William M. Briggs

Statistician to the Stars!

Page 394 of 707

The Red-Ball-In-The-Box Fallacy

Okay, friends, I need your help. One day last year, shortly after I had arrived in Taiwan, I awoke from a fitful, jet-lagged sleep at 2 a.m. with the argument which follows in mind. I arose with enthusiasm and enough sense to know that if I did not jot down some notes the argument would be forever lost. So I opened the blog and wrote a few paragraphs, but I soon tired and quit. I recall telling myself something like, “There. That’s enough. I’ll finish in the morning.”

I also remember how pleased I was with myself for thinking up the red-ball analogy. “That’ll show ’em!” There might have been chuckling. I went back to sleep. Next morning, I had business to attend to, and so did not look at my notes. And then time intruded. You can guess the sequel.

I would just toss it out as the product of a fervid dream, which it very probably is, except that (if I may be allowed) I wrote it so compellingly that I badly want to know what the point of it was. The damn thing ends with a cliffhanger, and I can’t wait to see how I was going to escape.

Can you help? It is obviously epistemological in nature; at that time I was working on some philosophy of probability; and given my state of mind, it is almost certainly a fallacy. But there is a tiny chance real meat exists. The idea is sound enough, though: people often incorrectly deny a thing is true even though they can see it because there exist a vast number of competing hypotheses. But as for the idea I had in mind and of the structure of the proof, who knows?

I am still heavily engaged learning the ropes at my new gig—a sheet is not a halliard is not a cable—so I will be slow answering comments.


In front of you is a large open box, painted white. Inside, you can see clearly a large, bright red ball. The question is: given this evidence, and assuming your brain and senses etc. are working flawlessly, is there a red ball in the box?

This is not a trick question. The answer is yes, there is a red ball in the box. Why ask such a simple, even stupidly obviously question? Well, let’s see.

Suppose some person now comes along and drops in the box a semi-see-through dirty white ball. It is the size of a playground marble, much smaller than the bright red ball, which is as big as a soccer ball. The question is again: given the previous and this new evidence, is there a red ball in the box?

The answer one might give is, “You’re boring me. Of course there is a red ball in the box.” This is the right answer.

But now suppose a second, then a third, then several more friends come along and each of them drops an ugly white marble into the box. Perhaps some of the white marbles differ ever-so-slightly from one another. But in the box they go. Further, none of them can be mistaken for the red ball, not even (because of the vast differences in size) by a blind man.

The box is more than large enough to hold all these items, but not so large that you can’t root around in it and examine its contents in a reasonable amount of time. The question is: given all the stated evidence, is there a red ball in the box?

How much of your patience have I sacrificed asking these damn fool questions? Most of it I imagine, and probably next even of all of it, because I’m going to disappoint you by saying there is no catch, no gotcha. The red ball is still there, it can be found and seen, just as the evidence says it can be.

I’ve presented the question as trivial and not worth asking. And it isn’t. Or at least it wouldn’t be if so many people insisted that the red ball isn’t in the box because there are ugly white balls in it, too.

What’s that? You don’t believe anybody would make such a mistake?

Descartes Knew More Than He Thought

Old Renes, Looking Very FrenchYou can only know what is true. For example, everybody knows the statement “If p is true then p is true” is true (just why in a moment). You can believe what is true or what is false. Thus many believe the statement “President Obama has never engaged in demagoguery” which is false (again, just why in a moment).

Rene Descartes famously wondered just what he really knew and what might be mere belief. That is, what is certain and what might be doubted. According to Stanford’s Encyclopedia of Philosophy:

Descartes understands doubt as the contrast of certainty. As my certainty increases, my doubt decreases; conversely, as my doubt increases, my certainty decreases. The requirement that knowledge is to be based in complete, or perfect certainty, amounts to requiring a complete absence of doubt — an indubitability, or inability to undermine one’s conviction. Descartes’ methodic emphasis on doubt, rather than on certainty, marks an epistemological innovation. This so-called ‘method of doubt’ will be discussed below

As nearly everybody who has attended at least one college course has heard, Descartes stripped away every piece of information for which he could express the least doubt. Again, the SEP (quoting Descartes):

I have convinced myself that there is absolutely nothing in the world, no sky, no earth, no minds, no bodies. Does it now follow that I too do not exist? No: if I convinced myself of something then I certainly existed. But there is a deceiver of supreme power and cunning who is deliberately and constantly deceiving me. In that case I too undoubtedly exist, if he is deceiving me; and let him deceive me as much as he can, he will never bring it about that I am nothing so long as I think that I am something. So after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind. (Med. 2, AT 7:25)

All will recognize this as part of the (in)famous cogito ergo sum. Although this is too summary, Descartes argued that this was the most fundamental or basic truth. Now, other truths existed or could be deduced from the basic truth, but the cogito (Descartes thought) begins it all.

