## What Statistics Really Is: Part Last

A picture I swiped from Wikipedia that kinda sorta has something to do with statistics.

Recapitulation: we have Pr(p|qm) where p is a proposition of interest, q the evidence we have compiled in the form of observations and so forth, and m is the “I believe”, i.e. the model, the whole-cloth thing which says, “I believe the uncertainty of p in the presence of q is characterized by this parameterized probability distribution.”

We began with p = “Tomorrow’s high temperature will be 72F” but learned that if our “I believe” was m = “A normal distribution with parameters a and b” then the probability of p is 0 no matter what a and b were and regardless of q. Bummer. So we switched to p = “Tomorrow’s high temperature will be greater than 72F”, which might be given a non-zero probability with this m.

But we haven’t yet found joy because we know nothing about a and b, which are absolutely required if we want to stick with this m. Bayesians make one guess, frequentists another. Both do so by appending m with extra “I believes” which justify the guesses.

Two crucial points. One, there is nothing universally or automatically wrong with adding “I believes”, for these might be true (with respect to other relevant evidence about q and p). People can guess correctly. Of course, they can also guess incorrectly. Two, regardless of the truth of the guesses (of both m and the parameters), assuming they and q are true produces the correct probability of p.

This second fact is difficult to keep in mind because critics of the probability of p will often attack the value of the probability itself, which (assuming no calculation errors, which are rare) are never wrong. What might be wrong are q, m, or the m’s appendages.

Example: q = “Just a certain proportion of Martians wear hats and George is a Martian”, m = “I believe this certain proportion is 0.6″, with p = “George wears a hat”, then Pr(p|qm) = 0.6, a correct and true deduction. Nothing in the world can make this conditional probability false (and all probability is conditional).

Q or m can be false. But if they are it has to be with respect to some other evidence, for truth, falsity, and probability are all conditional (even the most humble truth is known at least with respect to our intuitions). We could assess m by counting all Martians who wear hats and divide by the number of Martians. This proportion will either be 0.6 or not. No sample of Martians which don’t include all can falsify or verify m, though. No observation can falsify the first clause of q, but direct measurement can tell us about the second. And then there’s the observation that there are no Martians!

Back to temperature. We’ve used the observations (part of q) to make guesses of a and b and used m to deduce the probability of p; say, 93%. Now this 93% applies to p, of course, but suppose upon observation that p was false—this is another condition, not q or m, but a new observation, such that Pr(p|obs) = 0; that is, the temperature was less than 72F. The 93% was still true, but it doesn’t feel “close”.

It’s at this point we must “invert” the problem and ask what the observation says about q or m, and not about p (questions about p are trivially answered; either Pr(p|obs) = 1 or Pr(p|obs) = 0, assuming the observation is measured without error). That is, we want to know things like how true q or m are with respect to the observation. Three things are possible.

First, m (which includes the guesses of its parameters) might be true but q false or unlikely. For instance, suppose q contained measurements about temperatures in July for Tucson but we applied q and m for January in Detroit. A silly error no one would make? The opposite: misapplying the circumstance is the most common error, and also the most unrecognized. This is because q nearly always contains tacit premises, evidence which is in the investigators’ minds but which isn’t written in the formal statistical modeling phase, and these absences cause people to forget it’s there. This is the danger of shorthand notation.

Remember: the analysis is valid if m is true and for the kind of situations that match q. Psychologists, sociologists, educationists and the like abuse their models by forming q’s from (usually) nearby college students and then implying that q actually represents all human beings in all cultures in all times. This is why it is mandatory to fully specify q, a step which is rarely or never done. Some researchers take stabs at it, but unless their objects of study are controllable, in-the-small physical events (which don’t have volatile personalities), then they usually do a poor job of it.

Second, q might be true but m (or its parameters) might be false or unlikely. This is an easy situation because there are simple ways of verifying models, as long as the model is used regularly enough to build up evidence about its performance. Terms like “proper scores”, “calibration”, and the like come to play. Think of it as plotting instances of p against its predictions: if these line up, m is more likely true; if not, then not.

Funny thing about this is because most of statistical practice is psychotically fascinated about the parameters of m, this verification step is rarely done (I only ever see it in physics models). Researchers are confident that by telling you their guesses of parameters that they have told you everything you need to know, a practice which implies that m and q are—of course!—true.

