How do I choose parameters for my beta prior? Suppose today I'm going to flip a coin.  I believe that 9 of 10 flips will come up heads.  I flip the coin and 8 of 10 are heads.  Is my distribution of belief


*

*beta(9+8, 1+2)

*beta(1+9+8, 1+1+2)

*beta(m+8, n+2) where I choose m and n based on graphing beta(m,n) and deciding if it feels right.  (This is not a facetious answer, it's a real suggestion.)

*Hidden option 4.


If I use #1, I believe that is an improper prior, but Wikipedia claims some statisticians use them.
This choice is not obvious to me.
 A: If your prior belief is that 9 of the 10 coin flips will come up heads, then you want the expectation of your prior to be 0.9. Given $X \sim \mathrm{Beta}(\alpha,\beta)$ (for conjugacy in the beta-binomial model), then $E[X] = \alpha/(\alpha+\beta) = 0.9$, so you can use this as your first constraint. Obviously this leaves you with an infinite number of possibilities, so we'll need a second constraint which will represent your subjective confidence in this prior belief: the variance of your prior, $\operatorname{var}[X] = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\ $ 
Decide how you want to set your variance and solve the system of equations for $\alpha$ and $\beta$ to define the parameters for your prior. Justifying your choice of variance here may be difficult: you can always err on the side of a wider (i.e. less informative) variance. The wider you set the variance, the closer your prior will approximate a uniform distribution.
If you want a truly uninformative prior, you should consider using Jeffrey's Prior.
