pingouin.bayesfactor_binom(k, n, p=0.5)[source]

Bayes factor of a binomial test with \(k\) successes, \(n\) trials and base probability \(p\).


Number of successes.


Number of trials.


Base probability of success (range from 0 to 1).


The Bayes Factor quantifies the evidence in favour of the alternative hypothesis, where the null hypothesis is that the random variable is binomially distributed with base probability \(p\).

See also


Bayes Factor of a correlation


Bayes Factor of a T-test


Adapted from a Matlab code found at

The Bayes Factor is given by the formula below:

\[BF_{10} = \frac{\int_0^1 \binom{n}{k}g^k(1-g)^{n-k}} {\binom{n}{k} p^k (1-p)^{n-k}}\]



We want to determine if a coin if fair. After tossing the coin 200 times in a row, we report 115 heads (hereafter referred to as “successes”) and 85 tails (“failures”). The Bayes Factor can be easily computed using Pingouin:

>>> import pingouin as pg
>>> bf = float(pg.bayesfactor_binom(k=115, n=200, p=0.5))
>>> # Note that Pingouin returns the BF-alt by default.
>>> # BF-null is simply 1 / BF-alt
>>> print("BF-null: %.3f, BF-alt: %.3f" % (1 / bf, bf))
BF-null: 1.197, BF-alt: 0.835

Since the Bayes Factor of the null hypothesis (“the coin is fair”) is higher than the Bayes Factor of the alternative hypothesis (“the coin is not fair”), we can conclude that there is more evidence to support the fact that the coin is indeed fair. However, the strength of the evidence in favor of the null hypothesis (1.197) is “barely worth mentionning” according to Jeffreys’s rule of thumb.

Interestingly, a frequentist alternative to this test would give very different results. It can be performed using the scipy.stats.binom_test() function:

>>> from scipy.stats import binom_test
>>> pval = binom_test(115, 200, p=0.5)
>>> round(pval, 5)

The binomial test rejects the null hypothesis that the coin is fair at the 5% significance level (p=0.04). Thus, whereas a frequentist hypothesis test would yield significant results at the 5% significance level, the Bayes factor does not find any evidence that the coin is unfair.

Last example using a different base probability of successes

>>> bf = pg.bayesfactor_binom(k=100, n=1000, p=0.1)
>>> print("Bayes Factor: %.3f" % bf)
Bayes Factor: 0.024