Pr(X=51 at t=2 | X=1 at t=0) = Pr(X=51 at t=2 | X=4 at t=1) = 0.

In plain English, this describes a situation where it remains impossible for a variable to clear some threshold value by a given time even though its value increases somewhat over an earlier period. 

But say the following argument was made:

• We observed X=1 at t=0, and
• Now at t=1, we observe X=4.
• Therefore, Pr(X=51 at t=2) increased between t=0 and t=1.

Obviously, and as is made extra-clear by the initial example, the third statement does not follow as a matter of pure logic from the first two statements.  But don't we hear this sort of argument all the time?  Certainly it is very often a reasonable claim, but it presumes at least limited knowledge about the distribution of future outcomes given present data. 

The next argument, however, is in most cases probably less reasonable:

• We observed X=1 at t=0, and
• Also at t=0, we observed Y=4.
• Given these observations, we know that Pr(Y=51 at t=1) > Pr(X=51 at t=1).

When I spent several months as an activist in the Libertarian Party, I often heard this argument:  "Our candidates tend to get higher vote shares when they run for State Assembly than when they run for President.  Therefore, we should focus on campaigns for State Assembly, as that office is more winnable."  I thought this argument to be completely wrong; I argued that, as most voters don't know or care who runs for State Assembly, it is impossible for a third-party candidate to be elected, while the Presidential campaign draws enough attention so that a third-party candidate could win.  But my point today is more generalized:  this is, I believe, a type of argument that is often made and often accepted as more sound than it is. 

This all is surely elementary, but it is an example of a "logical fallacy", is it not?  If it is, is it important enough to get its own page on Wikipedia, or is it just a special case of the spurious relationship?  It seems different to me, but I am not a logician by trade.

UPDATE!  It occurs to me that this is perhaps not much more than a generalization and formalization of the slippery slope argument.  Even so, I still like it as a blog post.  It's end-of-semester crunch time*, so you can't be expecting too much around here.

*Jeremy:  I don't believe this is a sports metaphor, but I will defer to your judgment.  I know such metaphors are out of bounds, so edit if necessary!

And for the second time this year, I am happy to see that the Patriots have made the final four; this time, however, I hope they lose.

UPDATE:  I note that an ESPN web poll is giving the edge to the Patriots, with the Colts second.  It just goes to show that the size of the fanbase does not determine prices!

There are separate contracts on the winner of the general election, and these allow one to construct a crude implicit measure of each candidate's electability given their nomination.  Somewhat surprisingly given his reputation, McCain doesn't fare much better on electability than does Clinton.  Giuliani (9-10% to be elected, 13% to be nominated) clearly appears most electable, followed by Gore (7% to be elected, 11% to be nominated).  Obama, McCain and Clinton follow, with Romney appearing least electable.

So would Republicans be better off with Giuliani than with McCain, and would Democrats actually be better off returning to Gore than to Clinton?  (How deliciously ironic would it be if the latter question came in for serious public discussion during the primary campaign?!)

Apparently those willing to put money where their mouths are generally disbelieve that Bush can or will execute a successful "October Surprise" to boost himself by that date.

The prices on avian flu contracts at Tradesports offer encouraging news.  The BIRDFLU.USA.30SEP06 contract (on whether the U.S. government will announce that the H5N1 virus has been contracted by some American birds) looks to be headed for a zero-value expiration 20 days from now, down from a high of 65 last March.  Even better news is that, looking six months ahead, the BIRDFLU.USA.31MAR07 contract is only trading at around 30.

And speaking of disasters that were being hyped a year or so ago, check out the tanking price on CAT3.06SEASON.FL!

UPDATE:  Tyler Cowen reminds me that the chance of the virus mutating into another, more human-unfriendly form is a variable of prime concern.

[I]n 1996, the Iowa Electronic Markets (IEM) ran markets to predict the chances that different candidates would become the Republican Presidential nominee.  Other concurrent IEM markets predicted the vote shares that each party would receive conditional on the Republican nominee chosen.  ...[From the IEM prices,] Republicans could have inferred that Dole was a weak candidate and that his nomination would result in a Clinton victory.

Right now, major-party elders should be setting up prediction markets for 2008 so that they can use the early results to dissuade supporters of weak general-election candidates in time to prevent said candidates from attaining nomination.

However, if these markets are based overseas, they may be too thinly traded to be accurate.  The markets would certainly have more money on the line, and thus be more far more efficient or accurate, if the U.S. government would make it clear that the participation of American traders in these markets is legal.  For seriously, who but an American is going to stake a sawbuck on the candidacies of George Allen, Mark Warner, or others who are unknown on the international stage?

Perhaps this is an angle that can be used to persuade politicians fully to legalize prediction markets in the United States.  (Karl Rove, are you reading this?)