Sometimes I see people give subjective probability estimates in ranges. (eg 30-50%). My understanding is that this is intuitively plausible but formally wrong. Eg if you have X% credence in a theory that produces 30% and Y% credence in a theory that produces 50%, then your actual probability is just a weighted sum. Having a range of subjective probabilities does not make sense!
My friend disagreed, and said that there is formal justification for giving imprecise probabilities.
I don’t think I understand the imprecise probability literature enough to steelman the alternatives. Can someone who understand Bayesian epistemology better than me explain why the alternatives are interesting, and there’s an important sense, formally, for giving grounding to having ranges of subjective probability estimates?
When I do, I usually take it to mean something like ‘this is my 80% credible interval for what credence I’d end up with if I thought about this for longer [or sometimes: from an idealized epistemic position]’.
I overall think this can be useful information in some contexts, in particular when it’s roughly clear or doesn’t matter too much how much additional thinking time we’re talking about, exactly what credible interval it’s supposed to be etc.
In particular, I don’t think it requires any commitment to imprecise probabilities.
I do, however, agree that it can cause confusion particularly in situations where it hasn’t been established among speakers what expressions like ’30-50%′ are supposed to mean.
Hmm do you have a sense of what theoretical commitments you are making by allowing for a credible interval for probabilities?
A plausible candidate for a low-commitment solution is that idealized Bayesian agents don’t have logical uncertainty, but humans (or any agents implemented with bounded computation and memory) do.
An alternative framing I sometimes have is that I’ll have a “fair price” for my true probabilities, but for questions I’m more/less confused about, I’ll have higher/lower bands for my bid/asks in bets against general epistemic peers. I think this is justifiable against adversarial actors due to some analogy to the winner’s curse, tho I think my current intuitions are still not formal enough for me to be happy with.
My immediate response is that I’m making very few theoretical commitments (at least above the commitments I’m already making by using credences in the first place), though I haven’t thought about this a lot.
Note in particular that e.g. saying ’30-50%′ on my interpretation is perfectly consistent with having a sharp credence (say, 37.123123976%) at the same time.
It is also consistent with representing only garden-variety empirical uncertainty: essentially making a prediction of how much additional empirical evidence I would acquire within a certain amount of time, and how much that evidence would update my credence. So no commitment to logical uncertainty required.
Admittedly in practice I do think I’d often find the sharp credence hard to access and the credible interval would represent some mix of empirical and logical uncertainty (or similar). But at least in principle one could try to explain this in a similar way how one explains other human deviations from idealized models of rationality, i.e. in particular without making additional commitments about the theory of idealized rationality.
The discussion here might be related, and specifically this paper that was shared. However, you can use a credible interval without any theoretical commitments, only practical ones. From this post:
Give an expected error/CI relative to some better estimator—either a counterpart of yours (“I think there’s a 12% chance of a famine in South Sudan this year, but if I spent another 5 hours on this I’d expect to move by 6%”); or a hypothetical one (“12%, but my 95% CI for what a superforecaster median would be is [0%-45%]”). This works better when one does not expect to get access to the ‘true value’ (“What was the ‘right’ ex ante probability Trump wins the 2016 election?”)
This way, you can say that your probabilities are actually sharp at any moment, but more or less prone to change given new information.
That being said, I think people are doing something unjustified by having precise probabilities (“Why not 1% higher or lower?”), and I endorse something that looks like the maximality rule in Maximal Cluelessness for decision theory, although I think we need to aim for more structure somehow, since as discussed in the paper, it makes cluelessness really bad. I discuss this a little in this post (in the summary), and in this thread. This is related to ambiguity aversion and deep uncertainty.
I don’t tend to express things like that, but when I see it I tend to interpret it as “if I thought about this for a while, I expect the probability I’d end up with would be in this range, with moderate confidence”.
I don’t actually know how often this is a correct interpretation.
if you have X% credence in a theory that produces 30% and Y% credence in a theory that produces 50%, then your actual probability is just a weighted sum. Having a range of subjective probabilities does not make sense!
