Learning from correlated signals

Suppose I want to learn the value of a parameter $\theta \in R$. My prior is that $\theta$ is normally distributed with variance ${\sigma }_0^2$. I observe $n\ge 1$ signals $$s_i=\theta +{\epsilon }_i$$ of $\theta$. The errors ${\epsilon }_i$ in these signals are independent of $\theta$. They are jointly normally distributed with equal variances $\mathrm{Var}\left({\epsilon }_i\right)={\sigma }^2$ and pairwise correlations $$\mathrm{Cor}({\epsilon }_i,{\epsilon }_j)=\{\begin{array}{cc}1& \text{if}i=j\\ \rho & \text{otherwise}.\end{array}$$ I assume $-1/(n-1)\le \rho \le 1$ so that this distribution is feasible.¹

Observing $s_1,s_2,\dots ,s_n$ is the same to observing the sample mean $${\overline{s}}_n\equiv \frac{1}{n}\sum _{i=1}^n{s}_i,$$ which is normally distributed and has conditional variance $$\mathrm{Var}({\overline{s}}_n\mid \theta )=\frac{(1+(n-1\left)\right)\rho {\sigma }^2}{n}$$ under my prior. The posterior distribution of $\theta$ given ${\overline{s}}_n$ is also normal and has variance $$\mathrm{Var}(\theta \mid {\overline{s}}_n)={(\frac{1}{{\sigma }_0^2}+\frac{n}{(1+(n-1\left)\rho \right){\sigma }^2})}^{-1}.$$ Both variances are (i) decreasing in $n$ when $\rho <1$ and (ii) increasing in $\rho$ when $n>1$. Intuitively, if the signals are not perfectly correlated then observing more gives me more information about $\theta$. If they are negatively correlated then their errors “cancel out” and the sample mean ${\overline{s}}_n$ gives me a precise estimate of $\theta$.

The chart below shows how $\mathrm{Var}({\overline{s}}_n\mid \theta )$ and $\mathrm{Var}(\theta \mid {\overline{s}}_n)$ vary with $\rho$ and $n$ when ${\sigma }_0=\sigma =1$. If $\rho =-1/(n-1)$ then ${\epsilon }_1+{\epsilon }_2+\cdots +{\epsilon }_n=0$, and so $\mathrm{Var}({\overline{s}}_n\mid \theta )=0$ and $\mathrm{Var}(\theta \mid {\overline{s}}_n)=0$ because ${\overline{s}}_n=\theta$. Whereas if $\rho =1$ then signals $s_2$ through $s_n$ provide the same information as $s_1$, and so $\mathrm{Var}({\overline{s}}_n\mid \theta )=\mathrm{Var}(s_1\mid \theta )$ and $\mathrm{Var}(\theta \mid {\overline{s}}_n)=\mathrm{Var}(\theta \mid s_1)$.