Of course, it is true that “Because I think I therefore exist,” but in saying that, I (and Descartes) am actually admitting to more knowledge than just this statement. That is, I—and I say “I” because the cogito allows me to doubt “you”—am implicitly arguing that I know more than just that I exist. Can you see it? It is yet another lovely proof that empiricism is false. Empiricism, as Scruton defines it, is the desire that:

(i) All claims to knowledge are based on knowledge of experience: experience is the ‘foundation’ of knowledge.
(ii) Experience can provide such a foundation only if my beliefs about my present experiences are immune to error…
(iii) But my experience does provide a foundation, since the realm of experience is ‘set apart’ from the physical world: it is a realm off ‘privileged access’ where (and I alone) am sovereign…

Now it might appear that the cogito is known by experience, just as we know that the “demagoguery” statement is false from experience. I experience me and from that deduce that I must exist. But that sentence gives the game away—can you see it yet?—for how can I know how to deduce truths from experience? To make it clear: I start with the knowledge of me, but then I must use certain truths of logic to deduce from my observation that I in fact exist. But how did I know those truths of logic, the rules that allow and produce deductions?

I cannot know these truths from experience and there isn’t any way to bootstrap this knowledge to use in the proof of the cogito (and if there isn’t a way for me to do this, there is no way for anybody; there cannot be an infinite regress; the process has to begin somewhere). Just as I cannot know from experience that for any p that “If p is true then p is true” is true. I just know it (and other similar statements) are true.

Therefore, we must come pre-built with these (and other) truths, which is a long way of saying that truth exists and thus these truths must be acknowledged. But where do those pre-built truths come from? Well, the answer is obvious—which is why some desire empiricism.

A bit busy this week with my new gig. I’ll be a little slow answering comments.

Brain Atrophy Responsible For Religious Belief?

Hippo's campusReaders can help me choose the best metaphor. (1) A snowball which starts the size of a pea but gains in strength and speed as it rolls downhill, mindlessly consuming all in its path; or (2) An avalanche, a furious, powerful deluge which is set off by some small thing, mindlessly consuming all in its path.

The application is to fMRI and other imaging studies of the brain which show that some “aberrant” (i.e. non-leftist) behavior can be pinned to some small region inside the skull. These statistical studies, which were nonexistent just a short while ago, now pop up weekly. Each works cites the others, and in so doing hopes to convince by sheer mass. Like a snowball or avalanche. Call it Death by Correlation, or Attack of the P-Value.

We have seen dark triads, enjoyment of custom, faith, and many, many others, all said to be caused by quirks in the brains of conservatives and theists. Today’s example is “Being ‘Born-Again’ Linked to More Brain Atrophy.” Well, at least this is clear enough. Rotting, shriveling brains “linked” to faith.

The abstract of the peer-reviewed paper by Amy Owen et alia is instructive. It opens,

Despite a growing interest in the ways spiritual beliefs and practices are reflected in brain activity, there have been relatively few studies using neuroimaging data to assess potential relationships between religious factors and structural neuroanatomy.

That sentence is paradigmatic: “Despite interest in X, we have not yet seen work in subgroup Y” opens many papers. It suggests a bandwagon groaning under the weight of scientists with hyperactive pituitaries, or whatever cranial organ is responsible for copycat research—and according to these folks some such organ must exist.

As depressing as this beginning is, let’s push on. Our data-mining discovery-of-the-day comes from comparing “religious factors and hippocampal volume change using high-resolution MRI data of a sample of 268 older adults.” Main claim: “Significantly greater hippocampal atrophy was observed for participants reporting a life-changing religious experience.” And there you have it: withered brains produce faith. In case the connotation wasn’t obvious, the press article reminds, “Shrinkage of the hippocampus is also associated with depression, dementia and Alzheimer’s disease.”

(You know, it’s not often I brag or boast or plead, but I should receive some kind of humanitarian award for reading these papers so you don’t have to.)

Participants, with double the number of women, mean age near 70, were measured twice, a baseline and follow-up—follow-ups were not consistent: times “ranged from 2—8 years.” Participants were “those meeting DSM-IV criteria for major depressive disorder” and those “never-depressed”. They never tell us how many in this study were depressed and how many not. Ah, who needs rigor? This is statistics!

Affiliation was queried: non-born-again, born-again Protestant, Catholic, and Other. Only 19—just 19; a mere 19; 7%—listed no religion. Those claiming not to be born again were asked whether they had had “life-changing religious experiences” (LCRE).