Third, both m and q are false. These are difficult situations because one hardly knows where to begin. We’ll know something has gone kerplooey because the probabilities of p won’t match up to the real true states of p, but we won’t know why. We might automatically blame m, the usual step, and try to “fix” it, but since q is broken too, these efforts will largely be in vain.

Last word: all of what was said applies to all those probability models you—you—use daily, and not just in those that are written up formally in some obscure journal. Every time you make any judgment of uncertainty, you are applying “probability models” with some q in mind. You just weren’t aware of it.

## Probability Is Logic: Philosophy of Probability & Statistics Video Series

I’m starting—and even continuing, if there is interest—a new video series on the philosophy of probability and statistics.

Look at those retro production values! I boast of them following Julia Child who advised to say when anybody asked as pointed question about a meal, “It’s supposed to be like that.”

Probability is a matter of logic. It does not exist; it is not a physical, real thing. It measures how true one thing is with respect to another, which is to say, how true one proposition is given a set of premises or data. When premises change, so does the probability. This is demonstrated using divination stones, bwa bwei.

————————————————-

Note to computer geeks. I don’t have a camera, so I recorded this video with a Logitech webcam on my Kubuntu machine (the minimal Dell laptop). I tried ffmpeg, cheese, but settled on VLC for recording. I had endless difficulties with the audio being out of sync with the video, a very common problem. Except for the tip to install everything related to gstreamer, searching for solutions did not help. Everything failed.

Finally I recognized that the audio was jumping each time the computer dimmed its screen, which it did automatically at stated intervals. So I switched off of the power management and found joy.

I used kdelive to edit the video, and to the best of my ability compensated for the fluctuating white balance and other imperfections. Many flaws remain, as is obvious. Including the frightening mien of the presenter. But life is imperfect.

## On Intelligence & Religiosity

A girl with a flying spaghetti monster on her head. Wow, must she be smart!

Take a pencil and paper—do this—and write down the most intelligent people who have ever lived. Most brilliant in any field of endeavor, now. Who were the best of us? Make it at least twenty entries. Don’t peek below until you’ve finished.

Done? First thing is to count how many in your tally have died in the last, say, fifty to one-hundred years. Anything more than about ten-percent proves you are a product of a stunted educational system and that your opinions about what follows aren’t worth diddly. You folks, wounded as you are, just sit back and listen.

Those who you found yourselves counting women or non-whites, please click here and have a nice day.

Now that that’s settled, time for the test. How many of your luminaries believed in God? That’s right: most, probably all. What can we glean from this? First, that you could have multiplied this list many times and have come to the same conclusion. Second, that many, many of those far above us believed in a metaphysics shockingly disjoint from the one au courant.

Far from being humbled by these observations, modern skeptics might claim, “Culture! These fellows existed in times where it was considered acceptable and normal to be believers. Thus they believed; why, they even used their intelligence to justify their believing.”

That so? Well, today many of the bright claim to be non-believers, and if the skeptic is right about people being at least partly a reflection of their culture, then non-believers are so because of culture. It’s cool to be a non-believer, even mandatory in the sense that metaphysical talk is unwelcome at social gatherings, especially in academia. What’s more fun to join a campus group of “free thinkers” to giggle about flying spaghetti monsters and coming to agreement about how much smarter the group is than those foolish and stupid believers?

Enter the peer-reviewed paper “The Relation Between Intelligence and Religiosity: A Meta-Analysis and Some Proposed Explanations” by Miron Zuckerman, Jordan Silberman, and Judith A. Hall in the journal Personality and Social Psychology Review.

This was a “meta-analysis”, i.e. a disreputable conglomeration of other studies which hopes to prove what the individual studies could not. This meta-analysis “showed a significant negative association between intelligence and religiosity.” They mean “significant” in the wee p-value sense and not in any real consequence. And by “association” they mean linear correlation, the weakest and least generalizable of all statistical measures.

The individual studies cobbled into one were extremely heterogeneous, too, using a wide range of “intelligence” measures: GPA, syllogism tests, “Immediate free recall” exams, Peabody picture tests, and on and on. How did the authors compensate for these differences? Answer: they did not. This is proof enough the meta-analysis is of little to no worth. But academics are no easily discouraged, and onwards they plowed to measure religiosity. Which, again, was measured in a huge variety of ways and left uncorrected and uncontrolled in the meta-analysis. Thus there is no reason whatsoever to look at any of the numerical results, as they have no meaning.

