Bayes' Best Bet

Some time ago (too long ago for me find in the archives), someone over at The Corner (I think it was Ramesh Ponnuru) proposed, or at least described a Bayesian approach to determining who a party's "best bet" was: i.e., which candidate to vote for in the primary given their chances in the general election. For those unfamiliar with statistics, or those who didn't read that post some time ago, "Bayes' Rule" can be expressed in a number of different forms, but is a very fundamental building block for much of mathematics and statistical analysis. In the form relevant to this discussion, it can be stated as follows: Suppose there are two events, A and B. The probability that B occurs given that A has occurred is exactly equal to the probability of A occurring given that B has occurred, times the general probability of B occurring, divided by the general probability of A occurring. Or, expressed in somewhat sloppy mathematics: P(B|A) = P(A|B) * P(B) / P(A)
(The vertical line means "given".)

How does this apply to politics? First, let the event "A" be "Candidate Q receives his or her party's nomination." Then, event "B" is "Candidate Q wins the 2008 election." Thus, the "best bet" for your party of choice is formulated by answering the question: Given that candidate Q receives his party's nomination, what is the probability that he or she will be elected president? Or in the terms laid out above, "What is P(B|A)?"

So: how do we fill in the rest of the numbers? The most confusing is actually the simplest: P(A|B), or the probability that candidate Q received the nomination, given that they won the election. In our nice, neat, two-party system, a candidate can not realistically win the presidency without first winning a party nomination, so this probability is trivially 1.0, or a 100% probability for you non-math-types. That leaves us with two other numbers to find: P(B), the probability that the candidate will become president, and P(A), the probability that the candidate will receive his or her party's nomination.

Where, then, can we get these numbers? The author of my above-mentioned reference suggests that, rather than opinion polls, exchange/gamling sites such as Intrade serve as reliable estimates of these numbers (after all, when you put your money on the line, you're less likely to attempt to lie about the truth your profits depend on). Let's assume this is a reasonable assumption. Fortunately, Intrade itself provides us with exactly these two numbers. Without delving into how it works, suffice it to say that the price of a given candidate for either the nomination or the presidency, treated as a stock to buy or sell, is a good reflection (sufficient for this calculation) of the probability that that particular candidate will be, respectively, the nominee or the President. So, I ran these numbers, as of today, with some interesting results:

For the Republicans:

Candidate	P(A)	P(B)	P(B|A) x 100%
Giuliani	0.388	0.152	39.2%
Romney	0.241	0.087	36.1%
Thompson	0.181	0.071	39.2%
McCain	0.053	0.022	41.5%
Paul	0.060	0.022	36.7%
Huckabee	0.030	0.005	16.7%

And for the Democrats:

Clinton	0.674	0.461	68.4%
Obama	0.114	0.066	57.9%
Edwards	0.042	0.025	59.5%
Richardson	0.007	0.001	14.3%

The big (bolded, for your convenience!) punchline first: the Republican's best bet is to nominate, of all people, John McCain. And as if this surprises anyone other than John Edwards' secret lover (ht: K-Lo), Hillary is the Dems' best bet.

What's more interesting, at least superficially, is the rest of the numbers. Notable, with the sad exception of Huckabee and the very surprising inclusion of theronpaulexperience, is the overall closeness in viability of the Republican candidates -- while McCain is on top, the spread is very narrow (and very unfavorable for Republicans, but that's not what this is about). Also, seriously, Ron Paul having a better chance than McCain of getting the nomination (see below)? I smell a ronpaulswarm (say what you will about the nutjobs who follow him, but they are True Believers).

I say superficially, because it's important to look at the nature of numbers themselves. For Fred, Rudy, and Mitt, their numbers are sufficiently high that rounding errors are minimal. However, on Intrade, the smallest apparent gradation in price is 10 cents, which translates into a probability value of 0.001. Therefore, the smaller a candidate's numbers, the more likely that rounding errors exist in the overall calculation, and the more unreliable the overall "best bet" statistic becomes. Thus, both John McCain, Mike Huckabee, and Ron Paul's numbers should all be taken with a grain of salt, as should Bill Richardson's.

But if these were God's Own Truth, then the message is clear: if you're voting in the Republican Primary, and all you care about is a Republican victory, vote McCain. And if you wanna be extra-sneaky, switch parties and vote Democrat in the primary, for Bill Richardson!

One last thought: comparing McCain's numbers to Paul's numbers generates a somewhat counterintuitive result of Bayesian inference: both of their winning-thepresidency probabilities are the same, at 0.022. By virtue of McCain's lower chance of getting the nomination, he has a higher value as our "best bet." That's a feature worth mulling over for a while. People who don't understand that are likely the same ones who argue against the truth of the Monty Hall Problem.