A shortcut to get the result from integration of

? ? (q^c) * ((1-q)^d)

over the domain of q from 0 to 1 is simply computing the value of

? ? c! / ((d+1)*(d+2)*...*(d+c+1)),

where c and d are constants in both formulas.

This is useful when a Bernoulli parameter q has a Beta prior distribution, and we need to find its posterior given observed Bernoulli trials. The posterior distribution of q is in the following form:

? ? I * (q^c) * ((1-q)^d),

where I is the value computed from the above integration.

An even simpler but less intuitive way to get the posterior distribution is to recognize that the posterior is also a Beta distribution. Assume the prior Beta distribution has parameters a and b, that is to say

? ? q ~ Beta(a, b),

and we observe n Bernoulli trials with x successes. The parameters of the posterior distribution, denoted by a_new and b_new can be obtained from the following formulas:

? ? a_new = a + x

? ? b_new = b + n

Therefore, q has a posterior distribution of

? ? Beta(a_new, b_new),

and its mean is given by

? a_new / (a_new + b_new).