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If the variance of a random variable is proportional to its mean, then what is the best way of making a mixture distribution that will faithfully reconstruct a data set coming from a mixture model.

For e.g. say I have a data set containing say N values of (x_i,sigma_i), and say x's are clustered at a few values (one can plot a histogram of x to actually see this). Now if I want an approximate probability distribution function in x, I can assume that each x_i comes from a Normal distribution N(x_i,sigma_i), and then add up all the Gaussians, hence getting a Gaussian mixture model with equal weights. But the downside is that because the variance is proportional to the mean, all the samples with higher x values will have a lower contribution to the sum, and the result may become biased and not reconstruct the distribution of the data faithfully. For e.g it may not show peaks because of clustering at higher x values.

how can one get around this problem? is there a systematic way to do this..

thanks

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  • $\begingroup$ Do the underlying Gaussians have to have equal variances? $\endgroup$ – gung Mar 22 '16 at 12:21
  • $\begingroup$ well they don't have equal variances. The std deviations are sigma_i's. So if there are N such Gaussians there are N different sigma_i's. $\endgroup$ – gmm Mar 23 '16 at 6:07
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Have a look at Extreme Deconvolution by Bovy et al. (http://dx.doi.org/10.1214/10-AOAS439). It allows you to fit a GMM where each datum can have an arbitrary covariance matrix. The connection between mean and variance you have in your data is a special case of that situation.

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