## Choice of coefficients for a beta distribution (prior for binomial distribution)

The Beta distribution is the conjugate prior to the binomial test since the beta density function has a similar form to the binomial density. The Beta distribution, Be(a,b), has two parameters corresponding to shape (a) and scale (b).

These two values for the prior distribution, a and b, can be specified in JASP to specify the prior distribution. SPSS allows for separate values of a and b to be specified for the prior null and alternative distributions.

Jeffrey's prior of Be(0.5,0.5) is recommended as an uninformative prior for the binomial test.

Choice of informative Beta priors

The Beta expected value of a Be(a,b) distribution is a/(a+b). So that if we expect 9 tails in 10 coin tosses our expected value of the Beta distribution is 9/(9+1) = 0.9.

The variance of the beta distribution is ab/[(a+b)^{ 2} (a+b+1)] so that higher values of a and b are more informative priors having smaller variances.See here.