Quotation of a day

Following (i.e., stealing the idea from) Don Boudreaux, who posts interesting and thought provoking “quotations of the day” (e.g., today’s post on evolution), here’s an amusing bit from page 72 of Hamming’s Digital Filters:

The relationship of formal mathematics to the real world is ambiguous. Apparently, in the early history of mathematics the mathematical abstractions of integers, fractions, points, lines, and planes were fairly directly based on experience in the physical world. However, much of modern mathematics seems to have its sources more in the internal needs of mathematics and in esthetics, rather than in the needs of the physical world. Since we are interested mainly in using mathematics, we are obliged in our turn to be ambiguous with respect to mathematical rigor. Those who believe that mathematical rigor justifies the use of mathematics in applications are referred to Lighthill and Papoulis for rigor; those who believe that it is the usefulness in practice that justifies the mathematics are referred to the rest of this book…. Furthermore, since we are interested in the anatomy of the mathematics, we shall ignore many of the mathematically pathological cases. The fact that we are dealing with samples of a physical function implies that we are trying to understand a reasonable situation.

It makes on feel downright Newtonian. If it works, use it, foundations be damned.

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More than who in the what now?

Kirk Goldsberry explains an interesting new basketball shooting statistic he and a colleague have developed today. I’ll be discussing two strange statements that Mr. Goldsberry made. One is strange for logical reasons, and the other is strange for syntactic reasons.

In the introduction of the article, Goldsberry is expressing an amazing fact about LeBron James:

…However, consider the following ridiculous statistical couplet:

No player scored more points close to the basket than LeBron James last season.

No player converted a higher percentage of his shots near the basket than LeBron James last year.

Think about that. Not only did he outscore every player in the entire league within the NBA’s most sacred real estate, he converted his shots at the highest rate, too.

Okay, I’ve thought about it, and I don’t find it ridiculous at all. Unless James took substantially fewer shots close to the basket than did other NBA players (which even someone as NBA ignorant as me knows just can’t be the case), the fact that he was more accurate makes it essentially a mathematical necessity that he would outscore everyone else. This seems like an oddly innumerate bit in an otherwise relatively statistically sophisticated article.

The syntactically strange sentence is more fun. Goldsberry writes:

No player accumulated more points than expected than James.

It’s clear what he means – no one exceeded the number of expected points to a greater degree than did James – but, as written, this sentence is meaningless. I mean, I can garner some meaning from it, but it seems ill-formed for expressing that meaning (or any other).

It reminds me of the plausible Angloid gibberish sentences “More people have been to Russian than I have” and “In Michigan and Minnesota, more people found Mr Bush’s ads negative than they did Mr Kerry’s.

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Picking some more nits

Geoffrey Pullum has an interesting recent essay on the difficulties of pinning down exactly how old the cognitive revolution is. Naturally, I won’t be focusing directly on this topic.

Rather, I’d like to take this opportunity to bemoan the fact that Pullum is uncritically invoking Kuhnian philosophy of science:

Maybe revolution is not quite the right metaphor. I know Thomas Kuhn taught us that science develops through revolutions, the detailed work being done under the assumptions of the last one during periods of “normal science.” And it’s an exciting thought, the idea of an annus mirabilis when the whole conceptual world turns upside down, and what was formerly nonsense becomes accepted science (and vice versa), and old guys who don’t get with the program are left to face an embittered retirement. But I’m inclined to think it isn’t quite like that in this case.

I’d give him credit for challenging the idea that “it isn’t quite like that in this case” if I wasn’t already convinced that it’s never quite like that. As Larry Laudan wrote in 1986 (for the record, that’s 27 years ago) in Science and Values (p. xii, in the preface; emphasis mine):

In sum, this is a book about the role of cognitive values in the shaping of scientific rationality. Among recent writers, no one has done more to direct our attention to the role of cognitive standards and values in science than Thomas Kuhn. Indeed, for more than two decades, the views of Thomas Kuhn – and reactions to them – have occupied center stage in accounts of scientific change and scientific rationality. That is as it should be, for Kuhn’s Structure of Scientific Revolutions caused us all to rethink our image of what science is and how it works. There can be no on active in philosophy, history, or sociology of science whose approach to the problem of scientific rationality has not been shaped by the Gestalt switch Kuhn wrought on our perspective on science. This debt is so broadly recognized that there is no need to document it here. Less frequently admitted is the fact that, in the twenty-two years since the appearance of The Structure of Scientific Revolutions, a great deal of historical scholarship and analytic spadework has moved our understanding of the processes of scientific rationality and scientific change considerably beyond the point where Kuhn left it.

