Estimate gaussian (mixture) density from a set of weighted samples

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. Does anyone know how to modify the algorithm to account for the weights?

If not, can some one give me a hint on how to incorporate the weights in the initial formula of the maximum-log-likelihood formulation of the problem?

Thanks!

Answer 1

The usual EM algorithm can be modified for weighted inputs. Following along the Wikipedia presentation, you would use these formulas instead:

$a_i = \frac{\sum_{j=1}^N w_j y_{i,j}}{\sum_{j=1}^{N}w_j}$

and

$\mu_{i} = \frac{\sum_{j} w_jy_{i,j}x_{j}}{\sum_{j} w_jy_{i,j}}$

where $w_j \ge 0$ are the weights of the data points.