Editing Bayes' theorem (section)

==Interpretations==
<!-- NOTE: Before changing this image, be sure to seek consensus. The current consensus, as established at [[Talk:Bayes%27_theorem#RfC_on_illustration]], is that the image should stay. --> 
[[File:Bayes theorem assassin.svg|thumb|A geometric visualization of Bayes' theorem using astronauts, from the online game ''[[Among Us]]'', who may be suspicious (with eyebrows) and may be assassins (with daggers)]]
The interpretation of Bayes' rule depends on the [[Probability interpretations|interpretation of probability]] ascribed to the terms. The two predominant interpretations are described below.

===Bayesian interpretation===
In the [[Bayesian probability|Bayesian (or epistemological) interpretation]], probability measures a "degree of belief".{{cn|date=April 2025}} Bayes' theorem links the degree of belief in a proposition before and after accounting for evidence. For example, suppose it is believed with 50% certainty that a coin is twice as likely to land heads than tails. If the coin is flipped a number of times and the outcomes observed, that degree of belief will probably rise or fall, but might remain the same, depending on the results. For proposition ''A'' and evidence ''B'',
* ''P'' (''A''), the ''prior'', is the initial degree of belief in ''A''.
* ''P'' (''A''&thinsp;|&thinsp;''B''), the ''posterior'', is the degree of belief after incorporating news that ''B'' is true.
* the quotient {{sfrac|''P''(''B''&thinsp;{{!}}&thinsp;''A'')|''P''(''B'')}} represents the support ''B'' provides for ''A''.

For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see [[Bayesian inference]].

===Frequentist interpretation===
[[File:Bayes theorem tree diagrams.svg|thumb|Illustration of frequentist interpretation with [[Tree diagram (probability theory)|tree diagrams]]]]

In the [[Frequentist interpretation of probability|frequentist interpretation]], probability measures a "proportion of outcomes".{{cn|date=April 2025}} For example, suppose an experiment is performed many times. ''P''(''A'') is the proportion of outcomes with property ''A'' (the prior) and ''P''(''B'') is the proportion with property ''B''. ''P''(''B''&thinsp;|&thinsp;''A'') is the proportion of outcomes with property ''B'' ''out of'' outcomes with property ''A'', and ''P''(''A''&thinsp;|&thinsp;''B'') is the proportion of those with ''A'' ''out of'' those with ''B'' (the posterior).

The role of Bayes' theorem can be shown with tree diagrams. The two diagrams partition the same outcomes by ''A'' and ''B'' in opposite orders, to obtain the inverse probabilities. Bayes' theorem links the different partitionings.

====Example====
[[File:Bayes theorem simple example tree.svg|thumb|Tree diagram illustrating the beetle example. ''R, C, P'' and <math> \overline{P} </math> are the events rare, common, pattern and no pattern. Percentages in parentheses are calculated. Three independent values are given, so it is possible to calculate the inverse tree.]]

An [[Entomology|entomologist]] spots what might, due to the pattern on its back, be a rare [[subspecies]] of [[beetle]]. A full 98% of the members of the rare subspecies have the pattern, so ''P''(Pattern&nbsp;|&nbsp;Rare) = 98%. Only 5% of members of the common subspecies have the pattern. The rare subspecies is 0.1% of the total population. How likely is the beetle having the pattern to be rare: what is ''P''(Rare&nbsp;|&nbsp;Pattern)?

From the extended form of Bayes' theorem (since any beetle is either rare or common),

: <math display=block>
\begin{align}
P(\text{Rare} \vert \text{Pattern}) &= \frac{P(\text{Pattern} \vert \text{Rare})\,P(\text{Rare})} {P(\text{Pattern})}\\
[8pt] &= \tfrac{P(\text{Pattern}\vert \text{Rare})\,P(\text{Rare})} {P(\text{Pattern} \vert \text{Rare})\,P(\text{Rare}) + P(\text{Pattern}\vert \text{Common})\,P(\text{Common})}\\
[8pt] &= \frac{0.98 \times 0.001} {0.98 \times 0.001 + 0.05 \times 0.999}\\
[8pt] &\approx 1.9\%
\end{align}
</math>