Conditional mean for mixture of multivariate normal distributions

Question

If x = (x_1,x_2,...,x_n) is a vector whose components have a distribution that is a finite mixture of multivariate normals, is the expected value of x_1 still a linear function of the other components, as is the case when the components have a multivariate normal distribution?

Answer 1

If$$(x_1,\ldots,x_p) \sim \sum_{i=1}^k \pi_i {\cal N}_p(\mu_i,\Sigma_i)$$then$$f(x_1,\ldots,x_p)\propto\sum_{i=1}^k \pi_i \exp\{-(x-\mu_i)^\text{T}\Sigma_i^{-1}(x-\mu_i)/2\}\big/|\Sigma_i|^{1/2}$$and $$f(x_1|x_2,\ldots,x_p)\propto \sum_{i=1}^k \pi_i \varphi(x_{-1};\mu_i,\Sigma_i)\exp\{-(x_1-\mathbb{E}[X_1|x_{-1},\mu_i,\Sigma_i])^2/2\sigma_{i1}^2\}\big/|\Sigma_i|^{1/2}$$ meaning that the posterior expectation of $X_1$ given $X_{-1}$ is a non linear combination of the componentwise conditional expectations.