A probability range is useful for expressing unquantified model uncertainty. If there's some collection of salient models, you can sweep it for estimates of the target probability, and give the resulting range. Other people who share some of these models but not others will have an overlapping range. This doesn't mean that there isn't also a personal belief point estimate (and within-model point estimates for all the models), but a range has the nice property of abstracting out the weights of the models (you still need judgment calls on which models qualify). So a range can be better for developing and discussing the models than for decision making based on the probability estimates (where you do need to put weights on the models/hypotheses, to commit to particular beliefs and not merely in-principle understanding of what might make sense).
There is a valid probability-theory construction for reporting a range of probabilities as an answer: a credibility interval over a Beta distribution.
For example, if I am catching fish from a new lake that only has carp and bass in it, and I’ve caught 1 carp and 4 bass so far, the 95% interval for the proportion of carp in the lake is (0.04, 0.64). This is a probability range: if you ask me the probability that my next catch is a carp, I can answer with the mean of the distribution, but I can also answer with this interval. It has basis in theory and can be proved and derived and so on.
If I ask my friend if he‘s coming to my party this weekend, and he says “70, 80%”, maybe that’s a cop-out. But it’s also reasonable to consider it as an abbreviated form of “The 90% credibility interval over my party attendance on busy weekends like this one is 70%-80%”.
Said differently: asking someone to estimate a probability is asking for the mean of their Beta (or Dirichlet) distribution for the thing. But they're often not quite sure where that mean is, because it's hard to nail down, so instead they give you the credibility interval over it. This operation is probability-theory-valid.
So I think statements like
>every time you do update, you still end up with a precise value, not a range.
are too strong, because every point estimate of a probability is coming from an implicit Beta or Dirichlet distribution;
that distribution, not just a single point estimate, is what’s updating during an update;
and every such distribution that can output point probability estimates can also output credibility interval ranges over probabilities.
Credibility intervals make sense when you are describing an objective fact about reality, such as the proportion of carp or the frequency of being available for parties on weekends. Why would you use a credibility interval to describe your subjective probability, when you are necessarily perfectly informed to know your exact subjective probability?
The brute fact of the lake is something like “there are 203 carp and 781 bass, for a ratio of 203:781”. My estimates after catching a few fish are my subjective estimate. Someone who had caught more fish there would have a different subjective estimate.
Both the examples I gave seem to me to be subjective: my friend doesn’t actually have an excel spreadsheet in his head with all the parties he’s ended up going to, that he can then crunch to get a perfectly-calculated answer. Rather, he generates a feeling about how often he is able to attend on weekends such as these, and then gives his best guess.
Can you say more about this difference you’re describing? Or describe an example that meets your criterion of perfectly-informed-subjectivity?
In the fishing example, you're describing what the ratio of carp is. You don't know the actual ratio with certainty, so you have the 95% interval, and you need that interval because there is an actual real-world ratio of carp in the lake. Meanwhile, the 95% itself is a precise probability that the actual ratio is in that range, as far as you know. The 95% isn't describing the lake, but your subjective knowledge (and lack thereof) about the lake, so you only need the info you have to give a precise and correct 95%. If you're estimating the probability of catching a carp, you would just use the ratio if you knew it, but even though they're the same number, they're not the same thing since the ratio describes the lake and the probability describes what you know. Instead, you only know the best estimate for the ratio, so you use that. It's not guaranteed to be an accurate value to describe the actual ratio, but it is tautologically guaranteed to be an accurate value to describe the best estimate of it that you know.
In the party example, it's kind of fine for the answer to be a cop-out because it's more important to give a quick answer than to spend too long thinking about it, and possibly nobody cares about the probability at any greater precision than what the 70–80% range already has. But even if it is based on just a feeling, he could spend some time considering that feeling in more detail and come up with a precise number.
It feels like this post on LessWrong should mostly be preaching to the choir, but I have seen enough rationalists post probability ranges (mainly elsewhere) that maybe it is worth writing.
The probability of an event is not an objective fact about the world; it is a fact about your subjective uncertainty of it.
Hence, if you're not sure what the probability of something is, this is not because the world has some secret information that you don't have access to (the fact that you're estimating the probability in the first place is because of that), but because you just can't be bothered to put together all the relevant info you have and properly estimate and/or calculate the probability based on it. In a sense, not knowing what you need to take into account or how to do the math are just factors that add uncertainty to your world model, not something that stops you from estimating precise probabilities. If you really don't have any info whatsoever, such as if you have to estimate the probability of an event that is described to you in a language you don't speak so you don't even know what you're estimating, the probability is simply 50%, and any additional info you gain lets you update from there. You are allowed (and required) to do these updates based on a gut feel if you don't know how to do them "properly", and every time you do update, you still end up with a precise value, not a range.
There are some cases where posting a probability range is sensible for the purposes of communication. For example, if your probability of something is partially based on vibes and the vibes change daily for no apparent reason, maybe it makes sense to give your p(the thing) as a range from the lowest to the highest probability you've recently had, or if p(the thing) depends highly on some other thing, you might want to give your p(the thing) as a range from p(the thing | the other thing) to p(the thing | !the other thing) or vice versa. In these cases, it is usually helpful to explain why you're giving multiple probabilities, what they're based on, and how you then arrive from them to your actual probability, if you have the opportunity to. You could also give a probability range, especially in a highly public situation, because you want to seem like a reasonable and sane person both to the people who agree with your high end and the people who agree with your low end and don't want to reveal to either of those groups what you actually think. However, I strongly suspect that usually when people do post probability ranges, it is not because they're being strategic about communication but because they're being lazy about probability. Don't be lazy about probability if it's the probability of something that matters!