal distributions is 2.7 x 10-33. There is no way to say this simpler. Importantly, we are not allowed to interpret this probability if we do not imagine infinite repetitions.

Now, this p-value is less than the magic number so we, by force of will, say “This central parameter does not equal 0.” On to the next line!

The second line represents the change in the central parameter when switching from Campaign A to Campaign B. The “null” hypothesis here, like in the line above, is that this parameter equals 0 (there is also the implicit premise that the spread parameter of A equals B). The p-value is not publishable (it equals 0.19), so we must say, “I have failed utterly to reject the ‘null’.” Which in plain English says you must accept that this parameter equals 0.

This in effect says that our uncertainty in weekly sales is thus the same for either Campaign A or B. We are not allowed to say (though most would), “There is no difference in A and B.” Because of course there are differences. And that ends the frequentist hypothesis test, with the conclusion “A and B are the same.” Even though the boxplots look like they do.

We can do the classical Bayesian version of the same thing and look at the posterior distributions of the parameters, as in this picture:

The first picture says that the first parameter (the “(Intercept)”) can be any number from -infinity to +infinity, but it is most likely between 390 to 450. That is all this says. The second picture says that the second parameter can take any of an infinite number of values but that it most likely lives between -20 and 60. Indeed, the vertical line helps us quantify the probability this parameter is less than 0 is about 9%. And thus ends the classical or parametric Bayesian analysis.

We already know everything about the data we have, so we need not attach any uncertainty to it. Our real question will be something like “What is the probability that B will be better than A in new data.” We can calculate this easily by “integrating out” the uncertainty in the unobservable parameters; the result is in this picture:

This is it: assuming just normal distributions (still also assuming equal spread parameters for both Campaigns), these are the probability distributions for values of future sales. Campaign B has higher probability of higher sales, and vice versa. The probability that future sales of Campaign B will be larger than Campaign A is (from this figure) 62%. Or we could ask any other question of interest to us about sales. What is the probability that sales will be greater than 500 for A and B? Or that B will be twice as big as A? Or anything. Do not become fixated on this question and this probability.

This is the modern, so-called predictive Bayesian approach.

Of course, the model we have so far assumed stinks because it doesn’t take into account what we observed in the actual data. First thing to change is the equal variances; second is to truncate the data to ensure no sales are less than 0. That (via JAGS; not in the book) gives us this picture:

The open circles and dark diamonds are the means of the actual and predictive data. The horizontal lines shows the range of 80% of the actual data placed at the height where there is 80% of the predictive data below. Ignore these lines if they confused you. The predictive model is close to the real data for Campaign B but not so close for Campaign A, except at the mean. This is probably because our uncertainty in A is not best represented by a normal distribution and would work better with a distribution that isn’t so symmetric.

The probability that new B Sales are larger than new A Sales is 65% (from this figure). The beauty of the observational or predictive approach is that we can ask any question of the observable data we want. Like, what’s the chance new B sales are 1.5 times new A sales? Why that’s 4%. And so on.

In other words, we can ask plain English questions of the data and be answered with simple probabilities. There is no “magic cutoff” probability, either. The 65% may be important to one decision maker and ignorable to another. To stay with A or B depends not just on this probability and this question: you can ask your own question inputting the relevant information to you. For instance, A and B may cost differently, so that you have to be sure that B has 1.5 times as many sales as A. Any question you want can be asked and asked simply.

We’ll try and do some more complex examples soon.

Rioting Is An Ecstatic, Spiritual Experience: Or, Structural Sin Made Me Do It

“Rioting”—which is to say, looting, rampaging, vandalizing, engaging in wanton mayhem and violence, and generally acting very badly—“can be, literally, an ecstatic spiritual experience.” So says the is-he Right Reverend Peter Price, a man who is no less than the Bishop of Bath and Wells. (Note the Bish’s use of literally. PDF of report.)

Price’s comments have been widely reported, with opinion coalescing around the idea that Price has, literally, lost his mind. But this is unfair, because, literally, Price was merely quoting another churchman, one Father Austin Smith, who made his ecstatic comments after the Toxteth riots in the 1980s. In that feast of spirituality, “468 police officers had been injured, 500 people arrested and at least 70 buildings demolished.”

Anyway, Price continued, “Something is released in the [riot] participants which takes them out of themselves as a kind of spiritual escape.” Out of themselves and into shoe shops, where “participants” gleefully stole as many “trainers” as they could lay their thieving mitts on. Which shows that “participants” are not only immoral lawbreakers but that they also have appalling taste in footwear. (This is known in statistics as a correlation.)

