You probably already like imprecise probabilities
"The chance of rain is 0.50496847", said no one
The idea that our beliefs ought to be imprecise or indeterminate is potentially a big crux for how we should make altruistic decisions. Many people seem to deny that imprecision in our credences is ever rationally permissible, or at least required. But I think that, with a bit of reflection, most people would agree that it would be unreasonable to have arbitrarily precise beliefs about many real-world propositions. This is important because it means that all of the in-principle arguments against imprecision can be set aside, and all there is to argue about is how imprecise our beliefs ought to be in a specific circumstance. In particular, it opens the door to severe imprecision in beliefs about the effects of our actions on cosmos-wide welfare.
You ask your friend: “What do you think’s the chance that it will rain tomorrow?”
Your friend thinks for a moment and says: “That’d be 0.50496847.”
I’d guess that most readers will think that their friend has said something silly. How could they possibly have a probability to eight digits? It doesn’t seem like we should have precision greater than, say, one in a thousand at the very most. Specifying decimals beyond that is totally arbitrary.
Here are some arguments your friend might give in defence of all those decimals:
“People who guess more decimals perform better on average than people who don’t.”
“I’m throwing away information if I don’t use all the decimals that occur to me.”
“If your credences aren’t precise, you’re liable to be money-pumped.”
“If you don’t have arbitrarily-precise beliefs, you end up a radical skeptic / everything becomes permissible.”
“Not guessing more decimals is just giving up!”
Are you persuaded?
Aren’t precise people going to say that you should have precise credences the same way that you should have probabilities over all hypotheses and update instantaneously over evidence (as a Bayesian)? As in, the normative claim isn’t a claim about what should be done (insofar as what should be done could be done feasibly), but it is merely to say that this is how ideal reasoners behave and that we should approximate it.
Saying that this is a harsh standard, then, says very little about whether we could say that ideal reasoners would do it, and therefore we should approximate it, I think.
I think one critique here that does make sense is just to say that there is no actual evidence that can lead you, in principle, to precise probabilities for a rational agent. While I probably disagree with this (I think the principle of indifference and all your subtle priors over conditional probabilities will do a bunch of work here), I think this kind of critique is reasonable.
Thanks for the interesting post, Jesse.
A probability that it will rain tomorrow of 0.50496847 is almost exactly as accurate as a probability of 0.5 (it is slighly more or less accurate), and they are technically equally precise since they are both point estimates. I agree 0.5 feels reasonable, and 0.50496847 sounds silly, but I do not think this implies precise probabilities are fundamentally flawed.
0.50496847 implicitly communicates that having so many digits is relevant, whereas it is not. It would be hard to come up with real decisions where it matters whether the probability of rain is 0.5 or 0.50496847.
In addition, 0.50496847 implicitly conveys that the distribution describing the probability, i.e. its probability density function (PDF) (https://en.wikipedia.org/wiki/Probability_density_function), is very narrow, whereas this is not reasonable. In other words, 0.50496847 communicates an unreasonably high credal resilience (https://forum.effectivealtruism.org/topics/credal-resilience). For example, 0.5 could be interpreted as the mean of a distribution with 25th and 75th percentile of 0.4 and 0.6, whereas 0.50496847 could be read as the mean of a distribution with 25th and 75th percentile of 0.50496846 and 0.50496848. The 1st distribution is way more reasonable than the 2nd.