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Is it possible to learn a distribution over the parameters ($K=\Sigma^{-1}$ and $\mu$) of a Gaussian from noisy measurements of $X$? (Starting with some appropriate prior over the parameters)

I know the case where the measurements are perfect is established theory and can be found in many text books, but I cannot find a solution for this variation of the problem anywhere.

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If I understand your question, $X$ has a Gaussian distribution and that there is added noise independent from the data, so actually your observation is $X=Z+e$ where $e$ is independent and identically distributed Gaussian noise of variance $\sigma^2$ and $Z$ is your Gaussian data. In that case, the matrix that you can estimate is ${\boldsymbol\Sigma}={\boldsymbol\Sigma}_x+\sigma^2{\bf I}$, while your mean remains the same, assuming that the noise has zero mean. Basically, the optimal estimator in your case are the expectations ${\boldsymbol \mu}=\mathbb{E}(x)\approx \frac{1}{N}\sum_{n=1}^N x$ and ${\hat{\boldsymbol \Sigma}}=\mathbb{E}(xx^{\top})\approx \frac{1}{N}\sum_{n=1}^N x_nx_n^{\top}$. Nevertheless, you won't be able to estimate $\boldsymbol \Sigma_x$ and $\sigma^2$ alone.

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