“The tragedy,” Price says, is that

we have a large population of young people who are desperate to escape from the constrained lives to which they feel and appear to be condemned. Where hope has been killed off and with no prospect of escape, is it surprising that their energies erupt in antisocial and violent actions? In a consumer society, is it surprising that lusting after high-status goods is seen as a way to find meaning?

There is some truth here. And it is this: there is always somebody willing to excuse the repugnant behavior of one person as the fault of the political enemies of the excuser. Price’s enemies are—wait for it—rich people, “austerity measures,” and “social tensions”.

These culpable entities even caused innocent citizens who had just “come out to see what was happening” to become “quickly caught up in the thrill of the moment.” To excuse why these bystanders engaged in theft, Price hypothesizes that they were “just picking up things that had been discarded on the streets.” And keeping them and bringing them home. Just like dear old mom did not, we hope, teach them.

Now most of Price’s report is dull, written in pseudo-academese and contains phrases like “coveting is a big issue” and “The mobile nature of the events makes a locational analysis problematic in relation to those involved, and particularly those arrested.” It reads as if a cluster of senior clergy were sitting around the pub on wet afternoon when one suddenly announced, “Our duty is to issue a report”—and then they actually wrote it!

Because it’s in the nature of these documents, Price could not help himself from theorizing. The real cause of the riots was something called “structural sin which recognises how people on all sides of conflicts can face moral choices that are not between what is clearly right and clearly wrong but which are necessitated by circumstances in response to situations where much has gone wrong already.” In other words, pocketing an item dropped by a fleeing looter is not clearly right nor clearly wrong. Even lighting a shop on fire to watch it burn can be considered morally ambiguous if much has gone wrong already.

Price was not speaking gibberish, however. The idea of “structural sin” is well known and was developed inside something called “liberation theology,” which is often read as a codeword for theological Marxism (and if that isn’t a contradiction in terms, nothing is).

Structural sin is shared sin. Because of “unjust” economic, political situations, and “discriminatory lending practices,” the man who ecstatically wields the club upside the shopkeeper’s head is guilty of structural sin, but so are you who sat at home guilty because you contributed to the circumstances which caused the man to swing the club. Why, if it weren’t for you dutifully going to work each day and paying your taxes, these riots would never have occurred! Price says “flawed social structures” are responsible for “creating the conditions for sin to manifest itself.”

Sin seen in this way is like a gas which leaks out individuals and seeps through a community, concentrating here and there due to vagaries of wind. The only recourse is to bottle sin up by reinvigorating the welfare state and making risky loans. Indeed, if “austerity” is left in place, Price sees the possibility of future riots. “Nothing is inevitable — but the auguries are not reassuring.”

A Depressing [Real Sad] Churchill Quotation—Contest

As I this morning scanned Facebook, I ran across the image below. Read it through and pause for a moment before continuing.

Churchill quote

Notice anything strange? I was so intrigued by the editor’s use of brackets that I responded with the following:

Love the quotation [words cited from source]! An insightful [cap full o' thinkin'] analysis [figurin' out] of a morally stunted [short people] heretical [believing bad stuff about God] sect [I forget this one].

Naturally, it’s time for another no-prize contest. Who can come up with the most helpful edited quotation? Your audience are degree holding—this is not synonymous with educated—United States citizens. Extra points awarded if edits are in the voice of a sorority or fraternity member.

Here is my entry.

Four score and seven [many] years ago our fathers [not our real fathers; other peoples' fathers] brought forth [after thirdth] on this continent [happy land mass] a new nation, conceived [thought of] in liberty [forget this word: vote Obama], and dedicated [given] to the proposition [business deal] that all men [and women] are created equal [must pay their fair share].

Now we are engaged [living together] in a great civil war, testing whether that nation, or any nation, so conceived and so dedicated, can long endure [make it]. We are met on a great battle-field [area outside club] of that war. We have come to dedicate a portion [bit] of that field, as a final resting place for those who here gave their lives that that nation might live. It is altogether fitting [right size] and proper that we should do this.

But, in a larger sense, we can not dedicate, we can not consecrate [have sex with], we can not hallow [full of air] this ground. The brave men, living and dead, who struggled [worked hard] here, have consecrated [good grief!] it, far above our poor power to add or detract [insult]. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished [not done] work which they who fought here have thus far so nobly [lot of knowledge] advanced. It is rather for us to be here dedicated to the great task [minimum wage job] remaining before us—that from these honored [Facebook friended] dead we take increased devotion [follow on Twitter] to that cause for which they gave the last full measure of devotion—that we here highly resolve [see better] that these dead shall not have died in vain [self-centered]—that this nation, under God [or "god"], shall have a new birth of freedom–and that government of the people, by the people, for the people, shall not perish [Church place] from the earth.

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