Lots of people are following Nate Silver’s election tracking over at 538, especially his top-line estimate of the probability that Barack Obama will be re-elected in November. Silver has that number at 79.7% today. Sounds like good news for Obama. But it’s hard to get a gut feeling for what that number means. Yeah, it means Obama has a 4 in 5 chance of winning — but since the election isn’t going to happen 5 times, that proportion doesn’t quite engage the intuition.

Here’s one trick I thought of, which ought to work for baseball fans. The Win Probability Inquirer over at Hardball Times will estimate the probability of a baseball team winning a game under any specified set of conditions. Visiting team down by 4 in the 2nd, but has runners on 2nd and 3rd with nobody out? They’ve got a 26% chance of winning. Next batter strikes out? Their chances go down to 22%.

So when do you have a 79.7% of winning? If we consider the Obama-Romney race to have started in April or May, when Romney wrapped up the nomination, we’re about 2/3 of the way through — so let’s the 7th inning. If the visiting team is ahead by 2 runs going into the 7th, they’ve got an 82% chance of winning. That’s pretty close. If you feel the need to tweak the knobs, say the first two batters of the inning fail to reach; with two outs in the top of the 7th, bases empty and a 2-run lead, the visitors win 79.26% of the time, just a half-percent off from Silver’s estimate.

So: Obama 5, Romney 3, top of the 7th. How certain do you feel that Obama wins?

**Update**: (request from the comments) Silver currently has Obama with an 85.% chance of winning. That’s like: home team up 5-3, visitors batting in the top of the 8th, runner on first with one out.

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As a longtime Mariner fan, I don’t find this interpretation very reassuring.

Very interesting for me, I know little baseball and am curious about it.

It seems to me though that the analogy is a bit misleading. In the case of baseball statistics we are predicting the outcome of a single sample/match out of a “known” distribution, while in the case of election polling we are inferring (predicting) the distribution out of a sample, which we try to make large enough to be representative. Under usual assumptions (iid with finite moments) the sample (the empirical mean’s) distribution has variance Var X/n, where X is the preference of a voter and n the sample size. So polls are very accurate, actually the p-value for deciding whether Obama will win (null hypothesis: the mean of electoral votes for Obama is equal to or less than half the total), based on polls should be much lower than the .2 probability that Nate Silver gives. I am not sure how he computes that.

At election time we will sample a large number of independent voters/random variables, and we may assume we will have the “true” (mean) preference of a voter.

What may cause the outcome of the election to differ is more an actual change of the preference distribution over time (between now and election time) than current poll samples not being representative of current preferences. So this may be what Nate Silver takes into account, that opinions have historically evolved. Perhaps he extrapolates between now and election time by a random walk (with variance taken from historical data) starting from current polls (which favor Obama with a p-value much lower than .2) until election time, or using a more precise method, considering preference as a random process and estimating its parameters from historical data.

Coming back to the baseball analogy, I guess there is that kind of estimation when predicting match outcomes: if visiting team was ahead 2 runs in the 7th inning they won 82% of the time.

Ah… but now I see that the analogy is very good: an election is considered a single sample of the random process representing all elections, just like a baseball match is a single sample of the random process of all matches, which we both try to estimate. The uncertainty of measurements for matches, or elections is not relevant (it is actually 0 for the match, and almost 0 for elections, we just reasonably assume poll results represent true preference).

Now actually deciding on your final question must be computationally intensive and I’ll trust Nate Silver’s work. So I’m 80% sure Obama wins.

Thanks for the post.

How confident I feel in an Obama win really depends on his bullpen.

I can’t imagine leaving or tuning off a game with a 2-run differential going into the 7th… but of course, a team coming from behind in a baseball game is more interesting (and a lot less consequential) than a comeback in a presidential race.

and also whether the closer and 8-th inning guy(s) pitched the night before, or the night before that…

…while some of us can’t imagine watching as many as 6 innings of a baseball game…

[tried to include this link but it didn't work: http://xkcd.com/1107%5D

[argh, remove the "%5D" which used to be a closing "]“]

It’s the pitching, stupid!

that’s probably valid, although I might add some more baseball-related wrinkles and say that the “home” team (Romney) is facing some stiff relief pitchers in the 7th, 8th and 9th – maybe akin to Obama’s debating abilities? – and the “meat” of Romney’s “order” is up in the bottom of the 7th meaning that if they don’t make up the difference now – in the first debate or two? – the bottom of the order is coming up in the 8th and 9th against a good setup man and a great closer, i.e. Obama’s groundgame. Okay, this analogy may have officially become tortured but it was fun!

And whether his teammates are indifferently eating chicken and drinking beer in the clubhouse while his lead slips away.

Baseball is much more of a cerebral game than the other major sports. Action is less violent and limited. There is much more anticipation with each pitch. (Those who liken it to paint drying may need a prescription for their ADHD.) As Earl Weaver said: “You can’t sit on a lead and run a few plays into the line and just kill the clock. You’ve got to throw the ball over the goddamn plate and give the other man a chance. That’s why baseball is the greatest game of them all.”

[...] were down 2-0 in the ninth, their chance of winning per Fangraphs down to 5.8%. To keep up our comparison of baseball odds and political odds, that’s the same chance Nate Silver is giving Mitt Romney to win Minnesota. In other words, [...]

You were ahead of the curve on this one. Nate Silver now has Romney down by a touch down with 10 minutes to go in the fourth quarter. I think you should periodically update the baseball analogy, it is more intuitive to many folks than the statistical predictions.

7th inning and Obama is down plus JUST coming out with a plan….Odds aren’t so good now, rh?