Let (X_1,...,X_n) be sample from normal $\mathcal{N}(\theta,\sigma^2)$ distribution, where $\sigma^2$ is known constant, and prior for $\theta$ is $\mathcal{N}(\nu,\tau^2)$. Estimate $\nu$ and $\tau^2$ by MLE using the empirical Bayesian method.
I calculated that marginal distribution for $X$ is $\mathcal{N}(\theta,\tau^2+\sigma^2)$, so MLEs are $\hat{\nu} = \overline{x_n}$ and $\hat{\tau^2} = \overline{s_n}^2-\sigma^2$, where $\overline{s_n}^2$ is sample variance. I think this is ok.
But also, if I look at random variable $\overline{X_n}|\theta \in \mathcal{N}(\theta,\frac{\sigma^2}{n})$, then I get that marginal for $\overline{X_n}$ is $\mathcal{N}(\theta,\tau^2+\frac{\sigma^2}{n})$, so marginal for $X$ is $\mathcal{N}(\theta,n\tau^2+\sigma^2)$, so MLEs are $\hat{\nu} = \overline{x_n}$ and $\hat{\tau^2} = \frac{\overline{s_n}^2-\sigma^2}{n}$.
I guess I'm making some mistakes in the second case (I'm pretty sure that the calculations are correct), but I cannot see what is the problem, why I get different marginals for $X$ and what is wrong with my inference. Can someone help me with this? Thanks in advance.