Applying Bayes' Theorem in Deduction

The interpretation of Bayes' rule depends on the interpretation of probability ascribed to the terms. The two predominant interpretations are described below. Figure 2 shows a geometric visualization.

Figure 2: A geometric visualisation of Bayes' theorem

Bayesian interpretation

In the Bayesian (or epistemological) interpretation, probability measures a "degree of belief". 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 even 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.
• $P (A | B)$, the posterior, is the degree of belief after incorporating news that B is true.
• the quotient $\dfrac{P(B | A)}{P(B)}$ represents the support B provides for A.

Frequentist interpretation

Figure 3: Illustration of frequentist interpretation with tree diagrams

In the frequentist interpretation, probability measures a "proportion of outcomes". 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. $P(B | A)$ is the proportion of outcomes with property B out of outcomes with property A, and $P(A | B)$ is the proportion of those with A out of those with B (the posterior).

The role of Bayes' theorem is best visualized with tree diagrams such as Figure 3. 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

Figure 4: Tree diagram illustrating the beetle example. R, C, P and ${\overline {P}}$ 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 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 | 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 | Pattern)?

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

${\begin{aligned}P({\text{Rare}}\vert {\text{Pattern}})&={\frac {P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}})}}\\[8pt]&={\frac {P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}}\vert {\text{Rare}})P({\text{Rare}})+P({\text{Pattern}}\vert {\text{Common}})P({\text{Common}})}}\\[8pt]&={\frac {0.98\times 0.001}{0.98\times 0.001+0.05\times 0.999}}\\[8pt]&\approx 1.9\%\end{aligned}}$