Roger Kimball is causing a stink, a predictable yet enjoyable stink, by publishing Anthony Daniels’s review of an Ayn Rand biography in this month’s *The New Criterion*.

There are two enduring internet-subjects on which if any negative criticism appears, no matter how slight, can be counted on to generate tea-cup furies.

The first is Apple computer. For example, question the hubris of Steve Jobs, who last week introduced Apple’s tablet under a looming picture of a stone-carrying Moses descending Sinai, and legions of fanboys will descend upon your site and explain to you just how stupid you are, and why you will always be so since you cannot comprehend the simple logic of how the Israelites would have spent 50% less time wandering had they only been presented their Commandments via the iPad. (In colour!)

It used to be that any negative press of Ron Paul or Obama would produce the same attacks of splenetic fever. But Ron Paul is long dead (so I’ve heard) and the growing perception is that Obama has read from one teleprompter too many.

So we are left with Ayn Rand. Daniel knew the danger going in, which is why he took pains to present ideas of Rand’s which he thought were true. The first: “[S]he was among the first to appreciate that the notion of collective rights (a mirror image of racial discrimination) would ‘disintegrate a country into an institutionalized civil war of pressure groups, each fighting for legislative favors and special privileges at the expense of one another.'”

This was an empirical prediction which experienced has verified, and is therefore true, but not yet universally acknowledged.

Daniels also recommends “her observation that ‘Even if it were proved…that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual and it would be irrelevant to one’s judgment of him.'”

This is a (comforting) statement of philosophy and is false. Further, any statistician knows that it is false.

Suppose there are two group, M and N. And to avoid emotion, suppose M and N represent the sales (in dollars) of two rival products. The statement that the evidence shows the “incidence of…superior brain power is greater among the members of” a certain group is translated into “the evidence is that the probability of group M having higher sales than group N is greater than 50%.”

Writing this in traditional notion (for those comfortable with this) gives

Pr( Sales[M] > Sales[N] | Our Evidence) > 0.5).

Another way to state this is that if you had to guess which product, M or N, would have greater sales, you would maximize your chance of confirming your guess by saying “M.” This does not, of course, *prove* that the sales of M will be greater. N can still beat M.

Rand would say that the evidence that it is probable that M > N “would still tell us nothing about any given individual and it would be irrelevant to one’s judgment of him.” Here there is only one individual per group (just the sales M and N), but knowing who that individual is tells us something about that individual and is not irrelevant to our judgment of him.

You might object that Rand obviously meant more than one individual per group. So suppose sales of M and N are generated by salesmen. That is, M has a host of salesmen hawking it and so does N. The number of salesmen in each group need not be equal.

Our equivalent evidence is that the salesmen in M sell more than do the salesmen in N. That is, given this evidence, the probability that a salesperson from M outperforms a salesperson from N is greater than 50%. Notationally,

Pr( Individual[M] > Individual[N] | Our Evidence) > 0.5).

If *all*—pay attention to this “all”—we knew about the two individuals in front of us is that one sold M and the other sold N, then this *would* tell us *something* about these given individuals.

Our knowledge of what group these individuals belonged to *would* be *relevant* to our judgment of them. Our judgment is that, given the evidence we have of the two people in front of us, the guy that sold M is probably a better salesman than the guy who sold N.

This, again, does not prove guy M is better than guy N. We could learn new evidence that changes our perception: for example, guy M is drunk.

But in general, Rand’s statement is logically false.

Be careful to understand what we proved. We did *not* prove that, in comparing different groups of humans, there *will* be measurable differences. Whether or not there are is an empirical, and not a logical, question. This is what statistics is all about.

What we did prove was that *if* those differences are real, then that information would be relevant to judgments about members in different groups.

Categories: Philosophy, Statistics

It doesn’t seem to explain Dick Whittington and his cat.

.

vi!

The proposition assumes we have an unique kind of information, and only that information, on which to make judgments. While the logical syllogism you present may be valid, the underlying assumptions are not.

There is such a thing as crappy evidence, especially when dealing with complex real world phenomena. Bayesians in particular should be wary of bugs in the

a priorideck.So, how long ago did you move down from Canada, eh? Or were you just joshing the Apple-heads by the way you spelled “colour”?

