Quote on Nate Silver, and how to think about probabilities

If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d try to sell you my chance for 349 thousand dollars.

I think the bigger question is what exactly it means for the probability estimate to be wrong. The best I can figure is that it's not whatever it's called where exactly x% of the predictions that you're x% sure of are correct. In that case, Romney winning is evidence that it's more extreme than it should be, and Obama winning is evidence that it's less extreme. Whether or not it's evidence that it's wrong depends on whether you thought it was more likely to be more or less extreme beforehand.

I am not seeing the rationality in Klein's analysis.

"If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited." The difference is that presumably he [the speaker] could have a near infinite number of things happen to him. So picking one and giving it much greater odds than one could reasonably expect (given most contests that grant that kind of monetary rewards) does siginify a pretty unusual situation which he should be thrilled about.

However, for all intents and purposes, the winner of the election is a binary choice. (Let's say we have 999 of a 1000 units of probability to distribute between two candidates). I think it is a given that both candidates at the level of the general election are pretty excited to be there, given that they have radically greater odds than the rest of the eligible population.

But all that should be taken for granted by any adult with any familiarity with the system. Thus, a model that predicts one candidate over the other at 60/40, let alone 75/25, odds/confidence level is not any kind of good, exciting news for the other guy [Romney in Silver's model]. Of course Silver isn't ruling out the possibility. And the stakes are high, certainly greater than a million dollars. But nonetheless, 35% to win a million dollars when you didn't know the possibility existed is different than 35% to win the presidency when you are one of only two candidates.

Edit: Just thought of a better way to phrase the above--Whether news of 35% odds is good & exciting or bad & dispiriting depends on one's priors. I would assume the challenger in a pretty divided country would have had 40-45% odds to begin with and wouldn't be excited to update downwards.

Better suited to the Quotes Thread?

If you told me I had a 35 percent chance of winning a million dollars tomorrow, I’d be excited.

If you told me I had a 1 percent chance of winning a million dollars tomorrow, I'd be excited.

If you told me I had a 1/3 chance of winning a million dollars tomorrow, I'd be excited and I'd spend at least 1/3 of the day working out plans for using that money, with a clear conscience.

(I can't find the Sequences post that needs to be linked here, the one about how a TV anchor should allocate the time writing about the two possible outcomes for an event that will be decided the next day.) ETA: that's it, thanks!

The fun idea for scoring guessing games is to wait till the result is known, and then give each contestant points according to the logarithm of the probability that they had previously given the to outcome that actually occurred.

(Hmm, all the scores are negative; some are more negative than others. Perhaps the score keeper needs to give log n - log p points if he wants the numbers to be human-friendly. (The point of log n - log p is that if there are n outcomes you can give each a probability of 1/n and that scores zero. You hope to do better and make a positive score, but you can do worse (if something you think really unlikely actually happens)))

The clever mathematics behind this is Gibbs' inequality. When you come up with your probability distribution q, your expect score (relevant if the game has many rounds) is where p is the true but unknown distribution. Gibbs' inequality tells you that this will always be less than . In the short run you can get a higher score in a particular play of the game by combining overconfidence and good luck :-)

In the long run, you best hope for a high score is to do a good job of guessing p. Scoring log q sets up the game that way.

On the other end of the spectrum we have Elspeth Reeve coming to Nate Silver's defense while giving too much evidential weight to a Romney victory.

But Silver [takes a weighted average] because some pollsters have a better track record than others, and some have a clear partisan tilt, left or right. If his weighting is wrong, we'll know next week.

It's a little jarring, if not surprising, to see such a defense punctuated with such an off-base statement.

On a lighter note, Silver has publicly offered to bet Joe "It's A Tossup" Scarborough on the outcome of the election. I wish this sort of thing occurred more often.