if I had a data point-set as $X$ consisting of $\left\{x_{i} : i=1,2,....,N\right\}$
Gaussian clusters centers $V$ consisting of points $\left\{v_{a} : a=1,2,.....,K\right\}$
and their covariance matrices $\left\{\Sigma_a : a=1,2,.....mk\right\}$
the probability of the point $x_{i}$ given cluster $v_{a}$ is $p(x_{i}|v_{a})=$gaussian_distribution with ($v_a$-$x_i$) as a mean and $\Sigma_a $ as a covariance matrix
and probability of cluster $v_{a}$ given point $x_{i}$ is $p(v_{a}|x_{i})=\pi_{a,i}$
the probabilistic distribution model of all the data set X is $$ p(X,V)=\prod_{i=1}^N \sum_{a=1}^{K} \pi_{a,i}p(x_{i}|v_{a},\Sigma_{a}) $$ if I understand correctly then $$ p(x_{i}|V)=\sum_{a=1}^{K}\pi_{a,i}p\left(x_{i}|v_{a},\Sigma_{a}\right) $$ and $$ p(X|v_{a})=\prod_{i=1}^N \pi_{a,i}p(x_{i}|v_{a},\Sigma_{a}) $$ then why the sets probability for example $$ p(a|\{b,c,d\})=p(a|b)+p(a|c)+p(a|d) $$ and $$ p(\{a,b,c\}|d)=p(a|b)*p(a|c)*p(a|d) $$
1- $p(a,b,c)$ means $p(a\cap b\cap c) $ so $$ p(a|\{b,c,d\})=p(a|b\cap c\cap d)=p(a|b)+p(a|c)+p(a|d) $$
2- i didn't notice that the points in data points set are independent so $$ p(\{a,b,c\}|d)=p(a|d)*p(b|d)*p(c|d) $$