It is still interesting to exmaine the explanations the authors put to these numbers, as these tell us a great deal about the culture which drives disbelief, particularly the modern trend in which non-believers boast (endlessly) of their brilliance. Many atheists, proving their tone-deafness, even call themselves “brights”.

The authors of the paper claim “intelligent people are less likely to conform and, thus, are more likely to resist religious dogma”. The first clause is false. Intelligent people are people and, as we’ve already agreed, are as likely as other people to conform to the culture about them. On campus, this culture is atheistic and anti-religious.

The second “finding”—which you’ll notice cannot be derived from the data the authors use, but must be assumed—”intelligent people tend to adopt an analytic (as opposed to intuitive) thinking style, which has been shown to undermine religious beliefs.” This implies religious people do not adopt analytic thinking “styles”, which is false; and it is also false that thinking analytically undermines religious beliefs. We proved this with the lists we all wrote.

Their last speculation: “several functions of religiosity, including compensatory control, self-regulation, self-enhancement, and secure attachment, are also conferred by intelligence.” This is plain nonsense. Intelligence by itself does nothing, as should be obvious even to an academic. It’s the uses to which intelligence is put that matters.

The real finding? Academics never recognize their own biases.

Update I had thought it obvious, but…the lesson to be had in writing the names of our betters was humility. It was (I thought) not a definitive proof that the smartest, brightest, most intelligent, and most capable people of all time were believers that therefore God exists. But it should make us moderns (you would have supposed) far less cocky.

—————————————————————-

Thanks to Twitter user Intrepid Wanders ‏@intrepidwanders who told me of this study.

## Researchers Invent Exciting New Disease: Poverty Blindness

Test yourself today!

There must soon come a day at The Onion, the satirical newspaper, when the editors will gather in bewilderment and say to each other, “Fellas, there’s no use. The real headlines are more preposterous than any we can write. Time to close up shop, sit back, and watch civilization crumble from a safe distance.”

If the new peer-reviewed paper by William Ventres and Geoff Gusoff entitled “Poverty Blindness: Exploring the Diagnosis and Treatment of an Epidemic Condition“, published in that notable organ the Journal of Health Care for the Poor and Underserved, is any indication that day is coming soon, soon.

Now these gentlemen, as near as I can tell, appear to be great, big-hearted guys. I mean this. They are from El Salvador, a country which knows genuine, heart-breaking poverty, not the kind we have here where many of “the poor” have houses, cars, cell phones, enormous televisions, and much else. No, Ventres and Gusoff are concerned, as they should be, as we should all be, as we must be, about grinding, life-threatening, despair-inducing, lifetime poverty.

It is true and sickening that poverty, as they say, is “one of the central dramas of human history.” But is it also so that “Poverty is pervasive yet little noticed due to the marginalization of those who endure it”? This is doubtful; no, false. “The poor” are constantly in the public eye. (What’s missing are poor people.) But not in El Salvador, a place where evasion of the truly poor “is virtually impossible.”

Since this is so—since everybody sees poor people—but poverty is still present, there must be a reason for its continuance. Why? Because one of the central myths of the Enlightenment is that once somebody’s “awareness” of some problem is “raised”, i.e. once they are educated, the problem is supposed to evaporate as if by magic. If not, if the problem persists, it can only be because evil forces block the magic.

In this case, the evil is disease: “Poverty Blindness”. It, and the “closely related condition of Poverty Acuity Disorder (PAD)”. “People with Poverty Acuity Disorder see that poverty exists, but are unable to focus on its presence.”

Now “Absolute [Poverty Blindness] is rare”. Unfortunately “PAD in its various gradations exists in epidemic proportions.” What is a cause of PB? Money: “with significant monetary wealth are most often affected by PB, and the incidence of PB declines as accumulated net worth declines.”

What’s most worrisome is that this disease attacks not only individuals, but “Institutions and entire collective entities”, too. That finding alone is worthy of a Nobel prize, because no other known disease infects any but people one by one.

Poverty Blindness causes “lack of compassion”, leading to symptoms which include “lacking generosity, displaying distrust, being disobliged to help, exhibiting unethical behavior, and using common exculpatory mechanisms (including outright falsehoods).”

Is it catching? Yes. “Hereditary PB” exists, as does “Structural PB” (when entire societies are infected). And get ready for this. There is such a thing as “Academic PB“! And it come in two types.

