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Let's say I generated samples from a distribution $f$ whose functional form I do not know. I want to approximate this distirbution with a multi-variate normal.

We know that if samples are generated using a MVN, then the MLE for mean snd varaince are simply the sample mean and sample co-variance.

My question is that, in my case where I want to approximate with a MVN, would I just use the sample mean and sample co-variance as paramters for the MVN ?

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  • $\begingroup$ How did you generate samples without knowing the distribution? $\endgroup$
    – Peter Flom
    Commented Aug 11, 2019 at 13:30
  • $\begingroup$ Its the posterior distribution generated using MCMC for a Bayesian analysis. So I should say that it doesn't have a standard form. $\endgroup$
    – asifzuba
    Commented Aug 11, 2019 at 14:17
  • $\begingroup$ Do you have any reason to suppose that the posterior is at all close to MVN? If not it might be harder to do better than Laplace approximation to local modes. $\endgroup$
    – JeremyC
    Commented Aug 11, 2019 at 15:17
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    $\begingroup$ sorry, i don't know about the Laplace approxiamation to local models. We need a standard form for the distribution so that we can use these posterior estimates as priors in another Bayesian analysis. We thought that MVN would be a good approximation. The posterior distirbution, in our case, is a constrained multivaraite distribution, in the semse that all elements should lie on a simplex. $\endgroup$
    – asifzuba
    Commented Aug 11, 2019 at 16:55
  • $\begingroup$ means and variances alone don't determine a multivariate normal; you need the full covariance matrix. $\endgroup$
    – Glen_b
    Commented Aug 12, 2019 at 0:16

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