ddt said:
I have to backtrack on my own words again. For the expected number of EC votes, you can simply multiply the chance of winning in each state with its number of EC seats. It doesn't matter that the races in the various states are not independent - for that single number "expected value".
I don't mean to
but I don't think that is right.
Nate's core tool is
a simulation. Just one single run. In that one simulation there are no statistics
at all. Each candidate is allocated the deterministic (not statistical) EC votes that that simulation produced.
Next, he does that a gazillion times.
NOW, and only now, is when the statistical work is done. He has a distribution of EC votes per candidate and uses some statistic (mean, mode, etc. I don't know) to come up with his final probability. The absolute key here is that there is NO state-by-state averaging done.
Yes, that's right.
I think we largely agree on what Nate's doing. Only I would not say his core tool is the simulation. It's his model underlying that simulation. The process has the following steps.
First, there's the input data, of two kinds:
1) polling data from various polls, which is continuously added to by new polls
2) demographic data, which is static throughout the election process
Second, there's Nate's model which says how to interpret these data. The model says how to weigh each poll: some are better than others, some have an inherent bias to one of the parties. The model also interprets the demographic data into correlations between the outcomes in the various states. To take jt512's hypothetical example: if the demographies of CO and NM are exactly the same, the model translates this to that those states vote identical. This is exaggerated, the model is certainly more subtle than that, and probably also takes polling data into account for establishing correlation between the various states.
Third, the model produces 54 stochasts, to put it in mathematical terms, for each of the separate state elections. That is, you have probability distributions for each state what percentage each party will get. To make a crude layman analogy, you now have 54 dice with a "Clinton" side and a "Trump" side. Those dice are all differently weighted, and the correlations from the model say that some dice are connected. Those CO and NM dice - to carry on that example - have a 100% correlation, so they're effectively glued together. The OH and PA dice are more loosely tied together, so that whenever the OH die turns up "Clinton", the PA one will too.
Fourth, those stochasts are what he runs his simulation with. Each run of the simulation gives one discrete outcome, e.g., Trump wins OH with x% margin and Clinton wins PA with y% margin, and overall, Clinton has n EV and Trump m. He runs the simulation a gazillion times. Yes, that's simply a Monte Carlo run.
Fifth, all the numbers on the 538 page are the averages over those gazillion simulation runs. The probabilities per state that Clinton wins are simply the percentage of simulation runs she won, and the expected value of Clinton's EC vote is also the average number of EC votes over all of those simulations.
And this is a bit where I struggle with why Nate needs simulations at all. You can only run a simulation when you have a probability distribution to begin with. The probability distribution for, say, Ohio, already rolls out of his model and is plugged into the simulation algorithm. Basically, already at step (3) you can say "Clinton has a 65.1% chance of winning Ohio". I surmise it's in the correlations between the various states that his model is too difficult to simply be calculated and that he needs a Monte Carlo run. I admit to a bit of a bias against Monte Carlo runs, mainly because I see all too often people making simulations for trivial questions, like "what is the chance of throwing 7 with 2 dice?" which you can perfectly calculate with pencil and paper. But Nate is a professional statistician, he surely knows what he's doing.
Finally, to come back to my statement you objected to. Yes, that statement is true. Let's do that in proper mathematical terms, and define stochasts:
D
OH = number of Democratic electoral votes from Ohio
That's a stochast with outcome either 0 or 18. The chance it is 18 is those 65.1% that comes out of the simulation. Define the respective stochasts for all state races.
Then define the stochast:
D
USA = number of overall Democratic electoral votes
which is a stochast with discrete values between 0 and 538.
Then it obviously holds that:
D
USA = SUM (i in states) D
i
And then basic probability theory says about their expected values:
ED
USA = SUM (i in states) ED
i
It doesn't matter for the latter formula whether the various state stochasts are independent or not (they're not). That doesn't matter for the single value of "expected value", it does matter for the distribution.
