If the election is balanced (the mean of the distribution is a tie)...
That’s an important and non-obvious assumption to make.
a 50⁄50 chance for each voter gave a .00006 chance of tie, while 49.9/50.1 gave the 10^-90
So, in short, the 10^-90 figure is based on the explicit assumption that the election is not balanced?
That’s why the two methods you mention produce such wildy different figures; they base their calculations on different basic assumptions. One can argue back and forth about the validity or lack thereof of a given set of assumptions, of course...
That’s an important and non-obvious assumption to make.
Yes, I agree.
I’m much more sympathetic to the 10^-90 estimate in the paper than Gelman’s quote is; I think he misrepresents the authors in claiming they asserted that probability, when actually they offered it as a conditional (if you model it this way, then it’s 10^-90).
One can argue back and forth about the validity or lack thereof of a given set of assumptions, of course...
That is why I posted it as a comment on this particular post, after all. It’s clear that our subjective probability of casting a tie-breaking vote is going to be far less extreme than 10^-90 because our belief in the binomial idealization being correct puts a much less extreme bound on the tie-breaking vote probability than just taking 10^-90 at face value.
That’s an important and non-obvious assumption to make.
So, in short, the 10^-90 figure is based on the explicit assumption that the election is not balanced?
That’s why the two methods you mention produce such wildy different figures; they base their calculations on different basic assumptions. One can argue back and forth about the validity or lack thereof of a given set of assumptions, of course...
Yes, I agree.
I’m much more sympathetic to the 10^-90 estimate in the paper than Gelman’s quote is; I think he misrepresents the authors in claiming they asserted that probability, when actually they offered it as a conditional (if you model it this way, then it’s 10^-90).
That is why I posted it as a comment on this particular post, after all. It’s clear that our subjective probability of casting a tie-breaking vote is going to be far less extreme than 10^-90 because our belief in the binomial idealization being correct puts a much less extreme bound on the tie-breaking vote probability than just taking 10^-90 at face value.