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Is there an estimator to predict the variance matrix of a 2D distribution given the value of its density sampled on a regular finite grid? I'm not even sure that estimator is the right word to use here.

The idea is that you can take an image where the intensity of each pixel corresponds to the density of some underlying distribution which is close to Gaussian (but not necessarily Gaussian). Is there a statistically sound way to get the variance matrix of the underlying distribution from that data and a measure of the uncertainty of the variance matrix?

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