Specifying a beta prior distribution

Question

I need help trying to understand how values can be used in a beta distribution to represent priors/posterior probabilities. I'm aware that there are similar questions that have already been asked about beta distribution, but they don't really ask what I'm having trouble with.

In the following example we have:

p = a prior

D: Some data, with 16/20 success

and two parameters , (1.1,1.3).

I know that using R we can plot the data:

p <- seq(from = 0, to = 1 , len = 10)
plot(p,dbeta(p,1.1,1.3)

What I'm having trouble understanding is how the parameters relate to the rest of the information?

Do the parameters represent the prior probabilities, and if so, in order to find the posterior probability, could i just use the Bayes theorem?

Would I need to make an opinionated guess for the priors in order to find the posterior probabilities, or is that what the two parameters represent in this case?

Answer 1

The beta distribution has pdf $$p(\theta \,|\, a, b) \propto \theta^{(a-1)} (1 - \theta)^{(b - 1)}$$

The beta distribution can be used to describe prior beliefs about a proportion, $\theta$.

You can think of $a$ (called shape1 in dbeta()) and $b$ (called shape2 in dbeta()) in the prior as if they were previously observed data, in which there were $a$ successes and $b$ failures in a total of $a + b$ experiments.

For example, a $\text{Beta}(16, 4)$ prior corresponds to having 16 successes and 4 failures in previously observed data. The $\text{Beta}(16, 4)$ distribution is depicted below. The $\text{Beta}(16, 4)$ distribution is peaked at $0.8$ ($=16 / 20$), but values between $0.6$ and $1$ are also probable.