Let's say I have 1 success in 4 bernoulli trials, and I wish to plot the distribution of the parameter $p$ of the corresponding binomial distribution. I'm using R.
The probability of seeing 1 sucess and 3 failures in 4 tests for $p=0.25$ is, for these parameters:
> n <- 4 > p <- 0.25 > dbinom(1, n, p)  0.421875
To get the distribution for the parameter, I use a beta distribution $Beta(k+1, n-k+1)$. But when I try to calculate the value for $p=0.25$, I get a different result:
> k <- 1 > dbeta(p, k+1, n-k+1) >  2.109375
I tried to divide that by the function beta $B(k+1, n-k+1)$, but that did not work either:
> dbeta(p, k+1, n-k+1)/beta(k+1, n-k+1) >  42.1875
That's exactly 100 times the result I expected. I am certainly missing something - maybe something basic, I'm no expert.
Another thing that intrigues me:
dbeta region sums to 1 without dividing by the beta function.