Type I afflicts primarily scholars who work only with statistics disembodied from the people they purport to represent. Type II exists when financially well-endowed institutions of higher education, despite hosting poverty eradication initiatives among students, have overall cultures that ignore poverty.

This must be the reason so many statistical studies go off the rails!

There are other forms, but the worst and most intransigent manifestation is Malignant PB, which has two forms. “Gated MPB”, where the suffering isolate themselves, and “Savagely Capitalistic MPB” which applies to all those corporations whcih aren’t making really cool toys.

Our authors are developing a clinical test of PB, but they note that PB is associated with the maladies of “classism, racism, sexism…, and elitism.” Be on the lookout.

Treatment? Shock. No, really. “Exposure to catastrophic unforeseen events may be therapeutic.” If you suspect your patient suffers from PB, ship him off to Syria or decamp him to the side of an active volcano. Unfortunately, “the effects of these moments, however, rapidly fade from consciousness; such events are frequently considered temporary oddities of nature irrelevant to day-to-day life.”

That means the only true cure is—wait for it, wait for it—revolution. “Revolutionary efforts to restructure society…rapidly and unexpectedly bring poverty into one’s field of vision.”

What else can one say but ¡Viva la Revolución!

## Scientists Discover Men Enjoy Looking At Women’s Breasts.

The Mandarin word for bear if not given the proper tone sounds just like breast. Be careful at the zoo saying What a pretty bear!

Regular readers will recall there are two main kinds of bad statistics. First is when the technique has been done wrong or is misapplied. Errors of this kind comprise only half of all mistakes. The second, and more subtly nefarious, and just as pervasive, is where researchers announce they have used “science” to “discover” that which everybody already knew was true.

Nefarious because it strengthens or inculcates the bizarre and horrible fallacy that true knowledge can only come from science. That is, bad statistics of the second kind boosts scientism and makes scidolators of us all.

Our latest entry is Sarah Gervais, Arianne Holland, and the (given he has two female co-authors, presumably slavering) Michael D. Dodd in their peer-reviewed paper “My Eyes Are Up Here: The Nature of the Objectifying Gaze Toward Women” in the aptly named journal Sex Roles.

Here is the blockbuster opening sentence of the Abstract. Pay attention:

Although objectification theory suggests that women frequently experience the objectifying gaze with many adverse consequences, there is scant research examining the nature and causes of the objectifying gaze for perceivers.

Everything that can go wrong already has, which must set a mark or goal for other researchers to follow. Objectification theory? As the modern aphorism in the right-hand sidebar to this webpage indicates, “The love of theory is the root of all evil.” Only an academic could be puzzled enough that men look at women lovingly and in lust to create a theory of such behavior.

And then comes the “adverse consequences.” Like marriage? The joy, the bliss, the beautiful heartbreak from raising families? I can confess to you, my dear readers, that I first gave a serious eye to the female to whom I eventually plighted my troth. Of course there are also brutes and cads and construction-worker fashion critics, the men who, when they digress, should be instructed by gentlemen. But don’t forget those who gaze in rapturous silence. The mating process is imperfect. Human beings outside the academy understand this.

Our trio, relying on theory which comes before observations, pretend to believe two things which are blatantly false. First, that nobody knows men actually look at women in practice and that “data” is needed to confirm the theory. And second, that a theory is needed to explain this.

There is little point to surveying the “study” they did, but in brief, they used Photoshop to doctor the pictures of women to represent “cultural ideals of feminine attractiveness to varying degrees”. Now one wonders from where did they derive these cultural ideals except through the observations which they say have not yet happened? Never mind. Here are the body types:

high ideal (i.e., hourglass-shaped women with large breasts and small waist-to-hip ratios), average ideal (with average breasts and average waist-to-hip ratios), and low ideal (i.e., with small breasts and large waist-to-hip ratios).

Lo! Men preferred the hourglasses. A wee p-value confirmed this “finding”, or “discover”, if you prefer. That was the “main hypothesis.” Hypothesis forsooth!

And there were secondary “findings.” They “found that participants focused on women’s chests and waists more and faces less when they were appearance-focused (vs. personality-focused).” In other words, men gave the bodies of the pictures on the computer screen the once over before taking a gander at the faces. Who could have guessed? Well, everybody.

The researchers also were shocked—shocked!—to learn that women acted the same as men and that women were (to coin a word) judgmental. Golly.

