But the final probability is still just a single number.
But I suspect the GP's reasoning was getting interference from the valid point that poll numbers, which are not probabilities, are meaningless without confidence intervals or some other measure of uncertainty.
That uncertainty will then feed, via some model, into a probability estimate of a candidate winning. And that's just a single number.
For example, I can imagine a well engineered coin that has a 0.5 probability of heads with a 0.005 confidence (ie, we suspect that the true weighting between heads and tails is likely between 0.495 and 0.505 19 times out of 20) and I can contrast it with a hastily made coin that can still have a 0.5 expected probability with a 0.1 confidence (ie, we expect the true probability to be between 0.4 to 0.6, 19 times out of 20).
Coin A and Coin B have dramatically different impacts on our decision making. For example, selling insurance against 5 identical flips in a row is a much more expensive proposition for Coin B than Coin A.
I feel like I really should know this given my data science background, but sometimes the basics slip away.
We could do the same thing for the 2016 election, but we would have to specify exactly what we meant by a "repeat". Do we just let Hillary run for the 2020 election and see what happens? Or do we put back every atom to the exact position it had in 2014, so that the only divergences between the two elections are caused by quantum randomness? Or something in between, like looking at all elections where a demagogue outsider runs against an established insider?
Really it doesn't matter what definition of "repeat" we choose. Since the repeat won't actually happen, we can't be called to bet on it, so knowing the probability in that case isn't too useful. Whereas a coin actually can be flipped multiple times.
I could imagine some frameworks where confidence intervals in this way would be useful - ex. I have 3 fairly different world models that I think are equally likely, each has a P(election), what's the P(election|world) and get confidence intervals across world models rather than just summing over them to P(election). But I agree that for most common approaches that simple probabilities are most useful and clear.
But there's no meaningful way to repeat an election, so I don't think similar distributions or confidence intervals are useful in that case.
EDIT: We can imagine rerolling an election, but since we can't actually do it we don't have to bet on it so the information wouldn't be very useful to us.
"Bayesian perspective" covers a lot of territory, and your assertion depends on the situation and the modeling objective. If a probability is a model parameter (for instance, frequency of heads for a particular coin), then summarizing the posterior distribution on that parameter with a confidence interval can be a sensible thing to do.
Or do you mean as opposed to a credibility interval?
They probably should have just written that right there: "71%, better odds than two coins coming up heads."
But: in order to understand if the model is any good, you need to try to build a good model, describe your model, and describe what you know about the underlying processes. With words, visuals, and math. Which is pretty much what 538 does better than most.
The difference is how the probability changes in response to new information. Learning more about the coin won't change the probability of heads from 50%, but doing more exhaustive polls would have likely improved our prediction about the election.
However I don't think there's as much of a difference as you think. If we learnt more about how the coin was going to be flipped then that would certainly improve our estimate of the outcome. If we found out the exact way it would be flipped we could calculate the outcome exactly.
So we can't really compare the two situations quantitatively, since there's no way to match up like-for-like the information we could receive. But we can say, for each possible piece of information we could receive, how much we expect it to change our probability.