Couldn’t those people just not be able to sum/integrate over those ranges (yet)? I think about it like this: for very routine cognitive tasks, like categorization, there might be some rather precise representation of p(dog|data) in our brains. This information is useful, but we are not trained in consciously putting it into precise buckets, so it’s like we look at our internal p(dog|data)=70%, but we are using a really unclear lense so we can‘t say more than “something in the range of 60-80%”. With more training in probabilistic reasoning, we get better lenses and end up being Superforecasters that can reliably see 1% differences.
I mentioned this deeper in this thread, but I think precise probabilities are epistemically unjustifiable. Why not 1% higher or 1% lower? If you can’t answer that question, then you’re kind of pulling numbers out of your ass. In general, at some point, you have to make a 100% commitment to a given model (even a complex one with submodels) to have sharpe probabilities, and then there’s a burden of proof to justify exactly that model.
Eg if you have X% credence in a theory that produces 30% and Y% credence in a theory that produces 50%, then your actual probability is just a weighted sum.
Then you have to justify X% and Y% exactly, which seems impossible; you need to go further up the chain until you hit an unjustified commitment, or until you hit a universal prior, and there are actually multiple possible universal priors and no way to justify the choice of one specific one. If you try all universal priors from a justified set of them, you’ll get ranges of probabilities.
(This isn’t based on my own reading of the literature; I’m not that familiar with it, so maybe this is wrong.)
I do think everything eventually starts from your ass. Often you make some assumptions, collect evidence (and iterate between these first two) and then apply a model, so the numbers don’t directly come from your ass.
If I said that the probability of human extinction in the next 10 seconds was 50% based on a uniform prior, you would have a sense that this is worse than a number you could come up with based on assumptions and observations, and it feels like it came more directly from the ass. (And it would be extremely suspicious, since you could ask the same for 5 seconds, 20 seconds, and a million years. Why did 10 seconds get the uniform prior?)
I’d rather my choices of actions be in some sense robust to assumptions (and priors, e.g. the reference class problem) that I feel are most unjustified, e.g. using a sensitivity analysis, as I’m often not willing to commit to putting a prior over those assumptions, precisely because it’s way too arbitrary and unjustified. I might be willing to put ranges of probabilities. I’m not sure there’s been a satisfactory formal characterization of robustness, though. (This is basically cluster thinking.)
Each time you make an assumption, you’re pulling something out of your ass, but if you check competing assumptions, that’s less arbitrary to me.
Tangentially related:
Sometimes I see people give subjective probability estimates in ranges. (eg 30-50%). My understanding is that this is intuitively plausible but formally wrong. Eg if you have X% credence in a theory that produces 30% and Y% credence in a theory that produces 50%, then your actual probability is just a weighted sum. Having a range of subjective probabilities does not make sense!
My friend disagreed, and said that there is formal justification for giving imprecise probabilities.
I don’t think I understand the imprecise probability literature enough to steelman the alternatives. Can someone who understand Bayesian epistemology better than me explain why the alternatives are interesting, and there’s an important sense, formally, for giving grounding to having ranges of subjective probability estimates?
FWIW:
I think I do sometimes give ranges like ’30-50%’.
When I do, I usually take it to mean something like ‘this is my 80% credible interval for what credence I’d end up with if I thought about this for longer [or sometimes: from an idealized epistemic position]’.
I overall think this can be useful information in some contexts, in particular when it’s roughly clear or doesn’t matter too much how much additional thinking time we’re talking about, exactly what credible interval it’s supposed to be etc.
In particular, I don’t think it requires any commitment to imprecise probabilities.
I do, however, agree that it can cause confusion particularly in situations where it hasn’t been established among speakers what expressions like ’30-50%′ are supposed to mean.
Hmm do you have a sense of what theoretical commitments you are making by allowing for a credible interval for probabilities?
A plausible candidate for a low-commitment solution is that idealized Bayesian agents don’t have logical uncertainty, but humans (or any agents implemented with bounded computation and memory) do.
An alternative framing I sometimes have is that I’ll have a “fair price” for my true probabilities, but for questions I’m more/less confused about, I’ll have higher/lower bands for my bid/asks in bets against general epistemic peers. I think this is justifiable against adversarial actors due to some analogy to the winner’s curse, tho I think my current intuitions are still not formal enough for me to be happy with.
