What is the probability that the next president of the United States is an Independent?

Question

In his (fantastic, fascinating and sure to become a classic) book "Thinking, Fast and Slow", Daniel Kahneman asks the reader the question in the title. I was wondering how people would approach this problem. A similar question with some answers (one of them is mine) can be found here. However, these answers consider a fixed set of results (namely - success and failure). My answer and others don't take into account an unknown number of possible outcomes. So, I'm curious as to how one may overcome these issues. For example, the accepted answer uses the rule of 3 (explained in the comments). How one would adapt this to a varying number of outcomes?

One natural approach would be to call the event "a republican or a democrat won" a success and call all other events failure. Then the problem is reduced to the one cited above. Are there other ways to rephrase or approach the problem? This question is opinion based to the extent statistics is opinion based: You can have a frequentist solution, you can use Bayes' rule etc.

A couple of notes:

1. I don't care about the number, the reasoning is far more interesting.
2. I am not asking you to extend my (Bayesian) approach. I'm interested in your approach.
3. Carrot is the next president, so the question relates to the one after him.
4. Feel free to suggest tags.

Answer 1

It's important to understand that probabilities are really just functions on sample spaces (or rather sigma-algebras to be precise). The way we use language to express uncertainty often leads people to assume that there is some universally "true" probability out there and that it just happens to be unknown. However, there can only be a true probability if there is a universally agreed upon sample space and measure function. Of course, there is never universal agreement about these objects. This situation could be contrasted with "the" probability a coin toss lands heads. In that situation, there are two possible outcomes and everyone can agree that both have equal probabilities.

Anyone who volunteers an election probability is really just expressing their thoughts about the set of possible events and how to measure them. But we might wonder who happens to have the best probability. This seems to be the question you're getting at.

One formal way to quantify the quality of a probability assignment is to measure the size of the sigma-algebra from which it was derived. This is impossible in practice. Ultimately though, the best way to approach this problem is to start building out a sample space and provide probability assignments to the individual events. To do this, we need to start rattling off all the facts we can think of...like who the possible candidates are, how likely they are to run, how likely they are to win, etc. It's really tough to get information to make informed assessments about these events though.

Although organizations like FiveThiryEight and The Upshot do this sort of thing professionally, there's good reason to assume that their probabilities aren't the best available. They primarily use polling data to generate probabilities. On the other hand, gambling websites generate probabilities using polls and betting data. Betting data leverages information from a wide-range of people. This takes into account more information than any one person can come up with. Gambling sites don't explicitly post the probabilities of each party winning but they can (almost) be inferred from the odds they list.

Sadly, the next election is really far out. If you go to Paddy Power and look at the odds, you'll see some crazy listings. And if you try to infer probabilities from the odds, you'll see that they're leaving themselves a lot of room for error. Gambling websites just can't make precise inferences about the probabilities this early in the game. It's reasonable to assume no individual could do significantly better.