Ever read Richard von Mises? Ever heard of the frequency theory of probability?

If you think that being part of a group implies that you have the probabalistic attributes associated with the group, you are mistaken.

Post hoc, Dick Whittington.

.

Dick Whittington and his cat had an average of three legs. If I meet a three-legged man can I say, “You are Dick Whittington and I claim five pounds!”

Jim,

I have. Have you? Let me give a clearer example, acknowledging that my other ones were opaque.

You have two bags in which are contained blue and red marbles. Bag M has 80 red and 20 blue, bag N has the opposite: 20 red and 80 blue. Red marbles are better than blue. There is our evidence that group M has more of a better quality than group N.

Now suppose you were to draw one marble/individual out of each bag. What is the probability the individual/marble from M is “better” than the individual/marble from N? Which is equivalent to: what is the probability that M is red and N is blue?

Von Mises would get that right. Plus, he’d be happy that he still has some supporters.

Makes sense. If More January days are colder than those in August, assuming any given day in January will be colder than any given day in August is a reasonable assumption. Similarly, if group X averages bigger bazongas than group Y it seems a safe assumption that a randomly chosen member of X will have a bigger bazonga than one possessed by a member of Y. If these assumptions are invalid then what’s the point in studying the stats?

The problem with the Whittington average is you are misreading what is being said about group membership. Note that the probabilistic statement is a comparison as denoted by the use of

more, lessor end in-erin English. Even a frequentist should realize that the probability getting a sampleexactlyequal to the average — or any other specific value for that matter — is exactly zero.One way to gain support is to build some truss. This advice is undoubtedly a bit late for von Mises.

DAV,

Nobody like a bad pun better than I.

If all you want is average, then picking from the group with the better average is the way to go. No doubt about it…

Before you stands salesman

mfrom sales group M, and salesmannfrom sales group N. Group M outsells group N in terms of total aggregated sales. We don’t know the membership magnitude of M or N. We know nothing aboutmornexcept their group membership. (If M is much greater than N, then eachnmay outsell everymeven though the total sales of group M is greater than group N.)Groups M and N have an internal sales-frequency-per-salesman distribution. The distributions have sales frequency wings. Assume the M and N group sales frequency distributions overlap. We don’t know where

mornstand in their respective frequency distribution.If we know nothing of the sales frequency of either

morn, then no matter the total sales of M or N, we know nothing comparative about any individual pair ofmandnwe might face. That result obtains until we know the individual standings in their respective distribution.Likewise, with respect to the red and blue marbles. If we blindly pick from the respective 80/20 and 20/80 bags, we only know the color probability in each hand. That could inform us about which hand is most likely to have valuable contents. But we know nothing concrete about the relative value in each hand until we open our hands and see the color of the marbles.

Seeing the color of the marbles is like knowing the specific sales frequency of each salesman.

So, given

mandnfrom Groups M and N, and knowing nothing else, we know nothing specific about their relative standing or value. We only know likelihoods.Probabilities, after all, are statements about ignorance concerning specifics. Ayn Rand was right because knowledge of specifics takes the matter outside the purview of statistics.

The only exception comes if the sales-frequency-per-salesman distributions of groups M and N do not at all overlap. We’d then know the relative merit of

mandnwith no more information than their group membership. But then, of course, Gaussians have infinite wings, so strictly the no-overlap condition is a non-starter.Briggs,

Here is where you fail the test:

“If allâ€”pay attention to this â€œallâ€â€”we knew about the two individuals in front of us is that one sold M and the other sold N,(emphasis mine)then this would tell us something about these given individuals.Mises would have nothing of that. I suggest that you reread (read?) “The Definition of Probability”, from

Probability, Statistics and Truth(Mises 1957).Now Jim. Don’t let’s have that. Let’s have an argument why von Mises, or you, find my example misleading or incorrect.

Also, it is ungentlemanly of you to continue suggesting I have not read his work. I have, and I have found it wanting. Click on the “Statistics” tab on the left and read some of my other articles to see why.

