Using "prior intervals" instead of "prior distributions:" how should I update my beliefs?

The further I read Bayesian books, the clearer it becomes that traditional Bayesian inference has focused on very narrow problems: it requires users to completely specify a prior distribution over parameters.

Most of the time, however, users have only a coarse knowledge of reality. For instance, $$\theta \in [a, b]$$.

If my prior information is simply that $$\theta \in [a, b]$$, and I have observed some data $$X\sim P(X|\theta)$$, what is the rational way to update my beliefs about $$\theta$$?

• Is that not just a uniform distribution on the interval?
– Dave
Mar 19, 2021 at 17:39
• @Dave no, I don't have any knowledge of the relative probability of $\theta$ inside the interval. And as shown here (stats.stackexchange.com/questions/514557/…) lack of knowledge is not the same as a uniform distribution.
– user314217
Mar 19, 2021 at 17:40
• Are you sure you don't have any beliefs about $\theta$ beyond "$\theta \in [a, b]$"? Imagine you were forced to bet one dollar either for or against the proposition that $\theta \in [a + \frac{13(b-a)}{100}, a + \frac{14(b-a)}{100}]$. If you picked "against", that would suggest that you assign prior probability < 0.5 to the propositon. If you picked "for", then you probably lost a dollar, because most of the time, when you know a value is in a particular interval, it isn't between 13% and 14% along the interval. Mar 19, 2021 at 21:11
• @fblundun ok, say you give me this bet. Then I say I'm indifferent since you are forcing me. Then you give me another bet which "logically" I should take, given that I was indifferent to to that previous bet. But I say again I'm indifferent since you are forcing me. There will probably be no probability function that explain my bets.
– user314217
Mar 19, 2021 at 21:51
• @bayesian_newbie by "indifferent" do you mean you pick between "for" and "against" at random? In that case my point is that being indifferent to that proposition isn't a valid position, it's just a mistake on your end which will probably cost you if you randomly bet "for" rather than making the correct choice of betting "against". Mar 19, 2021 at 22:41

If you specify $$\theta \in [a,b]$$ then under a BL framework you might say $$E(\theta) = m = 0.5(a+b)$$ (the mid point). You would also then need to specify the variance. You might say that $$m + k \times var(\theta) = b$$. A sensible choice would be $$k=3$$ by Pukelshiems $$3\sigma$$ rule.
You can then perform BL updates in light of data which give you an updated mean and variance of $$\theta$$. This isn't the same as a posterior distribution but is a pair of posterior moments.