William M. Briggs

Statistician to the Stars!

Page 145 of 707

Probability’s Empirical Bias

Big Jule's dice have no spots. But probabilities are still possible.

Big Jule’s dice have no spots. But probabilities are still possible.

Bit of logic (a favorite example) from our teacher Lewis Carroll: given “All cats understand French and that some chickens are cats”, what can we say about the proposition “Some chickens understand French”?

We can say that it follows, that its probability is 1, that it is proved, that it is deduced, true. True conditionally, true given the premises, locally true.

But we also know it is false. Universally false, false because we observe that no cats understand French and that no chickens are cats. Universally or necessarily false, but conditionally true. And also conditionally false, conditional on different premises (such as “No cats understand French” and “No chickens are cats” and, also necessary, “No non-human animals, and very few human animals, understand French”).

Conditional truth is no trouble for logic, because logic is merely the study of the relation between propositions. Logic has nothing to say about the propositions themselves, not formally. Not about where they come from, their utility, their practicality or their lyricism. This is why the premises do not need to be empirical for logic to “work”.

For instance, given “All Martians wear hats and George is a Martian” it is conditionally true that “George wears a hat”—even though we know there are no Martians. Examples abound. See this site for a wealth of Carroll’s fun logic puzzles.

Probability, part of logic, is also concerned only with the relation between propositions and not the origin or usefulness of those propositions. We deduced “George wears a hat” had probability 11, just as we deduce the probability of “George wears a hat” is 3/4 given the premise “Three-quarters of Martians wears hats and George is a Martian.”

Probability, like logic, has no difficulty with non-empirical propositions. This is how we know it is (conditionally) true that “One creature must come out” given the premises “A gnome, fairy, and Godzilla are in a room and only one must come out”, and it is how we know the probability of “Godzilla comes out” is 1/3 given those same premises. We deduced the certainty of both conclusions (we are certain the probability is 1/3 in the second case).

Probability got its start answering empirical questions, mostly about gambling and people’s guilt (see Jim Franklin’s The Science of Conjecture: Evidence and Probability Before Pascal). Aristotle also relied on many empirical questions when delineating logic, but somehow logic came to be seen to be general, whereas probability hasn’t. Probability is empirical to many.

The empirical bias in probability is strong. It accounts for the frequentist fallacy that no probability exists except limiting frequencies of empirical events. A frequentist must remain mute about hat-wearing Martians, and about every other non-empirical proposition, including counterfactual ones. For example, historians often debate about whether there would have been a World War II in Europe if America had not joined World War I. Germany would have won WWI, these historians say, thus ensuring a kind of peace, or at least a Germany victory would not have created the conditions necessary for the National Socialists in that country and the International Socialists in Russia to come into existence.

Now those are perfectly understandable (and debatable) propositions, as are most “What if?” questions. And probability (and logic) can handle them, but no system which is empirically based can. Given (Stove’s example) “Bob is a winged horse” it follows that “Bob is a horse”, a statement which makes sense in logic and probability-as-logic, but a frequentist must pretend it is incomprehensible because there is no way to construct any empirical relative frequency.

More subtly, the empirical bias accounts for de Finetti’s and Ramsey’s error of defining probability as coherent gambles. This is backward. That coherent gambles are good decisions is result of probability, not its definition. (See this about Dutch Books.) There will be no “payoffs” for “events” which will never occur (or never not occur).

Besides, coherence is a weak criterion and mixes up consequences of evidence with evidence itself. A subjectivist can say the probability Godzilla comes out is 0.015764 (or any 0-1 number). As long as he also insists the probability Godzilla does not comes out is 1 – 0.015764, he is coherent. And you cannot prove his probability is wrong if probability is a gamble. No empirical evidence will ever be available.


1In measure theory “with probability 1” has a technical meaning which I do not here use. I mean this phrase in plain English.

