Bayesian point estimate of a random sample

Question

I am new to statistics and some concepts are not clear to me. I have a random sample that is distributed as a Binomial with parameters $k=2$ and $\theta$ unknow. Using a Bayesian approach I must give a point estimate for $\theta$.

I understand that being binomial, a priori I must use a beta distribution, and there is a relationship between the expectation and the variance with the beta distribution. I intend to obtain the expectation and variance of my initial data and with them obtain shape1, shape2, (I am working in R).

to get my a priori beta distribution, but shape1 and shape2 come out negative, which tells me that I am doing something wrong. Can someone guide me? Or if I'm doing it wrong, how should I proceed?

Answer 1

I intend to obtain the expectation and variance of my initial data and with them obtain shape1, shape2...

That is wrong, and contradicts the methodology of Bayesian analysis. If you use your data to determine the hyperparameters in your prior distribution then you are double-counting the data --- using it once to get the prior distribution and then using it again to update to get the posterior. The idea of Bayesian analysis is that you should formulate a prior belief that does not take account of your data. You then use Bayes rule to obtain a posterior belief that takes account of the data.

What you should do here is to choose a prior distribution (or a set of prior distributions if you want to do "imprecise" analysis) using some method that does not use your observed data. That may mean using some values to represent prior "ignorance" or it may mean using values that represent some a priori theoretical information about your problem.

Answer 2

It is called prior for a specific reason: