William M. Briggs

Statistician to the Stars!

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Bayesian Statistics Isn’t What You Think

A Logical Probabilist (note the large forehead) explains that the interocitor has three states.

Back to our Edge series. Sean Carroll says Bayes’s Theorem should be better known. He outlines the theorem in the familiar updating-prior-belief formula. But, as this modified classic article shows, this is not the most important facet of Bayesian theory.

Below we learn all probabilities fit into the schema \Pr(\mbox{Y}|\mbox{X}), where X is the totality of evidence we have for proposition Y. It does not matter how this final number is computed (if indeed it can be): it can sometimes be computed directly, and sometimes by busting X apart into “prior” and “new” information, or sometimes by busting X apart into ways that are convenient for the mechanics of the calculation. That’s all Bayes’s theorem is: a way to ease calculation in some but not all instances. An example is given below. The real innovation—the real magic—comes in understanding all probability is conditional, i.e. that it fits into the schema. As shown in this talk-of-the-town book.

This post is modified version of one that was been restored after The Hacking. All original comments were lost.

Bayesian theory probably isn’t what you think. Most have the idea that it’s all about “prior beliefs” and “updating” probabilities, or perhaps a way of encapsulating “feelings” quantitatively. The real innovation is something much more profound. And really, when it comes down to it, Bayes’s theorem isn’t even necessary for Bayesian theory. Here’s why.

Any probability is denoted by the schematic equation \Pr(\mbox{Y}|\mbox{X}) (all probability is conditional), which is the probability the proposition Y is true given the premise X. X may be compound, complex or simple. Bayes’s theorem looks like this:
\Pr(\mbox{Y}|\mbox{W}\mbox{X}) = \frac{\Pr(\mbox{W}|\mbox{YX})\Pr(\mbox{Y}|\mbox{X})}{\Pr(\mbox{W}|\mbox{X})}.
We start knowing or accepting the premise X, then later assume or learn W, and are able to calculate, or “update”, the probability of Y given this new information WX (read as “W and X are true or assumed true”). Bayes’s theorem is a way to compute \Pr(\mbox{Y}|\mbox{W}\mbox{X}). But it isn’t strictly needed. We could compute \Pr(\mbox{Y}|\mbox{W}\mbox{X}) directly from knowledge of W and X themselves. Sometimes the use of Bayes’s theorem can hinder.

Given X = “This machine must take one of states S1, S2, or S3”, we want the probability Y = “The machine is in state S1.” The deduced answer is 1/3. We then learn W = “The machine is malfunctioning and cannot take state S3”. The probability of Y given W and X is deduced as 1/2, as is trivial to see.

Now let’s find the result by applying Bayes’s theorem, the results of which must match. We know that \Pr(\mbox{W}|\mbox{YX})/\Pr(\mbox{W}|\mbox{X}) = 3/2, because \Pr(\mbox{Y}|\mbox{X}) = 1/3. But it’s difficult at first to tell how this comes about. What exactly is \Pr(\mbox{W}|\mbox{X}), the probability the machine malfunctions such that it cannot take state S3 given only the knowledge that it must take one of S1, S2, or S3? If we argue that if the machine is going to malfunction, given the premises we have (X), it is equally likely to be any of the three states, thus the probability is 1/3. Then \Pr(\mbox{W}|\mbox{YX}) must equal 1/2, but why? Given we know the machine is in state S1, and that it can take any of the three, the probability state S3 is the malfunction is 1/2, because we know the malfunctioning state cannot be S1, but can be S2 or S3. Using Bayes works, as it must, but in this case it added considerably to the burden of the calculation. In Uncertainty, I have other examples.

Most scientific, which is to say empirical, propositions start with the premise that they are contingent. This knowledge is usually left tacit; it rarely (or never) appears in equations. But it could: we could compute \Pr(\mbox{Y}|\mbox{Y is contingent}), which even is quantifiable (the open interval (0,1)). We then “update” this to \Pr(\mbox{Y}|\mbox{X \& Y is contingent}), which is 1/3 as above. Bayes’s theorem is again not needed.

Of course, there are many instances in which Bayes facilitates. Without this tool we would be more than hard pressed to calculate some probabilities. But the point is the theorem can but doesn’t have to be invoked as a computational aide. The theorem is not the philosophy.

The real innovation in Bayesian philosophy, whether it is recognized or not, came with the idea that any uncertain proposition can and must be assigned a probability, not in how the probabilities are calculated. (This dictum is not always assiduously followed.) This is contrasted with frequentist theory which assigns probabilities to some unknown propositions while forbidding this assignment in others, and where the choice is ad hoc. Given premises, a Bayesian can and does put a probability on the truth of an hypothesis (which is a proposition), a frequentist cannot—at least not formally. Mistakes and misinterpretations made by users of frequentist theory are legion.

