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I am trying to find the expression for the prior predictive density for a multivariate normal distribution with unknown mean and unknown variance.

In the short document Bayesian Inference for the Multivariate Normal is given for a single observable $w$:

enter image description here

As you can see, the expression for a single observable is a multivariate T-distribution. However, I want the expression for the entire data set $w_1, w_2, ..., w_N$. As you can see, it is stated:

If we sum up the sample density over the N observations we get the marginal likelihood, or model evidence [here, sample density refers to the prior predictive density for a single observable] [here, marginal likelihood refers to the prior predictive density for the entire data set].

I don't see how the expression for the entire data set can be the sum of N multivariate T-distributions. I suspect it is instead a single multivariate T-distribution. Perhaps by "sum", the author means something else.

It would be great if someone can give the expression for the entire data set.

Thanks, MM

P.S.

In the text book Bayesian Theory by Bernardo and Smith, they give the following table: enter image description here

My suspicion is that the expression for the marginal likelihood p(x1, x2, ..., xn) is obtained from p(x) by replacing k in the multivariate student t-distribution with k * n.

Am I right?

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    $\begingroup$ Did you want the prior predictive distribution, which uses no data, or the posterior predictive distribution, which uses data? $\endgroup$ Feb 14, 2021 at 5:26
  • $\begingroup$ I want the marginal likelihood p(w1, w2, ..., wn) obtained by integrating p(w1, w2, ..., wn|mu, sigma) over mu and sigma using a normal-Wishart prior for mu and sigma. $\endgroup$
    – user311292
    Feb 14, 2021 at 5:32
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    $\begingroup$ So you want the marginal distributions and not the predictive ones? You should edit your question title to reflect that. I would point out that $x$ in the above formulation is a vector of size $K\times{n}$. The multivariate marginal is already provided. That is what the table is showing you, the multivariate marginal density. If it had been marginalized down to one dimension, then the Wishart would have been a Gamma distribution and the k dimension Student's distribution would be the one dimensional version. The marginal predictive is there too. $\endgroup$ Feb 14, 2021 at 6:43
  • $\begingroup$ Thank you very much for your quick responses. I apologize for the confusion. I am not very experienced doing Bayesian analysis. I am still confused. I think x in p(x) is a k x 1 vector. I think what I want is p(z). I want an expression for the marginal likelihood or model evidence. $\endgroup$
    – user311292
    Feb 14, 2021 at 7:32
  • $\begingroup$ I would think "sum up" stands for integrating in this context $\endgroup$
    – Xi'an
    Feb 14, 2021 at 11:42

1 Answer 1

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I think what I was looking for is Eq. 234 in this document: enter image description here

I believe that this marginal likelihood is also often called the "model evidence" or the "prior predictive density".

For example, this document uses the term "prior predictive distribution" to refer to the 1-D version of Eq. 234: https://www.statlect.com/fundamentals-of-statistics/normal-distribution-Bayesian-estimation#hid6

Therefore, I will leave the title of this question as is.

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