Indeed, we are now in a position to state pretty unequivocally that Kuhn’s model of scientific change, as developed in Structure and elaborated in The Essential Tension, is deeply flawed, not only in its specifics but in its central framework assumptions. It is thus time to acknowledge that, for all its pioneering virtue, Kuhn’s Structure ought no longer be regarded as the locus classicus, the origin and fount, for treatments of these questions. It is time to say so publicly and openly, lest that larger community of scientists and interested laymen, who have neither the time nor the inclination to follow the esoteric technical literature of these fields, continues to imagine that Kuhn’s writings represent the last (or at least the latest) word on these matters.

Some simple math puts the origins of Kuhn’s ideas right around the time the so-called cognitive revolution began (though, as argued by Pullum, it’s not clear exactly when the cognitive revolution started, or even if it has a discernible beginning). It seems that Laudan’s nearly thirty year old call to move past Kuhn’s Structure either wasn’t heard or wasn’t heeded.

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Mischaracterizing Chomsky

Norbert ponders the first year of Faculty of Language, and in doing so links to the first post. I started reading FoL maybe six months ago, and I didn’t go back and peruse the archives, so I hadn’t previously seen this post.

Anyway, two things amused me about this post.

First, there’s an amusing case of structural ambiguity in the second paragraph (“It” refers to the blog and its stated purpose of rectifying ignorance about the generative enterprise):

It will partly be a labor of hate; aimed squarely at the myriad distortions and misunderstandings about the generative enterprise initiated by Chomsky in the mid 1950s.

Given the rest of the post, even if you didn’t know anything about Chomsky, the generative enterprise, or Norbert’s position with respect to either, you would know that the intention here was not to say that Chomsky initiated myriad distortions and misunderstandings about same. Nonetheless, I was amused by the a priori very reasonable parse that gives exactly this interpretation.

Second, I hadn’t realized that Norbert’s contentious relationship with Christina Behme started with the first post on FoL. She seems to have essentially infinite time on her hands to respond quickly, and often voluminously, to Norbert’s posts, but it surprised me to learn that she was there from the very beginning. I rarely read comments on most of the blogs I follow, but the comments on FoL are often interesting and worthwhile, and the Behme-Hornstein non-interactions provide amusing drama.

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Populations and individuals

Maggie Koerth-Baker posted a link at Boing Boing to a blog post purporting to explain why any particular finding in experimental psychology may or may not apply to any particular individual.

Actually, the author (Jamil Zaki) goes a lot farther than that:

Psychological studies are not about you.  They make few if any predictions about how you should live your life, how to tell if you’re an introvert, or anything else about you as an individual.

So, it’s not that it’s uncertain if such studies apply to you, it’s that they make “few if any predictions” about you.

The justification for this assertion is problematic, though. Zaki writes:

A typical study might include 200 people, dividing them into groups (say, people told to act generously versus those told to act selfishly), and demonstrate a statistically significant edge in happiness for one over the other.  Like a batting average, though, even strong differences across groups tell us virtually nothing about how generosity or selfishness would affect the happiness of any one person….

…psychological studies, without telling us about any one person, can tell us about how changes in behavior (again, think generosity) might affect the well-being of whole populations.

A strong difference across groups is, in the vast majority of cases, a difference in the means of some measured variable. Depending on how the individuals making up the groups are distributed, a group (mean) difference can tell us quite a bit about how generosity or selfishness (or whatever experimental manipulation we’re interested in) affects the happiness of any one person.

Of course, an experimenter has already observed how their manipulation affected the individual people in their study. And if we’ve got a reasonable model, we could, in principle, generate (probabilistic) predictions about more or less likely effects on a thus far unobserved individual. The predictions will be noisy, and they will be uncertain, but it seems too strong to say that measured group differences tell us “virtually nothing” about individuals.

If changes in behavior affect whole populations, then by necessity, they affect the individual members of those populations.