The problem with both philosophies is misdirection, the unreasonable fascination with questions nobody asks, which is to say, the peculiar preoccupation with parameters. About that, another time.

Summary Against Modern Thought: The Intellect Is Not The Senses

This may be proved in three ways. The first...

This may be proved in three ways. The first…

See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles. All posts are under the category SAMT.

Previous post.

Two more Chapters this week, and fairly easy ones (for those who have studied since the beginning). We come to an exciting Chapter next week: how does the immaterial intellect contact the body?

Chapter 66 Against those who maintain that intellect and sense are the same. (alternate translation) We’re still using the alternate translation this week.

1 Thinking that there was no difference between intellect and sense, some of the early philosophers were close to the persons referred to above. But that notion of theirs is impossible.

2 For sense is found in all animals, whereas animals other than man have no intellect. This is evident from the fact that the latter perform diverse and opposite actions, not as though they possessed intellect, but as moved by nature, carrying out certain determinate operations of uniform character within the same species; every swallow builds its nest in the same way. Therefore, intellect is not the same as sense.

Notes It is a good joke to say you never see swallows down the Home Depot on a weekend. See also this Ed Feser article “Da Ya Think I’m Sphexy?” about the determinate behavior of animals.

3 Moreover, sense is cognizant only of singulars; for every sense power knows through individual species, since it receives the species of things in bodily organs. But the intellect is cognizant of universals, as experience proves. Therefore, intellect differs from sense.

Notes The “common sense” takes the input from the disparate senses and paints a picture, a unified whole, which the intellect considers. The intellect knows universals, which cannot be sensed. As the next paragraph emphasizes.

4 Then, too, sense-cognition is limited to corporeal things. This is clear from the fact that sensible qualities, which are the proper objects of the senses, exist only in such things; and without them the senses know nothing. On the other hand, the intellect knows incorporeal things, such as wisdom, truth, and the relations of things. Therefore, intellect and sense are not the same.

5 Likewise, a sense knows neither itself nor its operation; for instance, sight neither sees itself nor sees that it sees. This self-reflexive power belongs to a higher faculty, as is proved in the De anima [III, 2]. But the intellect knows itself, and knows that it knows. Therefore, intellect and sense are not the same.

6 Sense, furthermore, is corrupted by excess in the sensible object. But intellect is not corrupted by the exceedingly high rank of an intelligible object; for, indeed, he who understands greater things is more able afterwards to understand lesser things. The sensitive power therefore differs from the intellective.

Notes Bright lights overwhelm, blinding insights do not (and now you understand the metaphor).

Chapter 67 Against those who hold that the possible intellect is the imagination. (alternate translation) We’re still using the alternate translation this week.

1 The opinion of those who asserted that the possible intellect is not distinct from the imagination was akin to the notion just discussed. And that opinion is evidently false.

2 For imagination is present in non-human animals as well as in man. This is indicated by the fact that in the absence of sensible things, such animals shun or seek them; which would not be the case unless they retained an imaginative apprehension of them. But non-human animals are devoid of intellect, since no work of intellect is evident in them. Therefore imagination and intellect are not the same.

Notes A mental picture produced by sensation of a possibility is not apprehension of a universal. As the next paragraph emphasizes. If you’re stuck for something incorporeal to use as an example, pick a number, any number.

3 Moreover, imagination has to do with bodily and singular things only; as is said in the De anima [3], imagination is a movement caused by actual sensation. The intellect, however, grasps objects universal and incorporeal. Therefore, the possible intellect is not the imagination.

4 Again, it is impossible for the same thing to be mover and moved. But the phantasms move the possible intellect as sensibles move the senses, as Aristotle says in De anima III [7]. Therefore, the possible intellect cannot be the same as the imagination.

Notes Phantasm, the mental image provided by distillation of the senses using the bodily apparatus. (Wow.)

5 And again. It is proved in De anima III [4] that the intellect is not the act of any part of the body. Now the imagination has a determinate bodily organ. Therefore, the imagination is not the same as the possible intellect.

6 So it is that we read in the Book of Job (35:11): “Who teaches us more than the beasts of the earth, and instructs us more than the fowls of the air.” And by this we are given to understand that man is possessed of a power of knowledge superior to sense and imagination, which are shared by the other animals.

Tolerance Brutes Have Esolen On The Ropes

What do you think about a guy who goes around writing things like this? Recalling, as you read, Jesus and the Lord God and the Holy Ghost are one:

Now we may glimpse the vast sweep of the condemnation of sodomy as leveled in Leviticus. “You shall not lie down with a man as with a woman,” says the Lord, for “it is to’evah,” typically translated as “an abomination.” But the word and its near relations in Hebrew suggest three things: going badly astray, that is, wandering to your confusion and ruin; repugnant filth, as of excrement and loathsome disease; and idol-worship, with its combination of the bizarre and the disgusting—think of Moloch and the charred little ones. To lie with a man as with a woman is to engage in unreality, un-creation; it is like fouling yourself with excrement, or like eating filth not meant for food; it is like falling in adoration of the idols that are tohu w’vohu, waste and void, like the emptiness of the world before God said, “Let there be light.”