Pat,

You were on the right track but slipped up at the beginning in your second assumption, which appears to allow that unequal numbers of salesmen in M and N could account for the group differences. This isn’t want Rand said. Her first key premise was, “Even if it were proved…that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others.” This is translated into, “Even if it were provedâ€¦that the incidence of men of potentially superior sales power is greater among the members of group M than among the members of group N.”

The key to thinking of this is in the second example I gave of the bags of marbles. I admit the salesmen example might be confusing.

Note: we are not asking to predict the group averages. We are interested in saying something about the two men in front of us. (Or the two marbles we’ll draw out.)

If 2% of the Brand X widgits are defective and 1% of Brand Y widgets are defective, there is enough information there to say, if the price is equal I would rather buy Brand Y. However, it would be false to say, If you have a brand X widget it will probably break.

Briggs,

Fair enough.

Please point me to the specific articles. That would be appreciated.

As I stated above, “If you think that being part of a group implies that you have the probabalistic attributes associated with the group, you are mistaken.”

Knowing the proabilities of a collective can never let you know anything about the individual (assuming that we are dealing with probability, so the set in not unified with the same attributes).

Put all the billard balls in an urn, and yes the probability of choosing the black ball is 1 in 15. But once you select your ball, it has either a 100% probability of being black or 0% probability of being black. It makes no sense to claim that the ball I hold, but cannot see, has the same characteristics as the probability of the collective.

Assume 1000 football fans, 700 Steeler fans and 300 Browns fans. That knowledge tells you nothing about the fan you select at random. It tells you info around a randomly selected subgroup of (say) 100. But nothing about the individual.

I consider your view (the mainstream, I admit) to be the Schoedinger’s Cat view of probability. Your selection exists in a quantum state until revealed.

BTW, I took “Von Mises would get that right. Plus, heâ€™d be happy that he still has some supporters.” to imply that Mises would have agreed.

Matt,

My read on the Ayn Rand quote is akin to your earlier analysis of men and women. I interpreted the quote to be referencing the tails of the mental ability curves. If there is greater variability in mental ability in Race A compared to Race B, but the mean mental abilities are identical, then you would have a situation consistent with Rand’s comment.

For my argument I’ll assume normal distributions of intelligence for both populations. The conditions that Rand’s comment implies are that the right extrema of population A (incidence of superior brainpower is greater) is greater than for population B. If the mean of the A mental ability is greater than the mean of B mental ability then I contend her statement is false.

If the mean of the A mental ability is identical to the mean of the B mental ability then I contend her statement is true.

My school days training said that for a statement to be true it must be always true, so I acknowledge that using my assumptions Rand’s statement is false. However there are conditions where her statement is true. Under those conditions the corrolary statement is also true:

‘Even if it were provedâ€¦that the incidence of men of potentially inferior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual and it would be irrelevant to oneâ€™s judgment of him.â€™

Earle Williams & Jim,

Not quite. Let’s repeat Rand: Even if it were proved…that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual and it would be irrelevant to one’s judgment of him.

Here is another way to paraphrase her statement: It is

knownthat the incidence, or proportion, of smart men is greater in group M than in group N. Given just that information,and no other, the probability a man from M is smarter than a man from N is greater than 50%.There is no special case or extension. Rand did not say, “Assuming the means of M and N are equal but their variances are not” or anything like that. She spoke of the

incidence. So, given her premise, her conclusion is logically false.Having the knowledge that the rate of smarter people in M does tell us something about the person in front of us and it relevant to our judgment of him.

Here’s another example. Premise: The incidence of advanced mathematical ability in nuclear physicists in higher than that of elementary school students. You have a U of M nuclear physicist in front of you and an elementary school student, too. Given our premise, the probability the nuclear physicist is better at math than the elementary student is greater than 50%. Knowing our man is a member of the “race” of nuclear physicists tells us something about that man’s mathematical ability, and knowing he is of that “race” is

relevantto our judgment of his mathematical ability. Even thoughit might be the casethat the elementary school student is another Gauss and our physicist the recipient of one government grant too many.Jim: Would von Mises agree with that?

I don’t have time to point to specific articles, but look under “Statistics” or search of “probability” or “logic.”