My immediate response is that I’m making very few theoretical commitments (at least above the commitments I’m already making by using credences in the first place), though I haven’t thought about this a lot.
Note in particular that e.g. saying ’30-50%′ on my interpretation is perfectly consistent with having a sharp credence (say, 37.123123976%) at the same time.
It is also consistent with representing only garden-variety empirical uncertainty: essentially making a prediction of how much additional empirical evidence I would acquire within a certain amount of time, and how much that evidence would update my credence. So no commitment to logical uncertainty required.
Admittedly in practice I do think I’d often find the sharp credence hard to access and the credible interval would represent some mix of empirical and logical uncertainty (or similar). But at least in principle one could try to explain this in a similar way how one explains other human deviations from idealized models of rationality, i.e. in particular without making additional commitments about the theory of idealized rationality.
The discussion here might be related, and specifically this paper that was shared. However, you can use a credible interval without any theoretical commitments, only practical ones. From this post:
This way, you can say that your probabilities are actually sharp at any moment, but more or less prone to change given new information.
That being said, I think people are doing something unjustified by having precise probabilities (“Why not 1% higher or lower?”), and I endorse something that looks like the maximality rule in Maximal Cluelessness for decision theory, although I think we need to aim for more structure somehow, since as discussed in the paper, it makes cluelessness really bad. I discuss this a little in this post (in the summary), and in this thread. This is related to ambiguity aversion and deep uncertainty.
I don’t tend to express things like that, but when I see it I tend to interpret it as “if I thought about this for a while, I expect the probability I’d end up with would be in this range, with moderate confidence”.
I don’t actually know how often this is a correct interpretation.
Couldn’t those people just not be able to sum/integrate over those ranges (yet)? I think about it like this: for very routine cognitive tasks, like categorization, there might be some rather precise representation of p(dog|data) in our brains. This information is useful, but we are not trained in consciously putting it into precise buckets, so it’s like we look at our internal p(dog|data)=70%, but we are using a really unclear lense so we can‘t say more than “something in the range of 60-80%”. With more training in probabilistic reasoning, we get better lenses and end up being Superforecasters that can reliably see 1% differences.
I mentioned this deeper in this thread, but I think precise probabilities are epistemically unjustifiable. Why not 1% higher or 1% lower? If you can’t answer that question, then you’re kind of pulling numbers out of your ass. In general, at some point, you have to make a 100% commitment to a given model (even a complex one with submodels) to have sharpe probabilities, and then there’s a burden of proof to justify exactly that model.
Then you have to justify X% and Y% exactly, which seems impossible; you need to go further up the chain until you hit an unjustified commitment, or until you hit a universal prior, and there are actually multiple possible universal priors and no way to justify the choice of one specific one. If you try all universal priors from a justified set of them, you’ll get ranges of probabilities.
(This isn’t based on my own reading of the literature; I’m not that familiar with it, so maybe this is wrong.)
Wait what do you think probabilities are, if you’re not talking, ultimately, about numbers out of your ass?
I do think everything eventually starts from your ass. Often you make some assumptions, collect evidence (and iterate between these first two) and then apply a model, so the numbers don’t directly come from your ass.
If I said that the probability of human extinction in the next 10 seconds was 50% based on a uniform prior, you would have a sense that this is worse than a number you could come up with based on assumptions and observations, and it feels like it came more directly from the ass. (And it would be extremely suspicious, since you could ask the same for 5 seconds, 20 seconds, and a million years. Why did 10 seconds get the uniform prior?)
I’d rather my choices of actions be in some sense robust to assumptions (and priors, e.g. the reference class problem) that I feel are most unjustified, e.g. using a sensitivity analysis, as I’m often not willing to commit to putting a prior over those assumptions, precisely because it’s way too arbitrary and unjustified. I might be willing to put ranges of probabilities. I’m not sure there’s been a satisfactory formal characterization of robustness, though. (This is basically cluster thinking.)
Each time you make an assumption, you’re pulling something out of your ass, but if you check competing assumptions, that’s less arbitrary to me.