Posted in SCIENCE!, statistical description, statistical modeling | 2 Comments

Three quick bits of silliness

Colin Berry doesn’t like Elysium. He complains that the characters are free of nuance and subtlety, and that despite ostensibly being about Serious Issues, the movie wallows in carnage and violence. I haven’t seen it, so I can’t comment on these criticisms other than to say that they are disappointing if true – District 9 was surprisingly fun and interesting, so I was thinking that Elysium would be, too.

can however comment on some very silly things that Berry says in his rant:

These kinds of films aren’t entertaining anymore; they’re offensive. Yet they sit side-by-side among countless clones, bullets in the chamber, cogs in wheels in the monstrous Movie Marketing machine — operating within the supercolossal Entertainment Industry — which only cares about money and formula and self-protection. No wonder the industry is dying. For our parts, as moviegoers, we are partly culpable for legitimizing these projects, in which innovation, risk, and surprise are next to nil, by continuing to buy their tickets.

1. To the extent that formula and self-protection matter to the entertainment industry, they matter because they help ensure money. It’s a for-profit business, movie making.

2. As far as I can tell, the industry isn’t dying. It’s actually doing quite well recently. The people looking to make money by making movies aren’t idiots.

3. Given that the industry is all about making money, and given that it’s doing a bang-up job at this, it seems to me that the ticket-buying public is entirely culpable for the objectionable content of so many of the biggest, most profitable movies. This is not to say that every person that buys a ticket for, say, the newest mumblecore project is to blame for The Avengers, Elysium, and the Fast and Furious franchise (among many, many other violent movies). But certainly the people buying tickets to those movies are at fault – they’re the whole reason those movies get made!

For what it’s worth, I agree with Berry that entirely too many movies are predictable and uninteresting. I rarely go to the theater to watch movies anymore for exactly this reason, not to mention the the ever increasing ticket prices and the fact that other moviegoers are, far too often, exceptionally obnoxious. But Berry’s “analysis” of the industry and why so many movies kind of suck just doesn’t cut any ice.

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A sad day

UPDATE: Bayes’ rule has been fixed on the xkcd page. Good (or “good”) thing I grabbed the original image to post here.

UPDATE 2: Relevant xkcd (ht Scott).

The venerable xkcd got Bayes’ rule wrong today:

seashell

Let O be “I’m near the ocean” and S be “I picked up a seashell.” The equation should look like this, instead:

    \begin{equation*}\Pr(O|S) = \frac{\Pr(S|O)\Pr(O)}{\Pr(S)}\end{equation*}

Of course, the point still stands, even if the joke is blunted for me. We can illustrate with some reasonable-ish values for various terms. Let’s say that the probability of picking up a seashell given that you’re near the ocean  is \Pr(S|O) = 0.35, that the probability of being near the ocean is \Pr(O) = 0.1, and that the probability of picking up a seashell when you’re not near the ocean is \Pr(S|\sim O) = 0.01.

Noting that \Pr(S) = \Pr(S|O)\Pr(O) + \Pr(S|\sim O)\Pr(\sim O) and that \Pr(\sim O) = 1-\Pr(O), we sub these numbers in to the formula above and get:

    \begin{equation*}\frac{0.35 \times 0.1}{0.35 \times 0.1 + 0.01 \times 0.9} \approx 0.8 \end{equation*}

So, even if you are (like me) way more likely to not be near the ocean than to be near the ocean, the paucity of seashell-picking-up opportunities far from the ocean makes it overwhelmingly more likely that, if you’ve picked up a seashell, you’re near the ocean.

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Falsifiability and interesting cowbell

The most recent few posts on Faculty of Language delve into some interesting questions in philosophy of science. The focus of these posts is falsifiability as a criterion for theory (or ‘proposal’, which is a kind of not-quite-theory theory) evaluation. They’re worth reading in their entirety, I think, but I wanted to briefly discuss a couple issues that deserve some elaboration.

In the first post in the series, Norbert writes (emphasis mine):

Lakatos talked about central belts versus auxilliary hypotheses…, but now pretty much any account of how the rubber of theory hits the road of experiment highlights the subtle complexities that allow them to make contact. This said, scientists can and do find evidence against particular models (specific combos of theory, auxiliary hypotheses, and experimental set-ups), but how this bears on the higher level theory is a tricky affair precisely because it is never the theory of interest alone that confronts the empirical jury. In other words, when something goes wrong (i.e. when an experiment delivers up evidence that is contrary to the deductions of the model) it is almost always possible to save the day by tinkering with the auxiliary hypotheses (or the details of the experimental set-up or the right description of the “facts”) and leave the basic theory intact.

