# What's special about 89% credibility intervals?

Piergianluca Porta Mana

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Piergianluca Porta Mana. What's special about 89% credibility intervals?. 2025. hal-05243742

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# What's special about 89% credibility intervals?

P.G.L. Porta Mana

Western Norway University of Applied Sciences <pgl@portamana.org>

7 July 2025; updated 26 September 2025

A curious information-theoretic property of 89% credibility intervals is discussed.

A *credibility interval* (or “credible”, “Bayesian”, “posterior”, “compatibility” interval or region) is a range where the true value of some real quantity lies with a given probability<fnref n="1" />. For instance, if [1.2, 7.0] is a 60% credibility interval for the quantity Q, then there is a 60% probability that the true value of Q is between 1.2 and 7.0 (note the difference in meaning from a *confidence interval*, whose definition is much more involved and less direct<fnref n="2" />). It is always best to report the full posterior probability distribution of a quantity of interest, so that the reader can calculate from it the probabilities of any desired ranges and make decisions<fnref n="3" />. But credibility intervals often provide useful summaries of the full distribution.

Credibility intervals of 50%, 75%, 90%, 95%, or even 99% probability are more or less commonly used in the Bayesian literature and given as defaults in software packages for Bayesian analysis. Also 89% ones have become quite fashionable since around 2016; they seemingly originate in McElreath's *Statistical Rethinking*<fnref n="4" />, first published in that year. They are used, for example, in the software package *bayestestR*<fnref n="5" />. McElreath justly points out<fnref n="6" /> that the specific probability value of a credibility interval should be chosen depending on the context; there is nothing special about values like 90% or 95%. To emphasize and remind us of this fact (and also to avoid confusion with confidence intervals, whose meaning is different) he uses 89%, which is an arbitrary value.

Or is it?

It turns out that 89% credibility intervals do possess a curious information-theoretic property, which may also motivate the use of 50% and 89% as defaults.

<footnote n="1">¹ Bernardo & Smith 2000 § 5.1.5 pp. 259ff.</footnote>

<footnote n="2">² MacKay 2005 § 37.3; Jaynes 1976.</footnote>

<footnote n="3">³ Raiffa 1970.</footnote>

<footnote n="4">⁴ McElreath 2020.</footnote>

<footnote n="5">⁵ Makowski, Ben-Shachar, et al. 2019.</footnote>

<footnote n="6">⁶ McElreath 2020 e.g. § 3.2.2 p. 56, § 4.3.5 p. 88.</footnote>
The true value of a quantity lies inside a p credibility interval with probability p, and outside with probability 1 − p. Our uncertainty about these two possibilities can be quantified by the Shannon entropy:<fnref n="7" />

$$
H(p) := -[p \log_2 p + (1-p) \log_2(1-p)] \text{Sh.} \quad (1)
$$

The value of 1 Sh (one shannon<fnref n="8" />) represents an equal uncertainty between
two alternatives.

Thus for a 50% credibility interval we have an uncertainty of 1 Sh
on whether the true value lies inside or outside it. This is the highest
level of uncertainty for such dichotomous alternative. It is useful to also
provide an interval with an uncertainty smaller than 1 Sh, so that we are
somewhat more certain that the true value lies inside than outside.

For example, for which interval is our maximal uncertainty *halved*
to 0.5 Sh? The answer is: for *an* 89.00% *credibility interval*. This result
is easily calculated solving *H(p)* = 0.5 Sh numerically. A more precise
solution is 88.9972%. Vice versa the entropy corresponding to the exact
probability 89% is approximately 0.499 916 Sh.

When we report a 50% and an 89% credibility interval we are therefore
reporting intervals having uncertainties of one shannon and of half a
shannon. Of course the arbitrariness remains.

For your amusement, some corresponding values of entropies and
probabilities are reported in table 1.

Bibliography

Bernardo, J.-M., Smith, A. F. (2000): *Bayesian Theory*, repr. (Wiley, New York). doi: 10.1002/9780470316870. First publ. 1994.

Harper, W. L., Hooker, C. A., eds. (1976): *Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Vol. II: Foundations and Philosophy of Statistical Inference*. (Reidel, Dordrecht).

ISO (2008): *ISO 80000-13:2008: Quantities and units 13: Information science and technology*. International Organization for Standardization.

Jaynes, E. T. (1976): *Confidence intervals vs Bayesian intervals*. In: Harper, Hooker 1976: 175-257. With discussion, comments by M. Maxfield and O. Kempthorne, and reply. Repr. with an introduction in Jaynes 1989 pp. 149-209; https://bayes.wustl.edu/etj/node1.html.

Jaynes, E. T. (1989): *Papers on Probability, Statistics and Statistical Physics*, repr. (Kluwer, Dordrecht). Edited by R. D. Rosenkrantz. First publ. 1983.

