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Consider a Gaussian sample $y_1, \ldots, y_n \sim_{\text{iid}} \mathcal{N}(\mu,\sigma^2)$ and treat it in the Bayesian way with the noninformative prior $\pi(\mu, \sigma) \propto \frac{1}{\sigma}$.

I have found some books providing the posterior predictive distribution of a single new observation (e.g. Marin & Robert, Bayesian Core). But I don't find any reference providing the posterior predictive distribution of multiple new observations. I can derive it myself (this is a multivariate Student distribution) but I am looking for a reference.

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