We start from the following
formula
P(\theta|k \;views, n \;visitors)=\left(\frac{n}{k}\right)
\frac{1}{P(Evidence)B(\alpha,\beta)}
\theta^{k+\alpha-1}(1-\theta)^{n-k+\beta-1}
where
n=794, k=12, \alpha=1.1, \beta=30
The probability of evidence for k successes in n trials would be
(see definition of probability for
Binomial distribution and
Binomial interference)
P(Evidence)=P(E)=\int^1_0p^k(1-p)^{n-k}dp
The definition of
Beta function is
B(\alpha,\beta)=\int^1_0 t^{\alpha-1}(1-t)^{\beta-1}dt
therefore we have the following statement
\left(\frac{n}{k}\right)\frac{1}{P(E)B(\alpha,\beta)}=
\frac{1}{B(\alpha,\beta)\int^1_0p^k(1-p)^{n-k}dp} = \frac{1}{C}
in other words our coefficient C is defined as
\begin{array}{lcl}
C &=& B(\alpha,\beta)\int^1_0p^k(1-p)^{n-k}dp \\
&=& \int^1_0p^k(1-p)^{n-k}dp\int^1_0t^{\alpha-1}(1-t)^{\beta-1}dt \\
&=& \int^1_0\int^1_0 p^k(1-p)^{n-k} t^{\alpha-1}(1-t)^{\beta-1}dpdt \\
&=& \int^1_0\int^1_0 t^k(1-t)^{n-k} t^{\alpha-1}(1-t)^{\beta-1}dt^2 \\
&=& \int^1_0\int^1_0 t^{k+\alpha-1}(1-t)^{n-k\beta-1}dt^2 \\
&=& \int^1_0 B(k+\alpha,n-k+\beta) dt \\
&=& B(k+\alpha,n-k+\beta) \int^1_0 dt \\
&=& B(k+\alpha,n-k+\beta)
\end{array}