I want to evaluate this multiple integral: $$ \iiint\limits_{ \sum_{i=1}^4 x_i=1,\ \ \ x_1,\, x_2,\,x_3,\, x_4\, \ge \,0 } (x_1+x_2)^{N1} (x_3+x_4)^{N2} (x_1+x_3)^{N3} (x_2+x_4)^{N4} \, dX$$
For context, this problem comes up when computing the marginal likelihood of data consisting of two dependent binary variables $(a,b)$. The conditional probability distribution for $B$ given $A$ is: \begin{align} P(B=0\mid A=0)=x_1+x_2; & & P(B=1\mid A=0)=x_3+x_4 \\ P(B=0\mid A=1)=x_1+x_3; & & P(B=0\mid A=1)=x_2+x_4 \end{align}
The marginal likelihood turns into this integral, where $N_i$ are the counts of $B\mid A$ in the data.
This looks like a dirichlet integral of type 1, but the problem is that I am unable to separate out the integral into integrals over fewer variables.
I have tried:
- Use dirac delta function to remove the simplex and then use laplace transform to find a closed form solution. However, it does not work here because the integrals are not separable.
- Using the variable substitution for beta variables as in this answer.
- Interpret the equation as product of two Beta functions: $B(N_1 +1, N_2 +1), B(N_3 +1, N_4 +1)$.
However, I am stuck because $x_i$ are shared across the terms.
Any ideas?