Estimate gaussian (mixture) density from a set of weighted samples - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:41:23Z http://mathoverflow.net/feeds/question/19011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19011/estimate-gaussian-mixture-density-from-a-set-of-weighted-samples Estimate gaussian (mixture) density from a set of weighted samples chrivo 2010-03-22T13:43:39Z 2010-04-07T01:25:40Z <p>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?</p> <p>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?</p> <p>Thanks!</p> http://mathoverflow.net/questions/19011/estimate-gaussian-mixture-density-from-a-set-of-weighted-samples/20579#20579 Answer by Neil for Estimate gaussian (mixture) density from a set of weighted samples Neil 2010-04-07T01:25:40Z 2010-04-07T01:25:40Z <p>The usual EM algorithm can be modified for weighted inputs. Following along the <a href="http://en.wikipedia.org/wiki/Mixture_model#Expectation_maximization_.28EM.29" rel="nofollow">Wikipedia presentation</a>, you would use these formulas instead:</p> <p>$a_i = \frac{\sum_{j=1}^N w_j y_{i,j}}{\sum_{j=1}^{N}w_j}$</p> <p>and</p> <p>$\mu_{i} = \frac{\sum_{j} w_jy_{i,j}x_{j}}{\sum_{j} w_jy_{i,j}}$</p> <p>where $w_j \ge 0$ are the weights of the data points.</p>