Estimate weighted variances in mixture models Given a generic mixture model $X$ of $k$ components, with distribution 
$$
f(x)=∑_i\pi_if_i(x),
$$
It is easy to show that the $k-th$ moment is just the weighted mean of the $k-th$ moments of the mixture components
$$
E_f[x^k]=∑_i\pi_i E_{f_i}[x^k] 
$$
However I can not find any result to estimate the weighted mean of the variances of the mixture components
$$
?=∑_i\pi_i Var_{f_i}[x] 
$$
Is there any known estimator for such a value? 
Consider that the $\pi_i$ are unknown and the the $f_i$ come from the same distribution.
Should I do any hypothesis about the distribution of the $f_i$? I would like to work in the highest possible generality..
Appearently it looks very simple but I can't come out with a result.
 A: This answer by whuber is close to what you are seeking. The variance of a mixture is not the weighted average of the variance since there is also additional variability due to the component uncertainty. If you start from
$$\mathbb{E}[X^2]=\sum_{i=1}^k \pi_i \mathbb{E}_{f_i}[X^2]$$
you can see that
$$\mathbb{E}[X^2]=\sum_{i=1}^k \pi_i \{\text{var}_{f_i}(X)+\mathbb{E}_{f_i}[X]^2\}=\sum_{i=1}^k \pi_i\text{var}_{f_i}(X) + \sum_{i=1}^k \pi_i\mathbb{E}_{f_i}[X]^2$$
while
$$\mathbb{E}[X]^2=\left(\sum_{i=1}^k \pi_i \mathbb{E}_{f_i}[X]\right)^2=\sum_{i=1}^k \pi_i^2 \mathbb{E}_{f_i}[X]^2+\sum_{i\ne j}\pi_i\pi_j\mathbb{E}_{f_i}[X]\mathbb{E}_{f_j}[X]$$
There is thus no direct moment expression linking $\sum_{i=1}^k \pi_i \{\text{var}_{f_i}(X)$ with an expectation under $f$. The quantity is a conditional variance in that
$$\text{var}_f(X)=\mathbb{E}_\pi[\text{var}(X|Z)]+\text{var}_\pi(\mathbb{E}[X|Z])$$
where $Z$ denotes the auxiliary variable "component of $X$", i.e.
$$X|Z=z\sim f_z(x)$$
In this representation,
$$\text{var}(X|Z=z)=\text{var}_{f_z}(X)$$
and
$$\mathbb{E}_\pi[\text{var}(X|Z)]=\sum_{i=1}^k\pi_i \text{var}_{f_i}(X)$$
while
$$\mathbb{E}[X|Z=z]=\mathbb{E}_{f_z}[X]$$
and
$$\text{var}_\pi(\mathbb{E}[X|Z])=\sum_{i=1}^k \pi_i \mathbb{E}_{f_i}[X]^2 - \left\{ \sum_{i=1}^k \pi_i \mathbb{E}_{f_i}[X]\right\}^2\,.$$
This essentially rephrases the answer by whuber!