The reminder about the Trinity was to show that Jesus, who is one with the Lord God, said some fairly, well, merciless things to say about certain acts, things the modern world would rather not hear about. Rather, the world doesn’t mind hearing things like this, or like anything, so much as has an intense dislike knowing there are actual people who believe these words.

Anthony Esolen, the author of the passage, believes the words. Diverse words, too, given their variance with the spirit of the day. And we all love diversity, don’t we?

Well, there’s diversity, as in Webster’s “multiplicity of difference; multiformity; variance”, fine things all (up to a point). And then there’s Diversity, as in rigid strict mandatory unbending ruthless quota-bearing uniformity. Capital-D Diversity is the not so much the idol of the age, but a blunt instrument of political power.

Esolen penned an essay on the Big D, which was given by its editors the appropriate title “My College Succumbed to the Totalitarian Diversity Cult“. His college is the ostensibly Catholic Providence College. In it he asked:

Is not diversity as it is now preached a solvent for any culture? That is, supposing that the people of a tribe in the interior of Brazil are compelled to accept cultural diversity for its own sake, rather than merely adopting and adapting this or that beneficent feature of another culture (something that people have always done), will that not mean that their own culture must eventually vanish, or be reduced to the superficialities of food and dress?

Is not diversity, as currently promoted, at odds with the foundational diversity built into the nature of the human race, the diversity of male and female, to be resolved most dynamically and creatively in the union of man and woman in marriage?

These were the beginnings of the questions and intelligent comments. And the start of his troubles.

You know what happened. A brace of brats filled their diapers and went bawling to the Providence President who, befitting his sober and stately and priestly office, promptly capitulated. Perhaps he couldn’t deal with the stink. Or perhaps he felt it his duty to toss, albeit softly, Esolen to the baying mob, this being the default and reflexive action of college presidents everywhere since the Sixties.

That’s the briefest summary. Read about the full hilarity here and here. Don’t miss the petulance of the Faculty, which is circulating the petition and document “Breaking the Silence“. It begins:

As PC Faculty, we pledge to break the silence around systemic racism and discrimination on Providence College’s campus. While we vigorously support free expression, recent publications on the part of PC faculty have involved racist, xenophobic, misogynist, homophobic, and religiously chauvinist statements.

The poor bloated pink-faced Faculty! I can’t speak for the reader, but my awareness has just been raised. No Siberian gulag came close to the horrors present at Providence. It’s worse than you might have thought. In Siberia, Stalin & Co. at least let dissidents forage for scraps of pine bark to eat. At Providence, that pestilential hotbed of racism and X-phobia, people have to pay some $60,000 per annum for the privilege to be abused—and that figure includes room and board. I wonder if they have meatless Fridays. Skip it.

“All Faculty, Briggs? Come now. There must be some Traditionalist Reactionaries remaining who haven’t self-emasculated and who defend poor Esolen.”

We know of one. A female creature (as Mike Royko would have said) by the name of Holly Taylor Coolman, who teaches theology. She gave an interview at Crux where she wondered aloud about the diminishing Catholic identify at Providence. About the row she said, “Our campus has seen increasing frustrations in the last few years, and I came to feel that a big blow-up was almost inevitable.” Asked if the disruption was between secularists and Catholics, she answered:

Not exactly. Another group immediately involved here are some of the people who tend to fall on the margins in our community-and also those supporting them. They have serious concerns about systemic forms of exclusion. (And here, too, are a number of concerns that I myself share.)

They can see, for example, that Providence College’s 100-year history includes almost nothing of the African-American experience, or of Hispanic culture and tradition. In the last few years, the college has made a concerted effort to recruit more students, faculty, and staff from underrepresented groups, but frankly, it hasn’t always succeeded in offering needed support once they arrive.

Coolwoman is right. A diligent search reveals no history of Hottentots, Maori, or Samoans at Providence. Damn few Finns and Latvians, neither. About horse lovers and other underrepresented groups, a count cannot be easily discovered, but it’s good money these folks were shunned.

But then, you didn’t see a lot of Irish learning to click in the Kalahari, an observed and undeniable fact which can only be, so Diversity theory assures, the result of systematic exclusion and racism. Probably sexism, homophobia, and every other manner of intellectual vice, too.

After all, should not a black man amble up to Providence to learn all about being black in the current climate instead of reading Shakespeare, Newton, Newman, Thucydides, Dante (translated by Esolen), Euclid, et cetera? Is not college about finding others who share your identify and reveling in that identity, however limited in place or time it is, making others acutely aware of that identity, and making that identity the sole basis and purpose of your life, and not about learning the best that was thought and said?

