“The share of large species that form large groups” and “the share of large-group species that are large” are two different numbers. Bayes’ theorem is the calculation that switches between these directions, and when you switch, how many large species there are to begin with strongly shapes the result.
Body size and group size: two questions
Among primates, 207 species have records for both body mass and group size. Here, species with a body mass of 10,000 g or more are called “large,” and species whose groups average 20 or more individuals are called “large-group” species.
For these data we can ask two questions that look alike but are different.
- Question 1
- If you pick a large species, what is the probability that it forms large groups?
- Question 2
- If you pick a large-group species, what is the probability that it is large?
Both questions count the species that are both large and large-group. What differs is what goes in the denominator. Question 1 asks about the share among large species; Question 2 asks about the share among large-group species.
Splitting 207 species into four groups
Body mass is on the horizontal axis and group size on the vertical axis, with one point for each of the 207 species. The dotted lines mark the two thresholds. Both axes use a log scale, so each tenfold increase takes the same distance.
Group size is the average group size recorded for each species. Data: PanTHERIA (Jones et al. 2009).
The 10 species in the upper right that meet both conditions are two species, the chimpanzee and the bonobo, plus eight Cercopithecidae species such as baboons (Papio) and macaques (Macaca). The lower right includes the gorilla and the orangutan, which live in small groups.
- P(large)
- —
- P(large group)
- —
- P(large group | large)
- —
- P(large | large group)
- —
Each square is one species. Dark squares are the denominator, and the orange squares among them are the numerator.
How it works: conditional probability and Bayes’ theorem
Here, a probability is the share of species meeting a condition when you pick one of the 207 species at random. There are 22 large species, so P(large) = 22 ÷ 207 ≈ 0.106.
A conditional probability P(B | A) is “the share of B when only the species meeting A are used as the denominator.” What sits to the right of the vertical bar is the group that forms the denominator.
The reverse, P(large | large group), is 10 ÷ 44 ≈ 0.23, only about half as large. The numerator is the same 10 species, but the denominator has changed from 22 species to 44.
Bayes’ theorem links the two. If you know the probability in one direction and the overall share of each condition, you can calculate the other direction.
Plugging in the numbers gives 0.4545 × 0.1063 ÷ 0.2126 ≈ 0.227, which matches the 10 ÷ 44 obtained by counting. In this formula, P(large) is called the prior probability and the resulting P(large | large group) the posterior probability: the probability of being large before and after learning that the species forms large groups.
The prior probability matters
If we change only the threshold for “large” and keep “large group” at 20 or more, the probabilities in the two directions move as follows.
| Threshold for “large” | P(large) | P(large group | large) | P(large | large group) |
|---|---|---|---|
| 2,000 g or more | 0.585 | 0.322 | 0.886 |
| 5,000 g or more | 0.420 | 0.333 | 0.659 |
| 10,000 g or more | 0.106 | 0.455 | 0.227 |
| 20,000 g or more | 0.019 | 0.500 | 0.045 |
P(large group | large) stays between 0.3 and 0.5, while P(large | large group) swings from 0.886 down to 0.045. Most of that change comes from how many large species there are to begin with, that is, from the prior probability.
If body size and group size were unrelated, the share of large-group species among large species would equal the overall share, P(large group) ≈ 0.21. This is called independence. With the 10,000 g threshold, P(large group | large) ≈ 0.45, so in these data the two are not independent.
Common misconceptions
“P(large group | large) and P(large | large group) come out about the same.”
In general they do not. In these data they are 0.45 and 0.23, a twofold difference. The two are equal only when the number of large species happens to equal the number of large-group species.
“Nearly half of large species form large groups, so a large-group species is probably large.”
Only 22 of the 207 species are large. Of the 44 large-group species, 34 are not large. Switching the direction while ignoring the original shares leads to badly wrong judgments. This is called base-rate neglect.
“A high conditional probability means one causes the other.”
That P(large group | large) is higher than the overall share only shows that the two tend to occur together. Whether large bodies lead to large groups, the reverse, or some other factor affects both cannot be decided from this table.
“0.45 means that 45% of large individuals live in large groups.”
What is counted here is species, not individuals. 0.45 means that 10 of the 22 large species form large groups. Counting individuals instead could give a different value.
Covered in the courses
The course sites are in Japanese and open to anyone. To go further, visit the page for each session.
Data: Jones, K. E. et al. (2009) PanTHERIA: a species-level database of life history, ecology, and geography of extant and recently extinct mammals. Ecology 90: 2648. Of 376 primate species, the 207 with records for both adult body mass and group size were used. Family names follow the course materials for Statistics I.
Which species have records may be biased, so the shares here are “shares among the 207 species with records.”