Or how about this argument? Premise: All humans are fallible. Ayn Rand was a human. Therefore, Ayn Rand was fallible.

Substitute “Briggs” into that and it’s just as true.

Briggs,

I’ll respond to your main argument in a bit.

Just keep in mind that I am not a Randian. And I am not defending her.

I’m simply responding to your theory of probability.

Matt,

I guess it depends on your reading of “superior”. If you read it as “smarter” then I accept your argument and prostrate myself before your elite reasoning skills.

If, like me, you read it as “really freaking smart”, ie. an arbitray measure such as an IQ of 150 and not a relative measure with respect to another population, then I refute your argument and break wind in your general direction.

Maybe I just have a problem parsing Rand’s statement.

Posit that the top 5% of men of race A have an IQ of 150 or greater. Posit further that the top 4% of men of race B have an IQ of 150 or greater. What can I know of John Q Random compared to Juan X Alazar?

Matt, the ‘unequal number of salesman comment’ in my post was a parenthetical aside, not an assumption impacting the analysis.

Please find that the analysis proceeds without that assumption. I remain unconvinced that we can know anything specific about individuals by way of statistical information, apart from their group membership.

(I’m deferring to Steve McIntyre’s rendering as your choice of first name. Hope that’s OK.)

Matt, you wrote that, “the probability a man from M is smarter than a man from N is greater than 50%.”

Followed by, “Having the knowledge that the rate of smarter people in M does tell us something about the person in front of us and it relevant to our judgment of him.”

The second statement does not follow logically from the first. The second should state, ‘Having the knowledge that the rate of smarter people in M does tell us something

probableabout the person in front of us and it relevant to ourprejudgment of him.’Any true judgment concerning specific properties can only follow from specific knowledge. Statistics do not give us knowledge specific to individual states.

Briggs,

Mises, as you well know, was a frequentist. So he would object from here, “Knowing our man is a member of the â€œraceâ€ of nuclear physicists tells us something about that manâ€™s mathematical ability…” As a frequentist, Mises would say that you know nothing about the individual, other than that he is in the class.

Briggs,

To change the subject: Any thoughts on value-added models in education (Bill Sanders having the one being used in Ohio — my state)?

DAV,

Unless you’re an amoeba “number of legs” has a discrete distribution so exact vales will have non-zero probabilities.

My statement was wrong on many levels including, it would appear, that of humour. Sorry.

Jim,

Hmm. I’m not sure; but only because I’m ignorant of the topic. Any good pointers?

Rich,

Your joke identified

thetruth. All measurements are discrete; all real things have non-zero probability.Rand: â€˜Even if it were provedâ€¦that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual…’

This is correct. An individual from the disfavored group could be an outlier on the upper scale or if from the favored group an outlier on the lower scale. We don’t know until we discover more information concerning the individual in question. The unknown individual obliterates the mean of his group and affirms its curve in so far as he is unknown. Statistics are group predictive but not individually prescriptive.

‘…and it would be irrelevant to oneâ€™s judgment of him.â€™

This is incorrect. Our knowledge of the statistically ascertained characteristics of the individual’s group will impact our judgement of him precisely because we know nothing else about him in and of himself.

Briggs,

Value added is the school accountability indicator that is currently in vogue. It purportedly shows student growth (the year-for-a-year stat).

Sanders claims that he can show student growth predictions (and, hence, teacher/school/district performance) that is independent of demographics.

His system is under wraps so there is little (that I’ve seen) on it. Yet it is being adopted across that nation. Ohio uses Sanders system via Battelle.

The problem I have with Rand is (and this is just personal experience with a small number of individuals):

Previously nice people who read and embrace Rand’s work have a high tendency to become selfish and rude. Whether this is the intended response or a gross misinterpretation of her supporters makes no difference.

Predictive value may give the best chance of choosing the right one, but saying it tells you anything about an individual is wrong, simply because the prediction can be wrong. I can’t get around this, at least not yet.

I disagree with Mr. Briggs’ argument. Knowing what is likely of an individual (based on his group’s performance) is not “knowing something [S] about” him. If S is false, then S is not known about him, or if it is true, nonetheless it is not known.

