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I want to analytically find the HPD interval for a posterior that has the normal distribution $\mathbb{N}(\mu, \sigma^2)$. So according to the definition of the HPD interval I am looking for expressions for the upper and lower limit such that the integral of the posterior over these limits is equal to $1-\alpha$. Would really appreciate some guidance on this as I am struggling.

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If the posterior is a $\mathcal{N}(\mu,\sigma^2)$ distribution, the HPD region is defined as $$\mathfrak{H}_\alpha = \{\theta;\ \varphi(\theta;\mu,\sigma)\ge k_\alpha\}$$ or $$\mathfrak{H}_\alpha = \{\theta;\ \sigma^{-2}(\theta-\mu)^2\le k'_\alpha\}= \{\theta;\ |\theta-\mu|\le \sigma k'_\alpha\}$$ meaning it is the symmetric credible region $$\mathfrak{H}_\alpha = (\mu+\sigma q_{\alpha/2},\mu-\sigma q_{\alpha/2})$$ when $q_{\alpha/2}$ is the $\alpha/2$-quantile of the standard $\mathcal{N}(0,1)$ distribution.

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