bilibili

Suppose that I have a coin with probability of heads p. I certainly know that p is fixed and does not change as I toss the coin. I would like to express my degree of belief in p and then update it as I toss the coin. Using a constant pdf to model my initial belief, the problem becomes a classic one and it turns out that my belief in p should be expressed with the pdf f(x)=(nh)xh(1−x)n−h after observing h heads out of n tosses. That's fine. But let's say I'm a super-skeptic guy that avoids accepting any statement with certainty, and I am aware of the issue of parametrization dependence too. So I dislike this solution and instead choose to attach beliefs to statements of the form S(f)= "my initial degree of belief is represented with probability density function f." Well this is not quite possible since the set of all such f is uncountable. However something similar to the probability density trick we use for continuous variables should do the job here as well. After observing some heads and tails, each initial belief function will be updated just as we did before, which will create a new uneven "density" distribution over S(f). When I want to express my belief that p is in between numbers a and b, now I have a probability density function instead of a definite number, which is a collection of all definite numbers from each (updated) prior. Now I can use the mean of this function to express my guess and I can even be skeptic about my own belief! This first meta level is still somewhat manageable, as I computed the Var(μ) = 1/12 for the initial uniform density over S(f) where μ is the mean of a particular f. I am not sure whether my approach is correct, though. Since the domain of each f is finite, I discretize this domain and represent the uniform density over S(f) as a finite collection of continuous random variables whose joint density is constant. Then taking the limit to infinity. The whole thing may not make sense at all. I'm just curious what would happen