Briggs might reply that “S is likely of him” is what is known about him, but this is just a way of saying that his membership in the group whose performance is known, is known. What is known about him is that he is a member of the group. What is known about the group is its performance.

Smoking Frog,

First, I congratulate you on your lifestyle choice.

Second, if you disagree with me, explain how I am wrong about the physicist/schoolchild example.

“Predictive value may give the best chance of choosing the right one, but saying it tells you anything about an individual is wrong, simply because the prediction can be wrong. I canâ€™t get around this, at least not yet.”Isn’t the whole point that you get the best chance of choosing the right one, not that you will certainly choose the right one? Aren’t we making these more probable choices so that we maximize our success rate in the long run? Hence Bishop Berkley’s “probability is the very guide of life”.

And is all that just to say that we do statistics just so we can be right more often than we’re wrong?

“First, I congratulate you on your lifestyle choice.” – Briggs

Smoking Frog was an important Mayan warlord of the 4th century. His real name or title was “Born of Fire,” but scholars used to think it was “Smoking Frog” because they misunderstood the hieroglyphics. “To be born” is an iguana looking upward, and “fire” is rising smoke. Of course, an iguana is not a frog, but maybe the scholars thought the glyph looked like a frog.

“Second, if you disagree with me, explain how I am wrong about the physicist/schoolchild example.” – Briggs

Since you think it is possible that the child is better at math than the physicist, you do not know that the physicist is better at it, so you must be claiming something less than this. What you are claiming you know is high likelihood of the physicist’s being better at math, but this is not knowing “something about” the physicist. It is knowing something about the group “physicists” and the group “schoolchildren.”

You do, in some sense, know something about the physicist’s math ability from the fact that he is called “physicist,” but this is not from probability and statistics. It is from (what philosophers call) the *intension* of the word “physicist,” i.e., what qualifies someone as a physicist. If you learned that this particular “physicist” knew little about math, you might decide not to classify him as a physicist. The same would be true if you learned that he was a witch doctor in an African country where witch doctors are called “physicists” because the people picked up the word from European colonials and misconstrued it.

The word “physicist” did not come about from the discovery of a cluster of features, and a decision that anyone within so many SD’s of means on these features, or anything like that, should be called “physicist.” In other words, “physicist” is not an arbitrary label for a statistical distribution.

Smoking Frog,

Did not know that. Thank you.

And while you offered a lot of words, you’re still off on the argument. You have forgotten Rand’s premise. To risk tedium, here it is again: “Even if it were provedâ€¦that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others.”

If that premise is accepted, as it is in my hypothetical physicist/student example, then the rest of what I say follows. You’re getting lost on the “Even if it were proved…” I am saying, for the sake of argument, that in a certain instance it

isproved. It makes no difference whether or not it really is true in real life (although I can’t imagine anybody disagreeing with it),givenit is true, then Rand’s conclusion is false.“And while you offered a lot of words, youâ€™re still off on the argument. You have forgotten Randâ€™s premise. To risk tedium, here it is again: ‘Even if it were provedâ€¦that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others.’

“If that premise is accepted, as it is in my hypothetical physicist/student example, then the rest of what I say follows. Youâ€™re getting lost on the â€œEven if it were provedâ€¦â€ I am saying, for the sake of argument, that in a certain instance it is proved. It makes no difference whether or not it really is true in real life (although I canâ€™t imagine anybody disagreeing with it), given it is true, then Randâ€™s conclusion is false.”

I don’t see how you can think I’ve forgotten Rand’s premise. My argument on your physicist/child hypothetical says that you do not know “something about” the physicist even though the premise is true. If it turns out that he’s poorer at math than the child, then very clearly you will not have known it, because it’s false. If it turns out that’s he’s better, you still didn’t know it, because the question of whether you knew it does not depend on whether it is true. Therefore, to be making sense, you must be claiming that you know something less than “The physicist is better than the child at math.” The only thing it can be is “The physicist is highly likely to be better than the child at math,” but I say that this is not knowing something about the particular physicist; rather it is knowing something about the group “physicists” and the group “schoolchildren.”

