Question: Assume I have a set of weighted samples, where each samples has a corresponding weight between 0 and 1. I'd like to estimate the parameters of a gaussian mixture distribution that is biased towards the samples with higher weight. In the usual non-weighted case gaussian mixture estimation is done via the EM algorithm.

Is there an implementation (any language is OK) that permits passing weights? If not, how can I modify the algorithm to account for the weights? If not, how to incorporate the weights in the initial formula of the maximum-log-likelihood formulation of the problem?

Answer: 
You can calculate a weighted log-Likelihood function; just multiply the every point with it's weight. Note that you need to use the log-Likelihood function for this.

So your problem reduces to minimizing $-\ln L = \sum\_i w\_i \ln f(x\_i|q)$ (see [the Wikipedia article](http://en.wikipedia.org/wiki/Maximum_likelihood#Principles) for the original form).
I was looking for a similar solution related to gaussian kernel estimation (instead of a gaussian mixture) of the distribution.

The standard [gaussian\_kde](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gaussian_kde.html#scipy.stats.gaussian_kde) does not allow that but I found a python implementation of a modified version here
<http://mail.scipy.org/pipermail/scipy-user/2013-May/034580.html>