>>12386958>if you choose a uniform prior for a binomial distribution, you just another binomial probability density equation posterior. For the parameter p you'll get a beta distribution, no? It couldn't be a binomial, as p is a continuous variable over 0, 1.
Remember, the likelihood function isn't technically a density –it's not a function of your data. It's a function of your parameter, in this case p. So while the likelihood for the parameter p, given N trials and K successes is:
This is not actually a probability density over p. To see this, fix n = 2, k =1, and integrate p over its support (again, [0,1]). We get:
If this were a density, we would have to get one, right? So what is the actual distribution over p? Well, with a non-informative uniform prior over [0,1], applying the Bayes rule we get:
And this is just an alternate form of the beta distribution. It's just that alpha = k+1 and beta = n-k+1.
Now, if you want to know why the beta function is parameterized with alpha -1, beta -1, look up the thread on stack exchange "why is there -1 in beta distribution density function"
