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How is to find the value of $k$ for the following equation which gives $100(1-\alpha)\%$ highest posterior density?

$$\int_{\theta:\pi(\theta|\mathbf x)>k}\pi(\theta|\mathbf x)d\theta=1-\alpha,$$

where $\pi(\theta|\mathbf x)$ is the posterior density for $\theta$, and $\alpha$ is the level of significance. Can anyone please show me a toy example from which I can understand how is to obtain the value of $k$?

user149054
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    In general you would need numerical methods (in effect try a $k$ and if it gives the wrong integral then try again with a better $k$) rather than analysis, though there may be shortcuts in some particular cases. – Henry Dec 10 '23 at 12:39
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    Examples worth reading include https://stats.stackexchange.com/questions/179678/finding-the-high-density-region-for-a-chi2-chi-squared-distribution and https://stats.stackexchange.com/questions/240749/how-to-find-95-credible-interval and https://stackoverflow.com/questions/42986446/determining-high-density-region-for-a-distribution-in-r and links from these questions – Henry Dec 10 '23 at 12:42

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