Now, over the course of follow-up, twenty-two—that is, 22, which is 1 more than 21—people were newly born again and 23 others had new LCREs. That is 55 total, folks, which is nearly three times as many who claimed no religion prior to the study. That must mean that many people merely reported switching their religious label. How did the researchers handle this ambiguity in their statistical models?

(I really am going to have to devise a graphic which indicates crickets chirping.)

Understand, dear reader: the evidence for this study is entirely statistical, the result of regression models producing publishable p-values. They created two models for change in (falsely assumed error-free) measures of the left and right hippocampus. For the LH model, every measure of religion, pre or post, gave negative model coefficients except “Other” (but that had an unacceptable p-value). This includes “None,” which I remind us means no religion. Same for the RH model, except “Other” was negative here; “None” was still negative. Negative means the item shrank the hippocampus.

Unfortunately for consistency in reporting, newly born-again status was not significant in either model; neither were new LCREs (baseline for both was). And do you know what? The authors did not report any kind of baseline comparison of hippocampus size and religious belief. Must have slipped their minds—which cranial organ was responsible for this error we do not yet know; further studies might tell us.

Now, participants were old. Yet, somehow, the model coefficient for “Duration in study” was positive for the LH model and negative for the RH model. How can this be? Aging causes the left but not the right hippocampus to grow? By golly, Owen has discovered that enrollment in fMRI studies encourages hippocampus growth! On the left side only, alas; the p-values aren’t there yet, but this is early days: more research is needed. Send donations.

As always, the most fun is had by reading the discussion, where authors allow their minds to range freely over possible explanations of the data. I was going to summarize these curious cogitations, but I quailed after reading, “Research on temporal lobe epilepsy indicates that features of hyper-religiosity may be positively associated with hippocampal atrophy, but findings are mixed.” Good grief. I had not the strength to continue.

But I did check: they hadn’t even an inkling that their results were a statistical artifact or due to a sloppy experimental protocol and even sloppier analysis. I blame myself and other statisticians for this. I really do.

Wearing A New Hat At JP Research

JP ResearchI am now a member of JP Research. What does this mean to you, my loyal readers? Little, except that my contact information for consulting has changed. New telephone and email: 650-559-5999, william@jpresearch.com. These are also listed on my Hire Me and Contact pages.

Story tips, personal correspondence, missives that begin “Briggs, you fool…”, and so forth still come to matt@wmbriggs.com.

At JP I’ll be doing the same kind of work, traveling to the same places and then some, and meeting new people. My locus in quo is unchanged. My teaching gig at Cornell remains. This blog will continue unmodified. I’ll almost certainly, however, be buying a new light grey Borcalino snap-brim fedora (teardrop crease). My collection of brown headgear is just-the-thing, except these chapeaux don’t always “go” with certain shades of blue, a major concern.

Rest assured that I have given this change serious consideration. Never, indeed, have I pondered longer or harder. There was even, at one point, actual hand wringing. But I came to the conclusion that the addition of a new hat to my stable is a necessity.

I know what you’re thinking: why grey and not blue? Well, blue hats are limiting. You can’t wear a blue hat with a brown, white, or tan suit. And you can’t even wear it with all shades of blue, especially darker ones. A blue hat is too jaunty: it is not a serious hat. Whereas a grey can be sombre or gay with the simple switch of the band. Borsalino

Not the stingy brim, either. These are Borsalino’s attempt to cash in on the hipster hat fad. And since all fads are doomed by definition, it is best to avoid them and save your money. Hipster hats are, I suppose, a gentle introduction to the world of adulthood, and so in this sense they can be encouraged. They are orders of magnitude better than a golf or baseball cap. And I believe that wearing a hipster hat might even increase the probability of wearing better clothes.

But if you’re going to spend the money, don’t blow it on a head covering that will look dated after one season. Which brings us to the well known men’s fashion rule: never skimp on shoes or hats. Nothing, but nothing, ruins the look of an outfit, no matter how carefully chosen or expensive, than to see it topped by a cheap hat or bottomed by rubber soles.

Incidentally a good, but expensive, pair of shoes, properly cared for will last forever, especially if they are a member of a rotation (try not to wear the same pair every day). Hats too can be Methuselan, as long as you don’t make a habit of doffing one by pinching the peak.

Speaking of homburgs: yes, but not for me. I am yet too young and anyway too long-faced. I want a derby, but I also want a moustache. The Powers That Be deny me both, sadly.

All these things and more will be on my mind as I begin my new life at JP.

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