But in reality, it is impossible to perfectly predict who will win; either that information literally doesn’t exist, or it exists in a way which we cannot access.
Another option: can you control (or influence) the outcome?
I agree with your final paragraph – I'm fine with assuming there is a true probability. That said, I think there's an important difference between how accurate a prediction was, which can be straight-forwardly defined as its similarity to the true probability, and how good of a job the predictor did.
If we're just talking about the former, then I don't disagree with anything you've said, except that I would question calling it an "epistemically good" prediction – "epistemically good" sounds to me like it refers to performance. Either way, mere accuracy seems like the less interesting thing of the two.
If we're talking about the latter, then using the true probability as a comparison is problematic even in principle because it might not correspond to any intuitive notion of a good prediction. I see two separate problems:
In this case, going from smart to very smart actually makes your prediction worse, even though you picked up on a real phenomenon.
I personally think it might be possible to define the quality of a single prediction in a way that includes the true probability, but but I don't think it's straight-forward.
[My inside view is pretty confident in this; my outside view is very not. Cross-posted from Grand, Unified, Crazy.]
Zvi’s post on Evaluating Predictions in Hindsight is a great walk through some practical, concrete methods of evaluating predictions. This post aims to be a somewhat more theoretical/philosophical take on the related idea of what makes a prediction “good”.
Intuitively, when we ask whether some past prediction was “good” or not, we tend to look at what actually happened. If I predicted that the sun will rise with very high probability, and the sun actually rose, that was a good prediction, right? There is an instrumental sense in which this is true, but also an epistemic sense in which it is not. If the sun was extremely unlikely to rise, then in a sense my prediction was wrong – I just got lucky instead. We can formally divide this distinction as follows:
But what do we mean by “true probability”? If you believe the universe has fundamental randomness in it then this idea of “true probability” is probably pretty intuitive. There is some probability of an event happening baked into the underlying reality, and like any knowledge, our prediction is good if it matches that underlying reality. If this feels weird because you have a more deterministic bent, then I would remind you that every system seems random from the inside.
For a more concrete example, consider betting on a sports match between two teams. From a theoretical, instrumental perspective there is one optimal bet: 100% on the team that actually wins. But in reality, it is impossible to perfectly predict who will win; either that information literally doesn’t exist, or it exists in a way which we cannot access. So we have to treat reality itself as having a spread: there is some metaphysically real probability that team A will win, and some metaphysically real probability that team B will win. The bet with the best expected outcome is the one that matches those real probabilities.
While this definition of an “epistemically good prediction” is the most theoretically pure, and is a good ideal to strive for, it is usually impractical for actually evaluating predictions (thus Zvi’s post). Even after the fact, we often don’t have a good idea what the underlying “true probability” was. This is important to note, because it’s an easy mistake to make: what actually happened does not tell us the true probability. It’s useful information in that direction, but cannot be conclusive and often isn’t even that significant. It only feels conclusive sometimes because we tend to default to thinking about the world deterministically.
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Eliezer has an essay arguing that Probability is in the Mind. While in a literal sense I am contradicting that thesis, I don’t consider my argument here to be incompatible with what he’s written. Probability is in the mind, and that’s what is usually more useful to us. But unless you consider the world to be fully deterministic, probability must also be in the world – it’s just important to distinguish which one you’re talking about.