<footnote n="7">⁷ ISO 2008 Items 13-24, 13-25.
penform&ievref=171-07-11</footnote>

<footnote n="8">⁸ https://electropedia.org/iev/iev.nsf/display?o</footnote>
<table><thead><tr><th>entropy/Sh</th><th>probability/%</th><th>entropy/Sh</th><th>probability/%</th></tr></thead><tbody><tr><td><strong>1.00</strong></td><td><strong>50.00</strong></td><td><strong>0.50</strong></td><td><strong>89.00</strong></td></tr><tr><td>0.95</td><td>63.09</td><td>0.45</td><td>90.59</td></tr><tr><td>0.90</td><td>68.40</td><td>0.40</td><td>92.06</td></tr><tr><td>0.85</td><td>72.40</td><td>0.35</td><td>93.42</td></tr><tr><td>0.80</td><td>75.70</td><td>1/3</td><td>93.85</td></tr><tr><td>0.75</td><td>78.55</td><td>0.30</td><td>94.68</td></tr><tr><td>0.70</td><td>81.07</td><td>0.25</td><td>95.83</td></tr><tr><td>2/3</td><td>82.60</td><td>0.20</td><td>96.89</td></tr><tr><td>0.65</td><td>83.33</td><td>0.15</td><td>97.85</td></tr><tr><td>0.60</td><td>85.39</td><td>0.10</td><td>98.70</td></tr><tr><td>0.55</td><td>87.27</td><td>0.05</td><td>99.44</td></tr><tr><td></td><td></td><td>0.00</td><td>100.00</td></tr></tbody></table>

<table><caption>Table 1 Some corresponding values of probabilities and entropies</caption><thead><tr><th>probability/%</th><th>entropy/Sh</th><th>probability/%</th><th>entropy/Sh</th></tr></thead><tbody><tr><td><strong>50</strong></td><td><strong>1.0000</strong></td><td>75</td><td>0.8110</td></tr><tr><td>51</td><td>1.0000</td><td>76</td><td>0.7950</td></tr><tr><td>52</td><td>0.9990</td><td>77</td><td>0.7780</td></tr><tr><td>53</td><td>0.9970</td><td>78</td><td>0.7600</td></tr><tr><td>54</td><td>0.9950</td><td>79</td><td>0.7410</td></tr><tr><td>55</td><td>0.9930</td><td>80</td><td>0.7220</td></tr><tr><td>56</td><td>0.9900</td><td>81</td><td>0.7010</td></tr><tr><td>57</td><td>0.9860</td><td>82</td><td>0.6800</td></tr><tr><td>58</td><td>0.9810</td><td>83</td><td>0.6580</td></tr><tr><td>59</td><td>0.9770</td><td>84</td><td>0.6340</td></tr><tr><td>60</td><td>0.9710</td><td>85</td><td>0.6100</td></tr><tr><td>61</td><td>0.9650</td><td>86</td><td>0.5840</td></tr><tr><td>62</td><td>0.9580</td><td>87</td><td>0.5570</td></tr><tr><td>63</td><td>0.9510</td><td>88</td><td>0.5290</td></tr><tr><td>64</td><td>0.9430</td><td><strong>89</strong></td><td><strong>0.5000</strong></td></tr><tr><td>65</td><td>0.9340</td><td>90</td><td>0.4690</td></tr><tr><td>66</td><td>0.9250</td><td>91</td><td>0.4360</td></tr><tr><td>67</td><td>0.9150</td><td>92</td><td>0.4020</td></tr><tr><td>68</td><td>0.9040</td><td>93</td><td>0.3660</td></tr><tr><td>69</td><td>0.8930</td><td>94</td><td>0.3270</td></tr><tr><td>70</td><td>0.8810</td><td>95</td><td>0.2860</td></tr><tr><td>71</td><td>0.8690</td><td>96</td><td>0.2420</td></tr><tr><td>72</td><td>0.8550</td><td>97</td><td>0.1940</td></tr><tr><td>73</td><td>0.8410</td><td>98</td><td>0.1410</td></tr><tr><td>74</td><td>0.8270</td><td>99</td><td>0.0808</td></tr></tbody></table>
MacKay, D. J. C. (2005): *Information Theory, Inference, and Learning Algorithms*, version 7.2 (4th pr.) (Cambridge University Press, Cambridge). https://www.inference.org.uk/itila/book.html. First publ. 1995. See also video lectures at https://videolecture.s.net/events/course_information_theory_pattern_recognition.

Makowski, D., Ben-Shachar, M. S., Lüdecke, D. (2019): *bayestestR: describing effects and their uncertainty, existence and significance within the Bayesian framework*. J. Open Source Softw. 4<sup>40</sup>, 1541. doi:10.21105/joss.01541, https://cran.r-project.org/package=bayestestR, doi:10.32614/CRAN/package.bayestestR.

McElreath, R. (2020): *Statistical Rethinking: A Bayesian Course with Examples in R and STAN*, 2nd ed. (CRC, Boca Raton, USA). doi:10.1201/9780429029608. First publ. 2016.

Raiffa, H. (1970): *Decision Analysis: Introductory Lectures on Choices under Uncertainty*, 2nd pr. (Addison-Wesley, Reading, USA). First publ. 1968.