I've been having trouble understanding how to simplify (as well as understand) the equation for what I'm calling the "Distribution Weight" of a Conditional Mixture of Gaussians distribution. Namely, this vector represents the proportion of data points that lie closer to one Gaussian than another.
My primary trouble is understanding how to solve for this variable given a dataset.
The equation is:
$$p(\mathbf x_A|\mathbf x_B)=\sum_{k=1}^{K}\pi'_kp_k(\mathbf x_A|\mathbf x_B),\quad\pi'_k=\dfrac{\pi_k\mathcal N\left(\mathbf x_B|\boldsymbol \mu_{kB},\boldsymbol{\mathbf\sum}_{kBB}\right)}{\boldsymbol\sum_k\mathcal N\left(\mathbf x_B|\boldsymbol\mu_{kB},\boldsymbol{\sum}_{kBB}\right)}$$
I am trying to solve for pi_prime_k . I have the given marginal dataset, and pi_k. How do you actually compute this?