Sustainability: Higher Education’s New Fundamentalism


The tile of today’s post is the same as the new book by the National Association of Scholar’s Rachelle Peterson and Peter Wood, released yesterday and which can be downloaded by all and one. I’ll later review that book, but today a report on the NAS’s conference on the same topic held in the darkening shadow of the United Nations Tuesday. Your Roving Reporter was there.

Peter Wood is also President of the NAS and was master of ceremonies. Woods forte is the joke most subtle. He said the fight against rising ocean levels was foreseen by Hamlet (“…to take arms against a sea of troubles…”).

Anyway, Wood said the sustainability movement got started when college presidents were lured into signing the President’s Climate Commitment, a device which elites use to signal to other elites their devotion to all things progressive. This document is usually signed by college leaders in absence of any meddling from actual professors, who might damper ardors, particularly on matters scientific.

The American Enterprise Institute was represented by their boss, Arthur Brooks, who reminded us of Bach’s answer to the question “Why do you do write music?”. Bach said, “For the glory of God and the good of mankind.” Brooks also rightly said that since around 1970 the percent of folks living in abject, starvation-level poverty has declined by some 80% because of capitalism and adherence the rule of law. Yet, strangely, both principles are under attack.

His advice to fight the rising tide of progressivism was not to fight against things, but to fight for people. It was his opinion, shared by most of the room, that the Republicans never lost their minority mindset. That’s partly true, but it’s better to say that Republicans are the party of slightly smaller government.

Herb London, Chairman of the NAS, railed against sustainability’s “censorious passion”, its insistence on “ideological conformity”. He said that at one time (only the oldest of old timers will remember this) colleges sought to study Western culture, but that now their “purpose is to repudiate” that culture. This accounts for the “soft totalitarianism” found at colleges, strengthened by the “practice of consensus, which is really the avoidance of confrontation.”

He mentioned some universities have banned trays in student cafeterias or have otherwise annoyed them with other trivial save-the-earth actions. Why? To make students pay a “psychological sustainability tax.” Nudging students, the favorite buzz phrase, is thought superior to old-fashioned indoctrination.

Sp!ked magazine was there. Have you never see it? I recommend their Frank Furedi. Editor Neil Ross spoke. He said the magazine believes “Western Enlightenment ideals are worth standing up for.” Perhaps they are, but it’s those ideals that have caused (or are) progressivism, so it’s not clear what standing up for them would do.

Ross was also against nudging: “Leave students to make their own decisions.” Also good, to an extent. Students making their own decisions in ignorance, say in picking what to study, has not led to improvements in universities. Yet Ross was spot on when he said that those who would nudge think they can perfect the species through science. This is scientism.

Finally, the star of the show, because she was the main author, Rachelle Peterson. She emphasized “sustainability” had two meanings: “wise maintenance of a grand inheritance” or “desperate survival [in the face of looming apocalypse]”. Guess which ones colleges have chosen?

She said the endless elbows in the ribs policies were designed to produce “socially optimal behavior”. Colleges now have “eco reps” who have the job of “shaming students” into doing what elites want them to. That raging “eco-morality” produces inconvenience nobody disputes. But what’s less understood is that inconveniences are a “primary goal”. “Deprivation” is seen as a good.

Why? Sustainability, as has often been noticed, is a religion. It has, Peterson said, “abstention, fasts, purity rituals designed to cleanse guilt and to improve mortal rectitude.”

Through the so-called precautionary principle, which she called an “eco-themed Pascal’s wager”, the religion demands that we “minimize risks at all costs”. But this always and necessarily leads to “regulating or eliminating.”

I asked Peterson what were the main drivers of sustainability. Besides those mentioned, she thought identification with progressive social goals was the strongest incentive. This seems true because throwing a scientific fact, more matter how solid or how forcefully hurled, against an sustainability activist never leaves a mark. Good science bounces right off of them.

Global Warming Is Anti-Science

Presiding Bishop Katharine Jefferts Schori  says the earth has "rights".

Presiding Bishop Katharine Jefferts Schori says the earth has “rights”. Image source.

