"Orthogonal" at the Supreme Court

Orin Kerr at the Volokh Conspiracy points to arguments in a U. S. Supreme Court case yesterday which used the word "orthogonal" in the technical-jargon sense defined, say, at the Jargon File. (See page 24 of the original transcript.) There's a follow-up here by Eugene Volokh, basically saying that there's no point in using big words if your audience doesn't understand them. (And the justices did stop to ask what the word meant.)

first let's kill all the lawyers

Language Log writes about the semantics of pork, inspired by this New York Times article. A current health care bill includes statements like

"any hospital that is co-located in Marinette, Wis., and Menominee, Mich., is deemed to be located in Chicago"

and other similar references that sound innocuous -- they're just defining a metropolitan area -- until you think, wait a minute, is that anywhere near Chicago? (I checked a map; it's 259 miles away.) And how many hospitals like that could there be? (As it turns out, exactly one.) Apparently hospitals in metropolitan areas get bigger reimbursements from Medicare for various procedures, on the theory that the cost of living for their employees is greater than that for rural hospitals.

Language Log goes on to point out that is an example of how there are two ways to define any particular set: by listing its elements, or by specifying a set of constraints. A mathematical example that immediately comes to mind is "the set of even primes". Or, somewhat more innocuously, the statement "let p be an even prime". You don't see this that often, because it's easier to say "let p equal 2". But I have seen it, in proofs of the form: "Theorem: All primes p have property x. Proof: let p be an odd prime. Then (proof). Alternatively, let p be an even prime. Then (simpler proof)."

Of course, when one specifies a set by giving constraints, there's always the problem that the set might be empty. What would happen if the health care bill in question said, say, "$100,000 should be distributed evenly between all hospitals within a quarter-mile of Isabel's apartment?" There are none. Who gets the money? (I suspect there is some conventional interpretation for this sort of thing. I don't think you'd see it in this sort of bill, but I can imagine, for example, a doting grandparent writing in their will "my grandchildren shall equally split my [large sum of money]" and then the grandchildren tragically die before the grandparent.) I've heard the apocryphal story of a student who goes into his PhD defense and says that he will be presenting results on a certain sort of algebraic structure satisfying the following eight conditions. One member of the committee interrupts and says that he can prove there are no such groups. The student doesn't get the PhD. Proving things about something that doesn't exist is considered worthless, no matter how ingenious the proofs might be.

jury duty: coincidences, and semi-juries

Today I had jury duty.

During the lunch break I went to Reading Terminal Market, where I spent $9.05 for lunch (it was a large lunch, because I didn't want to be sitting around hungry); my pay for one day of jury duty was $9. (I was not chosen for a trial.) Coincidence? Probably, because I wasn't thinking "I'm spending my nine bucks on lunch" when I was walking around choosing where I would buy it.

Then I wandered down to a bookstore and found myself flipping through Steven Landsburg's book More Sex Is Safer Sex: The Unconventional Wisdom of Economics. . (I didn't buy it; it seemed interesting, but not $26 worth of interesting.) In particular, this book suggests that the jury system is broken (the link goes to the Freakonomics blogs, where he was interviewed a few weeks ago). The basic idea is that jurors have no incentive to do a good job. This is clearly true, although I'm not sure how to incentivize the jury system. When I got home I ran across an entry in the Freakonomics blog which mentioned that book. Coincidence? Maybe. Maybe not. (And I thin I saw the original Landsburg interview a few weeks ago and forgot about it, which may have primed me to be more likely to look at that specific book.)

Yet another coincidence: I went to high school with the judge's son. (I don't think this is how I got out of serving.) I also went to high school with the son of one of my panel-mates. It occurred to me as I was heading in this morning that if a few hundred people were called today, the chances I'd know one of them were not bad; there are probably about a million adults in Philadelphia, of whom I know a few hundred. I don't think anyone I know was there today (if so, I didn't see them) but as I said there were parents of people I knew. In some ways Philly is the largest small town in the country.