As I indicated, that discussion disregards the possibility that you know something about the physicist other than by knowing the statistical distributions of math ability in physicists and schoolchildren. Yes, you do. You know that he is called “physicist,” and that “knows a lot about math” is a distinguishing feature of physicists, and you trust that he has been correctly classified as “physicist.” We very often say that we “know” something which we take on trust. For example, if your wife were to say, “There’s a schoolchild in our front yard,” you would not expect to find that actually it was an adult or a dog. You would consider that you know that a schoolchild is in your front yard.

CORRECTION

“If it turns out that heâ€™s poorer at math than the child, then very clearly you will not have known it, because itâ€™s false.” – Smoking Frog

I meant to write, “If it turns out that he’s poorer at math than the child, then very clearly you will not have known that he is better, because it’s false.”

I do know something about the physicists and children, froggy old man. That is what the premise states.

“I do know something about the physicists and children, froggy old man. That is what the premise states.” – Briggs

I didn’t say you didn’t. My argument assumes the premise, and that means assuming that you know something about physicists and children. What you don’t know is something about any given physicist, if this means knowing something based only on knowing the statistical distributions.

Matt, your physicist and schoolchild example is wrong for two reasons. First, in re-defining your comparison from two adults to adult/child, you have added a new qualification to your argument (with the new content implicitly inviting a disparaging bias toward the schoolchild entry). That violates the original rules of debate (and adds a rhetorical device). Second, your new argument says nothing about the quality of your physicist’s Ph.D.

With no specific information, no one can decide about the degree of native mathematical ability of either party, physicist or schoolchild. All that’s known is their group membership.

No matter your philosophical distance, it’s time for you to gracefully concede that Ayn Rand was right about her particular.

Pat Frank,

Nope. Change the physicist and schoolchild to races M and N, where you KNOW (that’s the premise, after all) that more in M are smarter than those in N. Same result.

Rand, however infallible she might have been in every other thing she said in her life, was wrong about this.

â€˜Even if it were provedâ€¦that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual and it would be irrelevant to oneâ€™s judgment of him.â€™

I’ve always understood this statement to be more about saying that given group averages we still ought to judge people as individuals.

I grant you that it’s probably true that a given individual in the lower average group is therefore lower but is that sufficient reason to prejudge them.

Secondly, I get the impression that she is comparing two things. In the first part she is specifically talking about brain power. In the second she is being more general. People have lots of characteristics and the fact that people have superior intelligence doesn’t give much insight into say their athletic ability.

Matt, you wrote: “Change the physicist and schoolchild to races M and N, where you KNOW (thatâ€™s the premise, after all) that more in M are smarter than those in N. Same result.”

Your claim was that you knew a randomly selected individual

mof class M was smarter than a randomly selected individualnof class N, merely because the class M had a greater number of smarter individuals.However, your own premise allows you to know

nothingspecific aboutmwith respect tonapart from group membership. I showed above that you made a logical non-sequitur in stepping from probable to specified.Your conclusion would work if you had premised that

mis a member of M,iffmis smarter than allnin N.I have no particular use for Ayn Rand’s philosophy, by the way. I was encouraged to read her “Objectivism” years ago, and quickly found that she assumed what should be demonstrated.

Pat Frank,

No sir, your additional premise is

differentthan Rand’s original (after your iff; Rand said “more” not “all”). You have not showed a non sequitur, and are making the same error repeatedly. Plus, you add the premise “randomly”, which is not necessary.I challenge you to find any other logician who can agree with you. If you can,. then we’ll continue to discuss this. Otherwise, I acknowledge that my ability to teach this topic is inadequate.

â€˜Even if it were provedâ€¦that the incidence of men of potentially superior brain power is greater among the members of certain races than among the members of others, it would still tell us nothing about any given individual and it would be irrelevant to oneâ€™s judgment of him.â€™

We’ll take a simple example. The incidence of extreme tallness (however you define it) is greater among men than women.

How much are you prepared to bet that you are taller than a woman who has called to make an appointment to consult you?

I hope you won’t stake all your assets on this bet!

I think Rand was correct. Probabilities tell us only about what is true on the long run, not about individual instances.