It is not anti-science to say that humans influence the climate, because, of course, humans, and every other species on the planet, influences the climate. At the least, humans move through the atmosphere, which influences it and hence the climate. Only a science denier would deny this.

It is an semi-open question how much humans, and each other species, influences the climate. What is not an open question, indeed it is a question as closed as can be, that the global-warming-of-doom promised by organizations like the IPCC has failed to materialize in the ways these groups have promised. Only science deniers deny this.

We know the IPCC (and etc.) is wrong because the warming they have swore would happen did not happen. We know that the models on which the IPCC relies did not perform these past two decades as well as a simple “model” stating that next year will be like this year. Climate models have no skill.

Only a science denier would deny this.

So what do we make of the New York Times, Natural Resources Defense Council, FEMA, the Episcopalian Church? These organizations deny science. And not only that, they would have everybody else deny them too.

The New York Time’s Justin Gillis, who is, as regular readers know, unteachable on this subject, is a True Believer. He said recently, “It is a lie to say that global warming poses no danger”.

A lie.

He must know the scientific fact that climate models have no skill, and therefore should not be used and cannot be trusted. Yet he says to acknowledge this scientific fact is a lie.

A lie. Fighting words.

FEMA has bruited a plan “making it tougher for governors to deny man-made climate change.” Says the report, “the agency will approve disaster-preparedness funds only for states whose governors approve hazard-mitigation plans that address climate change.”

No pinch of incense in the fire, no federal support. Which many would say—I say—is a good thing. Anything that restrains the reach and power of Leviathan is to be encouraged. But that is a political argument, and not a scientific one. Yet FEMA, which should know the science, which must know the science, must deny that science, is denying the science, to make their own, opposite political argument. So for them, at least, global warming is politics.

In the same report, one Becky Hammer, an attorney with the Natural Resources Defense Council, an activist organization, said, “If a state has a climate denier governor that doesn’t want to accept a plan, that would risk mitigation work not getting done because of politics”.

Since the science is clear, and settled, that the IPCC models have no skill, it can only be that the phrase “climate denier” means “one who won’t play along”, or something similar. This necessarily follows, because knowing that IPCC models have no skill is not consonant with believing in a climate of doom. Why? Because there is no evidence for the climate of doom except through these models, and we have proved the models have no skill.

Once again, proof that global warming is politics—at least for the NRDC.

Earlier this week, Katharine Jefferts Schori, Presiding Bishop of the Episcopal Church (yes, they still exist), said, “people who reject climate science are turning their backs on one of God’s most generous gifts: knowledge.”

She said, “Episcopalians understand the life of the mind is a gift of God and to deny the best of current knowledge is not using the gifts God has given you”. And she said, global warming was a “moral issue, in terms of the impacts on the poorest and most vulnerable around the world already.” Finally, she said the earth had “rights.”

It is through God’s gift of knowledge we know the IPCC models have no skill and should not be trusted. Yet Schori, herself once a scientist, denies that science, turns her back on it, even. Why? The clue is in her quip, “World Ends: The Poor Disproportionately Effected.” In other words, for Schori, global warming is politics.

Since the science that bad theories make for bad forecasts is settled and, indeed, basic, it should be known by every working scientist. Why don’t those scientists then admit this? Why aren’t they taking people like Gillis, Hammer, and Schori to task publicly when they say false and unscientific things?

Could it be that global warming is political even for scientists?

A Probability Non-Paradox


Before you is a box in which is a slip of paper on which is written either ‘0’, ‘1’, ‘2’, or ‘3’. Given that premise, what is the probability of X = “‘3’ is written”?

Right: it’s 1/4.

Notice, incidentally, the probability does not tell us how that number came to be written on the paper, nor why the paper is there, nor how you will “draw” the paper from the box, information which is any case is irrelevant. All that we deduce from the premise is that writing exists and you are partially but not wholly ignorant of what it is.

