bfactor_to_prob: Turn Bayes Factors Into Posterior Probabilities

In pedro-teles-fonseca/pcal: Calibration of P-Values for Point Null Hypotheses

Turn Bayes Factors Into Posterior Probabilities

Description

Update 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 turns Bayes factors into posterior probabilities using a vectorized version of the following equation from \insertCitebergerDelampady1987;textualpcal:

P(H0 | x) = (1 + ( (1 - π) / π) (1 / B(x) ) ) ^ ( -1 )

where B(x) is a Bayes factor in favor of the null hypothesis (given the data x), π is the prior probability of the null hypothesis and 1 - π is the prior probability of the alternative hypothesis.

If bf is a vector of Bayes factors (in favor of the null hypothesis) and prior_prob is a vector with the prior probabilities of those hypotheses then bfactor_to_prob(bf, prior_prob) updates prior_prob to posterior probabilities. The posterior probabilities of the alternative hypotheses can be obtained with 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. Warning messages are thrown if there are NA or NaN values in bf or in prior_prob.

References

\insertAllCited

See Also

• bfactor_interpret for the interpretation of Bayes factors.

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 hypothesis as a function
# of the prior probability:
# -----------------------------------------------------------------
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)

pedro-teles-fonseca/pcal documentation built on Nov. 4, 2022, 3:01 p.m.