Esolen doesn’t think so. That fine gentleman is aware that you don’t need to go to college to know what you already know or believe what you already believe and can’t be talked out of. If all you care about is “social justice”, skip college, go right into “activism”, or stare at your identity in a reflection at the lake like that Greek fellow, and save yourself, or your parents, a bundle.

Problem is, Esolen is surrounded by social justice warriors who, though they lack in intellect, fortitude, and cleverness, are great in number and abound in indefatigable self-righteousness. The strain of defending Truth and Commonsense might be getting to him. This we gather from his recent essay in The Catholic Thing. Pardon the extensive quote.

Because of recent events at the school where I teach, Providence College, I have come to see that the winning side of the so-called culture wars has no interest in rational or equable conversation about the neuralgic issues of our time. I use the word interest advisedly. They have nothing to gain by it.

We can ask, till we are exhausted from asking, what they mean by “marriage,” if the thing is not rooted in the fundamental biology of the human race, and exactly what justifies any boundaries at all wherewith they suppose they can limit the definition. If man and man, why not man and woman and woman?…

It won’t matter. The aim was never rational coherence, or even a concern for the common good. The aim was power: to get what they wanted, to keep it, and to crush those who would question their right to it.

So they have the power now, power gained not by argument, whereof there has been very little, but by a combination of political force, mass media sentimentalism, public lassitude, and an anti-culture of licentiousness and the neglect of children.

Why bother to argue?…

Why indeed? Arguing only exposes you as an enemy of The People, and targets you as problem that must be dealt with—as Esolen learned with his Diversity essay. Argument and question are taken as political attacks, which, in a sense, they are. The Faculty who have surrendered to the World “are by nature no better and no worse than anyone else. It’s just that they have, whether they acknowledge it or not, exchanged the God of heaven for a god of prestige and power. Politics is the god.”

As long as you possess the “right” politics, you are like the pagan who has secured divine favor by the “right” sacrificial rituals.

You may then do as you please. You may, for example, go out of your way to ruin reputations and careers and turn families upside down; all justified, all for the good of the “cause.”

It is obvious our friend needs cheering up. What practical things can we do to help?

Pray, for one. It is still and always the best weapon. If you’re up for more earthly activities, you can buy his books. His Dante is excellent, and his The Politically Incorrect Guide to Western Civilization is great fun. There are many more.

Bonus Embattled conservative professor refuses to be silenced: ‘They did not kill the lion’.

Stream: Fake News of the Decade? Or, Does Russia Rule The World?

This is posting here a day late, because I had already promised the Vox Day post yesterday. Yet there is still some fun to be had at the Stream: Is the Trump/Russia ‘Dossier’ the Fake News of the Decade?

Fake news or conspiracy theory? Or the most epic troll since Dan Rather was conned into accepting forged documents about George Bush? Or a hilarious amalgam of all three?

All elements of this story are as yet unknown, but what is unfolding has the makings of historical high comedy. Here’s a rundown.

Buzzfeed, a website whose specialty is celebrity tittle-tattle, asinine quizzes such as “Which ‘Pixie Hollow Fairy Are You?“, and get-skinny-quick-by-petting-cats articles, published a document, which they gave the graduated title of dossier, which purported to show how Russia, under the devious and genius scheming of Vladimir Putin, had been grooming and bribing Donald Trump for many years, and blackmailing him by threatening to reveal perverted sexual practices, so that Trump would be induced to enter the US Presidential election, win it by secret dirt supplied by Russian intelligence agents, and so place the once United States of America under the control of a foreign government.

Yes, really.

Go there to read the rest.

The last lines of the piece: “The story isn’t over. The news on why Steele wrote the document, if he wrote it, and why, including who paid him for it, is bound to generate even more fun.”

Assuming (it’s not a stretch) that Steele wrote the document, how did he arrive at the contents? Did he make it up whole cloth, filling in bits with suppositions he knew could never be checked because many of the events took place long ago in Russia? Was he given dirt by Republican NeverTrumpers and assured by them it was true? Was Steele himself duped by actual Russians who couldn’t believe their luck?

Did Steele, or whomever, do it to scam McCain and other NeverTrumpers? Would a real live MI-6 agent really think fictional scribblings would fool real live CIA and FBI agents? Maybe he thought the document would never be made public, because the Republicans who asked for it during the election would never release it, since it would paint the GOP in a bad light. But now that it has become public, Steele has scarpered, to use a Britishism. Is he now in Moscow sharing a flat with Edward Snowden?

If Steele was duped, are we going to hear of tales of a wigged Debbie Wasserman Schultz faking a Russian accent whispering in Steele’s ear? “Listen very carefully…I shall say this only once!

Addendum: see also this.

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