While waiting to be selected, it occurred to me that the voir dire procedure is set up so that no individual juror who is selected was biased. We were a panel of 50 for a sexual assault case, from which fourteen jurors were essentially selected; although I wasn't counting, I would guess that there were no more than twenty people who satisfied the following three conditions:

• 1. doesn't possess a strong technical background (we had to write our occupations on the forms that were distributed; it seemed that all the people who were asked about their occupation by the judge were either people in technical fields or people who worked as lawyers, police officers, etc.);


• 2. does not claim that jury duty would pose an extreme hardship;


• 3. had not been sexually assaulted or had someone close to them sexually assaulted. It's often said that one in four people are sexually assaulted during their lifetime.

Still, would it have been such a horrible thing to have a sexual assault victim on the jury? A randomly selected panel of fourteen would probably have had at least one. I don't see why each individual juror has to be unbiased in order for the group as a whole to be unbiased.

Finally, some math. In Landsburg's book he suggests the following: break each jury up into two half-juries of six. If they come to the same conclusion, that's the verdict; he wasn't clear on what to do if they came to opposing conclusions. (Presumably it would be treated like current hung juries are.) In this study by Bruce Spencer it's suggested that juries are "right" about 88% of the time. This got me thinking -- how likely does this mean an individual juror is to be "right" about the verdict? If we assume that jurors make their decisions independently, that majority rules (which is a bit ingenuous because juries in criminal cases have to be unanimous), and throw out 6-6 results, it turns out each individual juror has to come to the right decision with probability 63.6% to recover this 88% probability. This is related to the post I made a couple weeks ago about the World Series; if one team is slightly better than another, they have a decent chance of winning a single game but not so good a chance of winning a whole series. The teams here are, of course, "guilty" and "not guilty".

So what does this say for the half-jury suggestion? Let's say that each juror, indepedently, has a 63.6% chance of being right, and there's a jury of six. Say the defendant is guilty. Then the probability that all six jurors will say this is (.636)⁶ = .066; the probability that five think he's guilty and one thinks he's innocent is 6(.636)⁵(.364) = .227; the probability of the 4-2 split is 15(.636)⁴(.364)² = .325. So the probability of one of these three results is 0.618; the probability of having a 3-3 split is 20(.636)³(.364)³ = 0.248. So the probability of a half-jury finding the defendant guilty is (.618)/(1-.248) = .821. Not surprisingly, this is less than the chance of a full jury finding the defendant guilty.

But the chance that both half-juries find the defendant guilty is (.821)² = .674; the chance that they both find him innocent (even though he did it!) is (1-.821)² = .032. So the probability of finding the defendant guilty, given that there's a verdict at all, is .674/(.674+.032) = .955. In the end, this plan achieves much greater accuracy at the expense of increasing the number of hung juries. It seems worth considering, though. (Incidentally, you can't beat the hung jury problem by changing the sub-jury size; either at least one sub-jury is of even size or there's an even number of sub-juries, since 12 is even.)

I suspect, though, that this sort of thing would be rejected as being unnecessarily complicated. But the current voir dire process is byzantine enough that that hardly seems like a legitimate complaint.

edit (Tuesday, 9:16 AM): Landsburg has commented to this entry. In particular he points out that my assumption that juries would have the same accuracy in the arrangement with two half-juries as in the current system is incorrect; jurors would have more incentive to be accurate in his proposed system. This is true because in his proposed system the jurors are rewarded when both juries agree. But what I intended to show was that even without such a reward, his system still leads to a greater proportion of correct verdicts.

edit (Tuesday, 2:16 PM): Richard Dawkins suggested in 1997 that "Two juries of six members, or three juries of four members, would probably be an improvement over the present system". He also points out that jurors don't act independently, which is true; in my original analysis I was suggesting that even though jurors don't act independently, we'll assume that they act independently up until the moment they begin deliberation. This assumption is of course not true, but it was only a crude analysis.