He follows this up with a discussion of physicists figuring out the appropriate energy range to look for evidence of the Higgs boson. I won’t claim to know much at all about the standard model in physics, but I don’t think that physicists probing multiple energy ranges in the search for the Higgs boson is a good example of saving a theory by tinkering with auxiliary hypotheses. A better example would be, say, if physicists didn’t find any plausible Higgs-related signatures in their particle collision data, so they revised their (auxiliary to the standard model) hypotheses about how the particle detectors function.

But there is a larger point to make, namely that even if you have a picture perfect illustration of theory-saving auxiliary-tinkering, this is a far cry from the claim that “it’s almost always possible” to empirically insulate a theory this way. A single instance does not a general fact make.

Furthermore, even if you could, in general, change auxiliary hypotheses to save a theory, it doesn’t follow that you should do so. Larry Laudan discusses this in detail in his essay Demystifying Underdetermination, which is highly recommended.

The other issue I wanted to touch on is Norbert’s conflation of personal taste and more objective criteria of theory evaluation in his cowbell proposal (follow-up here). The basic idea here is that theories that are interesting are good, theories that are boring are not, and whether a theory is interesting or boring is unrelated to that theory’s truth or falsity.

Having read lots of Larry Laudan’s work on philosophy and history of science, I’m convinced that theories are (appropriately) evaluated on multiple dimensions. Norbert touches on some (more or less) objective criteria (e.g., whether or not a theory makes surprising predictions), but his notions of interesting/boring and “explanatory oomph” seem to be very subjective and share some of the problems that his notion of elegance (also) exhibits.

Specifically, to the extent that we can get useful definitions of ‘interestingness,’ ‘boringness,’ or ‘explanatory oomph,’ my guess is that they will be functions of more objective (and readily, rigorously defined) theory evaluation criteria that have real epistemic value. Norbert seems to be getting, at least in part, at the fecundity of a theory, with maybe a dash of explanatory scope thrown in for good measure, but when he writes things like “de gustibus very much disputandum est,” he is clearly, deliberately invoking subjective criteria.

This seems to me to be a lousy theory of theory choice. It’s not even really a theory, more just a post hoc description. Saying that GB theory became less interesting, more boring, and lost explanatory oomph seems, at best, to be saying something about syntacticians’ personal opinions of the theory. Such opinions may well be based on epistemically valuable criteria, but the interestingness/boringness is produced by these criteria, and it’s these criteria that function scientifically to enable theory evaluation, not the interestingness/boringness per se.

For example, people may be finding GB more boring because it has reduced fecundity, or maybe because there are inconsistencies between it and other, neighboring cognitive theories. (I get a sense of the latter when, e.g., Norbert writes that, in the minimalist program the faculty of language “is a congery of domain general powers (aka operations and principles) with a small dollop of linguistic specificity thrown in.” GB was nothing if not highly linguistically specific.) But if you’re going to convince anyone that GB should be abandoned for the prettier, younger Minimalism, you won’t get very far just by arguing that the former is boring and the latter interesting.

All that said, there’s much I agree with in the FoL posts (naive falsificationism is bad, m’kay?), and whether you agree with Norber or not, they’re thought provoking and good fun to read.

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lme4, fixed effects, and SEs

When you fit an lmer model in R, it’s easy to see the fixed-effect parameter estimates and the standard errors associated with them (i.e., just type the name of your fit.object or feed it to summary()), but, at least at first glance, it’s not so easy to extract both of those sets of numbers to generate tables or figures.

Well, it turns out that if you have the arm package, getting the SEs is easy, too. To get the fixed effect estimates, just use fixef(fit.object), and to get the SEs (with arm installed), just use se.fixef(fit.object).

I don’t really have anything else to say about this, but I figure having a post devoted to it may be helpful to someone at some point, thereby counteracting, if only in a small way, the generally awful ‘official’ R help responses to questions about this kind of thing.

UPDATE: Edited to make use of the fact that there is a built in way to make code look like code that I had somehow overlooked.

Posted in SCIENCE!, statistical modeling | 2 Comments