If we assume a beta prior, which includes a non-informational prior that follows a uniform distribution, then for binomially distributed data, the posterior distribution is also a beta distribution. This enables us to easily determine the high-density interval.
Topics
References
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., Rubin, D. B. (2014) Bayesian data analysis, 3rd Ed. CRC Press
https://statisticalsupportandresearch.files.wordpress.com/2017/11/bayesian_data_analysis.pdf
Marin, J-M and Robert, C. R. (2014) Bayesian essentials with R. 2nd Ed. Springer
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https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/index.html
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Reich, B. J., Ghosh, S. K. (2019) Bayesian statistics methods. CRC Press