Converting Prior Knowledge into a Bayesian Prior Suppose I flip a coin 10 times and get 7 heads - based on a Binomial Distribution, I can create a Likelihood Function and estimate the probability of getting a head is 0.7.
Now, suppose someone tells me that they have very strong evidence to believe that the probability of getting a head is 0.9. Suppose I decide to use a Beta Distribution(alpha, beta) as a prior.
My Question: I know that the Expected Value of the Beta Distribution is 'alpha/(alpha + beta)'. If I want to use the Prior Information that was provided to me - does this mean that I have choose values of 'alpha and beta' such that 'alpha/(alpha+beta)' = 0.9? For example - based on the n = 10 samples I observed, in this case, would alpha = 9 and beta = 1?

If I understand correctly, your problem is not defined correctly:

*

*The likelihood is a Binomial distribution (you flip the coin)

*The Prior is a Beta distribution (representing the distribution of
the parameter p of the Binomial)

*Beta is a conjugate prior for Binomial, it means the posterior is still a Beta distribution.

Here you can find the following details:

Therefore, in order to identify the "believe", that is a Beta distributions, you need 2 parameters:

*

*alpha & beta (your problem is solved)

OR

*

*mean & variance, in order to solve this equation:

$$\mathbf E[X]=\frac\alpha{\alpha+\beta}$$
$$\mathbf V[X]= \frac{\alpha\beta}{(\alpha+\beta) ^2(\alpha+\beta+1) }$$
as reported here: Calculating the parameters of a Beta distribution using the mean and variance.
Therefore: Variance is missing.