But enough. Because we are now at the last sentence of the abstract, where all the errors above are compounded and multiplied. “Implications for objectification and person perception theories are discussed.” Person perception theories? This at least explains what academics do with their plentiful free time. They make up stuff to study.

## What Statistics Really Is: Part II

10 points for whoever can spot the mistake in this formula. (Yes, there is one).

I claimed, and it is true, that all statistical problems could be written $\Pr(p|q)$, where p is a proposition of interest and q is our evidence, or premises, or data, or data-plus-model, whatever you like to call it. Recall q is a compound proposition, including the data and whatever other knowledge we assume or possess.

I also claimed that q often contains “I believes”, in the form of “I believe the uncertainty in p is represented by this parameterized probability ‘distribution’.” Regardless whether these beliefs are true, as long as there are no calculation errors, $\Pr(p|q)$ is the true probability—because it assumes q is true, but does not seek to prove it. This is no small distinction; it must be kept continuously in mind or mistakes will be (and are) made. (More on this in the last Part.)

So let’s separate the “I believes” from q and call them m (for “models”). Thus we have $\Pr(p|qm)$ where q is as before sans the arbitrary model. Now, we don’t always need models. The example I showed last time didn’t need one. Here is another where a model is not needed. Example: p = “At least 7 4′s will show in the next n throws” and q = “We have a k-sided object (where k is at least 4) which when tossed must show only one side, with sides labeled 1, 2, …, k.” We deduce the probability of p directly (it is binomial).1

It turns out, at least in theory, that we can always deduce probabilities when p and q speak of finite, discrete things, which are really all the things of interest to civilians.2 Mathematicians, statisticians, and the odd physicist, however, insist on stretching things to limits to invoke continuity. Noble tasks, worthy goals and the only real mistake these folks make in pursuing them is anxiousness. Because the “I believes” are usually stated in the continuous, infinite forms as if given to us from on high and are not themselves deduced or inferred from the evidence on hand. And—as one of my favorite jokes has it—that’s when the fight started.3

The m’s, the “I believes”, are the cause of (rightful) contention between the two main sects of statisticians, the frequentists and Bayesians. Give you an example: p = “Tomorrow’s high temperature will be 72F”; q is any sort of data we have on the subject, and m = “The uncertainty in p is characterized by a normal distribution with parameters a and b.” The parameters of this model, as they are in most, are themselves continuous and unobservable; well, they are just fictions necessary to compute the probability of p.

Which in this case is 0 regardless of the value of a and b. That’s because a normal distribution, like all continuous distributions, give 0 probabilities to all single observables. (Don’t forget this probability is true assuming q and m.) This is why we can’t ask normal questions of normals (a pun!). You can see this is the point where adherence to a lovely theory can screw with reality. Anyway, if we want to use continuous distributions we must change our propositions so that they become answerable: let p = “Tomorrow’s high temperature will be greater than 72F”. This will have some non-zero value no matter what a and b are.

And just what are a and b? Nobody knows. There is no evidence in q or m to tell us. But since knowing what they are is absolutely necessary to solve the problem, we have to make some evidence up. Bayesians start talking about “flat” or “non-informative” or “improper” priors; some like to say “maximum entropy!” (the exclamation mark is always there). This move baffles the frequentists who say, and say truly, “You’re just making it up! How do you know it’s the right answer for this problem?” The Bayesian demurs and starts discussing “objectivity” and so forth, all different names for the same maneuver he just pulled.

So the frequentists go their own way and say, “I don’t know a or b either, so I’ll just guess them using one of several functions, or test their values against this null hypothesis.” Now it’s the Bayesians turn to demand accountability. “But you have no idea if your guesses are right in this problem! And, anyway, nobody in the world believes your so-called null hypothesis.” The frequentists retort, “Well maybe we don’t know if the guesses are right in this instance, but they will be if we do problems exactly like this an infinite number of times. And nobody ever believes null hypotheses, sort of.”

The steaming opponents—who you will have noticed ignore that both made up m out of whole cloth—leave the field of battle and head back to their encampments to produce their guesses which—surprise!—are usually not that different from each other’s. This is partly because all or almost all statisticians start as frequentists and only see the light later, so everybody uses the same kind of math, and partly because there’s usually a lot of good, meaty knowledge in q to keep people from going too far astray.

But the criticisms of both are right: from the arbitrariness of the m to the arbitrary guesses of the parameters, there’s a lot of mystery. Both sides are guessing and don’t like to say so.

