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The likelihood contributes precision $n$ and precision-weighted mean $n\overline{X}$, while the prior contributes precision ${\tau }^2$ and precision-weighted mean ${\tau }^2\theta$. By normal-normal conjugacy with known observation variance, the posterior isThus the posterior mean isWhen $\theta =0$, its expectation under true parameter $\mu$ is $n\mu /(n+{\tau }^2)$ and its variance is $n/(n+{\tau }^2{)}^2$. Hence its frequentist mean squared error is

As $n\to \infty$,whileThus the credible interval and confidence interval have asymptotically equal centres and half-widths, and both contract around the true parameter. Their interpretations remain different: the confidence statement concerns repeated samples, whereas the credible statement concerns posterior probability. This is normal-normal conjugacy with known observation variance.