Different set up. In a new box (or even the same!) are three marbles, each either white or black. Given that premise, what is the probability of X = “The number of white marbles is 0 (or 1, or 2, or 3)”? First note that we deduce there may be no white marbles, or no black ones, or any combination of the two, as long as there are 3 in total.

The probability may not be as obvious, and indeed the formal mathematical proof begins with the hypergeometric “distribution”, noting the logical equivalence of constants, and carrying all this forward. You can take my word—or Laplace’s, a more eminent authority, and the man who first derived it—that the calculations produce a probability of 1/4 for 0 white marbles (and 1/4 for each of 1, 2, or 3), which might be in line with your intuition, as suggested by the first example.

Again notice that there isn’t any word about “drawing” the marbles out, how the marbles got their color, or anything else to do with causes. Though some thing or things must have caused the color and number of the marbles, but we know it not.

A third set up. In a new new box are three marbles, each either white or black, and we’re going to draw out three of them. Given that premise, what is the probability of X = “The number of white marbles is 0 (or 1, or 2, or 3)”?

Let’s enumerate. Given this premise, we could see any of the following sequences, and number of white marbles:

  1. W1W2W3, 3
  2. W1W2B3, 2
  3. W1B2W3, 2
  4. B1W2W3, 2
  5. W1B2B3, 1
  6. B1W2B3, 1
  7. B1B2W3, 1
  8. B1B2B3, 0

This indicates the probability (given the premises) of 0 whites is 1/8, of 1 white is 3/8, of 2 whites is 3/8, and of 3 whites is 1/8.

Something has changed. The second example, conditional on very similar premises to the third, gave a probability of 1/4 for each possibility, while the third example gives varying answers. What gives? “Paradox!” answer some. Howson and Urbach, in their influential Scientific Reasoning: The Bayesian Approach, argue from the apparent paradox to a justification of subjective probability (p. 59-62 in the second edition; pdf first edition). Besides being wrong, this is defeatist. “We don’t know what the probability should be so we can make up whatever feels best” can scarcely be a satisfactory answer. (Though it is to all “subjective Bayesians”.)

Still, it’s odd. Similar premises give completely different answers. To see what’s happening, let’s change the premise in the first example so that the slip of paper has the marking “W1W2W3“, or the marking “W1W2B3“, or, etc. Given that premise, what is the probability of the marking “B1W2B3“? Easy: 1/8—and it’s the same for any marking. The probability would also be 1/8 were the premise that the paper had the marking ‘1’, ‘2’, etc., ‘8’.

So what makes writing the labels 0 through 3 the same (in probability) as “In a new box are three marbles, each either white or black”? And why are they dissimilar to “In a new new box are three marbles, each either white or black, and we’re going to draw out three of them”?

Cause; rather, our knowledge of causes. All we require of the markings in the first example is that they be distinct. All? Well, not quite all. We also require that the label, whatever it is, be written in advance, by some (unknown, unspecified) cause. That cause fixes the labels or balls in advance. Our knowledge of cause in these cases is extremely limited: we only know that a cause, or causes, must have been present, and we know the outcome.

But there is no way to think of drawing out marbles without envisioning some kind of drawing-out cause. If you are practiced at simulation, this will make sense—and all of us are practiced at simulation. Think coin flips. There is no way to imagine, or rather to manufacture, a string of three flips, or three anythings with dichotomous outcomes, that does not make reference to physical causes.

Our understanding of causes in the two situations is a real, and huge, difference. At least for the very small. Once we get going and start taking observations, (it can be shown) the two views “collapse” to the same, especially for “large N” (many observations).

This is not the only system in which the measurement process dramatically changes our perspective, as the not inapt comparison to quantum mechanics reveals. Anything-we-know-not-what could have fixed the labels/constituents of the box, whereas not just any old thing could make a string of white/black (0/1, etc.) to emerge from a box. Far from being a paradox, the differences in probabilities highlight the importance of measurement and the knowledge which comes with it.

Conclusion? There is no problem with logical probability, a.k.a. probability as argument.

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