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I am using a Gaussian model with a conjugate Normal-Inverse-Wishart (NIW) prior, as described here. The advantage of this approach is that the marginal likelihood $p(y)$, which is what I am interested in, is available in closed form.

My problem is that the results seem to be dependent of the NIW hyper-parameters (I have no prior information), with some of the dangers being described here.

As an alternative, I am considering bootstrapping my data in order to obtain $m$ estimate of the mean and covariance. Then I could calculate the marginal likelihood:

$$ p(y) \approx \frac{1}{m} \sum_{i=1}^m p(y|\hat{\mu}_i, \hat{\Sigma}_i). $$

Would this prior be an approximation to an Empirical Bayes prior, something else or just nonsense?

Thank you

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Have you considered simply applying a scaling to the covariance matrix, as suggested by Andrew Gelman?

See also this paper on the scaled inverse Wishart.

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  • $\begingroup$ That's pretty much what I was doing originally. Then I tried to get a prior that depended only on the data (by bootstrapping), in order to avoid choosing a prior and its hyper-parameters. $\endgroup$ – Matteo Fasiolo Oct 30 '14 at 20:19

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