The alternative? Restate the problem in discrete, finite terms and then use q to deduce the probabilities in p—if they even exist as single numbers, which most times they don’t. For most applications this would be enough. For instance, do we really care about 72F in particular? Maybe the temperature at the levels (‘below 60′, ‘between 60 and 70′, ‘between 70 and 75′, ‘above 75′) are all we really care about. After all, we can’t make an infinite number of decisions based on what the temperature might be, only a finite number. This moves gives us only four categories, some good observations in q, and we won’t be adding anything arbitrary. Everything is deduced starting with premises that make sense to us, and not to some textbook.

Well, this works. And if we really are enthusiastic, we work out all the math and then, and only then, take things to the limit and ask what would happen.

See this poorly written paper for an example of the typical “unknown probability of success”.

Next, and last, time: how do we learn about q?

———————————————————

1I’m not going to prove it here, but we don’t need information about “uniformity”, “symmtery”, “priors” or any of that stuff. See the statistics and probability philosophy papers for more details. Just believe it for now.

2I’m not proving this here either, but if you disagree I challenge you to state a measure of interest not of the categories listed about that isn’t discrete and finite.

3My wife and I were out to eat and there was a drunk at the next table. My wife said, “That’s the guy I used to date before we were married. He started drinking the day we broke up and hasn’t stopped since.” “My God,” I said, “Who would’ve thought a guy could go on celebrating for that long!” And that’s when the fight started.

## What Statistics Really Is: Paradox Digression

One dox, two dox, a pair of…

We’re taking a small digression to answer a question put by Deborah Mayo in Part I, pointing to this article on her site. Mayo should be on everybody’s list because she has good critiques of orthodox Bayesian statistics (which I don’t follow; we’re logical probabilists here), and because she has many named persons in statistics who comment on her articles. The material below is worth struggling through to see the kinds of arguments which exist over foundations.

Loosely quoting Mayo, a hypothesis (proposition) h is confirmed by x (another proposition) if $\Pr(h|xd) > \Pr(h|d)$ where d is any other proposition (this will make sense in the example to come). The proposition is disconfirmed if $\Pr(h|xd) < \Pr(h|d)$. If $\Pr(h|xd) = \Pr(h|d)$ then x is irrelevant to h. Lastly, h’ means “h is false,” “not h,” or the complement of h.

Mayo (I change her notation ever-so-slightly) says “a hypothesis h can be confirmed by x, while h’ disconfirmed by x, and yet $\Pr(h|xd) < P(h'|dx)$. In other words, we can have $\Pr(h|xd) > \Pr(h|d)$ and $\Pr(h'|xd) < \Pr(h'|d)$ and yet $\Pr(h|xd) < \Pr(h'|xd).$” In support of this contention, she gives an example due to Popper (again changing the notation) about dice throws. First let d = “a six-sided object which will be tossed and only one side can show and with sides labeled 1, 2, …”, i.e. the standard evidence we have about dice.

Consider the next toss with a homogeneous die.

h: 6 will turn up

h’: 6 will not turn up

x: an even number will turn up.

$\Pr(h|d) = 1/6, \Pr(h'|d) = 5/6, \Pr(x|d) = 1/2.$

The probability of h is raised by information x, while h’ is undermined by x. (It’s probability goes from 5/6 to 4/6.) If we identify probability with degree of confirmation, x confirms h and disconfirms h’ (i.e., $\Pr(h|xd) > \Pr(h|d) and \Pr(h'|xd) < \Pr(h'))$. Yet because $\Pr(h|xd) < \Pr(h'|xd)$, h is less well confirmed given x than is h’. (This happens because $\Pr(h|d)$ is sufficiently low.) So $\Pr(h|xd)$ cannot just be identified with the degree of confirmation that x affords h.

I don’t agree with Popper (as usual). Because $\Pr(h|d) = 1/6 < \Pr(h|xd) = 2/6$ and $\Pr(h'|d) = 5/6 > \Pr(h'|xd) = 4/6$. In other words, we started believing in h to the tune of 1/6, but after assuming (or being told) x, then h becomes twice as likely. And we start by believing h’ to the tune of 5/6, but after assuming x, this decreases to 4/6, or 20% lower. Yes, it is still true that h’ given x and d is more likely than h, but so what? We just said (in x) that we saw a 2 or 4 or 6: h’ is two of these and h is only one.

