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Suppose we observe $n$ independent random variables $X_1, \dots, X_n$. Suppose also that the mean and variance of each $X_i$ is unknown. If $$X_i \sim \mathrm{N}(\mu_i,\sigma^2_i)$$ then a conjugate prior for $\mu_i,\sigma^2_i$ is $$\mu_i,\sigma^2_i \sim \mathrm{N\Gamma^{-1}}(\xi_i,\lambda_i,\alpha_i,\beta_i).$$ I am interested in the case where the variance $\sigma^2$ is common to all the variables. That is $$X_i \sim \mathrm{N}(\mu_i,\sigma^2).$$ Does a joint conjugate prior exist for $\mu_1,\dots,\mu_n,\sigma^2$ and, if so, can anyone point to references which might be helpful in determining it? Any hints would be appreciated too.

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To check for the existence of a conjugate prior, the likelihood must be expressed as an exponential family: $$(X_1,\ldots,X_n) \sim \sigma^{ -n} \exp \frac{-1}{2\sigma^2} \left\{ \sum_{i=1}^n (\mu_i-x_i)^2\right\}$$ Which leads to $$\mu_1,\dots,\mu_n|\sigma^2 \sim \mathcal{N}(\xi_i,\alpha\sigma^2)$$ and$$\sigma^{-2}\sim\mathcal{G}(\beta,\gamma)$$as acceptable conjugate prior, since updating to posterior means changing \begin{align*} \xi_i &\quad\text{ in } &\{\alpha^{-1}\xi+x_i\}\{\alpha^{-1}+1\}^{-1}\\ \alpha &\quad\text{ in } &\{\alpha^{-1}+1\}^{-1}\\ \beta &\quad\text{ in } &{\beta+n}\\ \gamma &\quad\text{ in } &{\gamma+\{\alpha+1\}^{-1}\sum (x_i-\xi)^2} \end{align*} or something similar.

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  • $\begingroup$ Thank you for your clear answer. When going from prior to posterior did you need to write out Bayes' theorem or is there a trick to get the updated parameters in this case? $\endgroup$ – Estacionario Sep 25 '18 at 17:49
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    $\begingroup$ That's the whole point about conjugacy, when looking at the product prior x likelihood you can identify these hyperparameters. $\endgroup$ – Xi'an Sep 25 '18 at 18:37

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