bfactor_to_prob {pcal}

Turn Bayes Factors Into Posterior Probabilities

Description

Update the prior probabilities of models/hypotheses to posterior probabilities using Bayes factors.

Usage

bfactor_to_prob(bf, prior_prob = 0.5)

Arguments

bf

A numeric vector of non-negative values.

prior_prob

A numeric vector with values in the [0,1] interval. If length(bf) == 1 then prior_prob can be of any positive length, but if length(bf) > 1 then the length of prior_prob can only be 1 or equal to the length of bf.

Details

bfactor_to_prob computes the posterior probability of the null hypothesis using the following equation from Berger and Delampady (1987):

P(\textrm{null} \, \textrm{hypothesis}|\textrm{data}) = \left(1 + \frac{1 - {null\_prob}}{null\_prob} \times \frac{1}{bf}\right)^{-1}

where bf is a Bayes factor if favor of the null hypothesis and prior_prob is the prior probability of the null hypothesis. The alternative hypothesis has prior probability 1 - prior_prob and posterior probability 1 - bfactor_to_prob(bf, prior_prob).

The prior_prob argument is optional and is set to 0.5 by default, implying prior equiprobability of hypotheses. prior_prob can only be of length equal to length(bf), in which case each prior probability in prior_prob will be updated using the corresponding element of bf, or of length 1, in which case it will be recycled (if length(bf) > 1) and each element of bf will update the same prior_prob value.

Value

If length(bf) > 1 then bfactor_to_prob returns a numeric vector with the same length as bf, otherwise it returns a numeric vector with the same length as prior_prob.

References

Berger JO, Delampady M (1987). “Testing precise hypotheses.” Statistical Science, 2(3), 317–335.

See Also

• bfactor_interpret and bfactor_interpret_kr for the interpretation of Bayes factors.

• bfactor_log_interpret and bfactor_log_interpret_kr for the interpretation of the logarithms of Bayes factors.

• bcal for a p-value calibration that returns lower bounds on Bayes factors in favor of point null hypotheses.

• pcal for a p-value calibration that returns lower bounds on the posterior probabilities of point null hypotheses.

Examples

# With a Bayes factor that is indifferent between the null and the alternative hypotheses:
bfactor_to_prob(1)

# Same as above but the null hypothesis has high prior probability:
bfactor_to_prob(1, .99)

# Posterior probability of the null as a function of different prior probabilities:
bfactor_to_prob(1, seq(.5, 1, .1))

# With Bayes factors that favor the null hypothesis:
round(bfactor_to_prob(seq(2, 50, 2.5)), 3)

# Same as above but the null hypothesis has low prior probability:
round(bfactor_to_prob(seq(2, 50, 2.5), prior_prob = .01), 3)

# Posterior probabilities obtained with Bayes factors that favor the alternative hypothesis:
round(bfactor_to_prob(seq(0, 1, .05)), 3)

# Same as above but the null hypothesis has high prior probability:
round(bfactor_to_prob(seq(0, 1, .05), prior_prob = .99), 3)

# Application: chi-squared goodness-of-fit test,
# lower bound on the posterior probability of the null hypothesis:
x <- matrix(c(12, 41, 25, 33), ncol = 2)
bfactor_to_prob(bcal(chisq.test(x)[["p.value"]]), prior_prob = .9)