“Does x (in the presence of d) confirm h?” is a separate question from “Which (in the presence of x and d) is the more likely, h or h’?” The addition of x to d “confirms” h in the sense that h, given the new information, is now more likely.

No problems so far, n’est-ce pas? And Mayo recognizes this in quoting Carnap who noted “to confirm” is ambiguous. It can mean (these are my words) “increases the probability of” or it might mean “making it more likely than any other.” Well, whichever. Neither is a difficulty for probability, which flows perfectly along its course. The problems here are the ambiguities of language and labels, not with logic.

No real disagreements yet. Enter the so-called “paradox of irrelevant conjunctions.” Idea is if x “confirms” h, then x should also “confirm” hp, where p is some other proposition (hp reads “h & p”). There are limits: if p = h’, then hp is always false, no matter which x you pick. Ignore these. As before we can say p is irrelevant to x if $\Pr(x|hd) = \Pr(x|hpd)$. Continuing the example, let p = “My hat is a fedora”; then $\Pr(x|hd) = 1$ and so is $\Pr(x|hpd) = 1$.

The next step in the “paradox” is to note that if x “confirms” h in the first sense above, then $\Pr(x|hd)/\Pr(x|d) > 1$. In our example, this is 1/(1/3) which is indeed greater than 1. So we’re okay. Now we assume p is irrelevant, so $\Pr(x|hpd) = \Pr(x|hd)$. Divide this by $\Pr(x|d)$, then because $\Pr(x|hd)/\Pr(x|d) > 1$ so too does $\Pr(x|hpd)/\Pr(x|d) > 1$. Ho hum so far; just some manipulation of symbols.

Then it is claimed that x, since it “confirmed” h, must also “confirm” hp. Well, this is so. Then Mayo says (still with my notation):

(2) Entailment condition: If x confirms T, and T entails p, then x confirms p.

In particular, if x confirms (hp), then x confirms p.

(3) From (1) and (2), if x confirms h, then x confirms p for any irrelevant p consistent with H.

(Assume neither h nor p have probabilities 0 or 1).

It follows that if x confirms any h, then x confirms any p.

That’s the “paradox.” I don’t buy it. Like most (all?) paradoxes, there was a trip up in evidence along the way.

In our example, in (2), h does not entail p, but hp does entail p. What does entail mean? Well, $\Pr(p|hp) = 1$. The paradox says x confirms p just because hp entails p. Not a chance.

What’s happened here is the conditioning information, which is absolutely required to compute any probability, got lost in the words. We went from “x and hp” to “x and p”, which is a mistake. Here’s the proof.

If x confirms h, then $\Pr(h|xd) > \Pr(hp|d)$ (using the weaker sense of “confirmed”). Because p is irrelevant to h and x, then $\Pr(x|pd) = \Pr(x|d)$ and $\Pr(h|pd) = \Pr(h|d)$ and $\Pr(x|hpd) = \Pr(x|hd)$. But if p is confirmed by x, then it must be that $\Pr(p|xd) > \Pr(p|d)$. But $\Pr(p|d)$ doesn’t exist: it has no probability. Neither does $\Pr(p|xd)$ exist.1 What does wearing a hat or not have to do with dice? Nothing. You can’t get there from here. This is a consequence of p’s irrelevancy.

So p can’t be confirmed by x in the usual way. What if we add h to the mix, insisting $\Pr(p|xhd) > \Pr(p|hd)$? Not much, because again neither of those probabilities exist. You can’t have inequalities with non-existent quantities. And when we “tack on” irrelevant p, we’re always asking questions about $\Pr(hp|xd)$ or $\Pr(hp|d)$ and not $\Pr(p|xd)$ or $\Pr(p|d)$.

Result? No paradox, only some confusion over the words. Probability as logic remains unscathed. If anybody thinks the paradox remains, she should try her hand at stating the paradox purely using the probability symbols and not the mixture of words and symbols. The exercise will be instructive.

See the necessary comment by Jonathan D and my reply. Looks like JD found the mistake actually starts earlier in the problem.

————————————————————–

1Thinking every probability has a unique number is a mistake subjectivists make. They’ll say “Well I believe $\Pr(p|d) = 0.14779$” or whatever, but what they really have done is inserted information and withheld it from the formula, i.e. when they make statements like that they’re really saying $\Pr(p|qd) = 0.14779$ for some mysterious q that forms their belief. Given q that probability might even be right, but $\Pr(p|qd)$ just is not $\Pr(p|d)$. Still no paradox.