| dfba_beta_bayes_factor {DFBA} | R Documentation |
Bayes Factor for Posterior Beta Distribution
Description
Given a beta posterior distribution and given a prior for the variate, computes the Bayes factor for either point or interval null hypotheses.
Usage
dfba_beta_bayes_factor(a_post, b_post, method, H0, a0 = 1, b0 = 1)
Arguments
a_post |
The first shape parameter for the posterior beta distribution. Must be positive and finite. |
b_post |
The second shape parameter for the posterior beta distribution. Must be positive and finite. |
method |
One of |
H0 |
If method="interval", then the H0 input is vector of two values, which are lower and upper limits for the null hypothesis; if method="point", then the H0 input is single number, which is the null hypothesis value |
a0 |
The first shape parameter for the prior beta distribution (default is 1). Must be positive and finite. |
b0 |
The second shape parameter for the prior beta distribution(default is 1). Must be positive and finite. |
Details
For a binomial variate with n1 successes and n2 failures, the
Bayesian analysis for the population success rate parameter \phi is
distributed as a beta density function with shape parameters a_post
and b_post for a_post = n1 + a0 and b_post = n2 + b0 where a0
and b0 are the shape parameters for the prior beta distribution. It is
common for users to be interested in testing hypotheses about the population
\phi parameter. The Bayes factor is useful to assess if either the null
or the alternative hypothesis are credible.
There are two types of null hypotheses – an interval null hypothesis
and a point null hypothesis. For example, an interval null hypothesis
might be \phi \le .5 with the alternative hypothesis being \phi > .5,
whereas a point null hypothesis might be \phi = .5 with the alternative
being \phi \ne .5. It is conventional to call the null hypothesis H_0
and to call the alternative hypothesis H_1. For frequentist null
hypothesis testing, H_0 is assumed to be true, to see if this
assumption is likely or not. With the frequentist approach the null
hypothesis cannot be proved since it was assumed in the first place.
With frequentist statistics, H_0 is thus either retained
as assumed or it is rejected. Unlike the frequentist approach,
Bayesian hypothesis testing does not assume either H_0 or H_1; it
instead assumes a prior distribution for the population parameter
\phi, and based on this assumption arrives at a posterior
distribution for the parameter given the data of n1 and n2 for the
binomial outcomes.
There are two related Bayes factors - BF10 and BF01 where
BF01 = 1/BF10. When BF10 > 1, there is more support for the
alternative hypothesis, whereas when BF01 > 1, there is more support
for the null hypothesis. Thus, in Bayesian hypothesis testing it is possible
to build support for either H_0 or H_1. In essence, the Bayes
factor is a measure of the relative strength of evidence. There is no
standard guideline for recommending a decision about the prevailing
hypothesis, but several statisticians have suggested criteria. Jeffreys
(1961) suggested that BF > 10 was strong and BF > 100
was decisive; Kass and Raffrey (1995) suggested that BF > 20 was
strong and BF > 150 was decisive. Chechile (2020) argued
from a decision-theory framework for a third option for the user to decide
not to decide if the prevailing Bayes factor is not sufficiently large.
From this decision-making perspective, Chechile (2020) suggested that BF > 19
was a good bet - too good to disregard, BF > 99 was a strong
bet - irresponsible to avoid, and BF > 20,001 was virtually certain.
Chechile also pointed out that despite the Bayes factor value there is often
some probability, however small, for either hypothesis. Ultimately, each
academic discipline has to set the standard for their field for the strength
of evidence. Yet even when the Bayes factor is below the user's threshold for
making claims about the hypotheses, the value of the Bayes factor from one
study can be nonetheless valuable to other researchers and might be combined
via a product rule in a meta-analysis. Thus, the value of the Bayes
factor has a descriptive utility.
The Bayes factor BF10 for an interval null is the ratio of the posterior
odds of H_1 to H_0 divided by the prior odds of H_1 to H_0.
Also, the converse Bayes factor BF01 is the ratio of posterior odds of
H_0 to H_1 divided by the prior odds of H_0 to H_1;
hence BF01 = 1/BF10. If there is no change in the odds ratio as a
function of new data being collected, then BF10 = BF01 = 1. But, if
evidence is more likely for one of the hypotheses, then either BF10 or
BF01 will be greater than 1.
The population parameter \phi is distributed on the continuous interval
[0,1]. The prior and posterior beta distribution are probability density
displays. Importantly, this means that no point has a nonzero probability
density, even as the probability mass for any mathematical point is zero.
For this reason, all point null hypotheses have a probability measure of
zero, but can have a probability density that can be different for prior and
posterior distributions. There still is a meaningful Bayes factor for a point
hypothesis. As described in Chechile (2020),
BF10 = [p(H_1|D)/p(H_1)][p(H_0)/p(H_0|D)]
where D denotes the data.
The first term in this equation is 1/1 = 1. But the second term is of
the form 0/0, which appears to undefined. However, by using L'Hospital's
rule, it can be proved that the term p(H_0)/p(H_0|D) is the ratio of prior
probability density at the null point divided by the posterior probability
density. This method for finding the Bayes factor for a point is called the
Savage-Dickey method because of the separate contributions from both of those
statisticians (Dickey & Lientz, 1970).
Value
A list containing the following components:
method |
The string of either |
a_post |
The value for the posterior beta first shape parameter |
b_post |
The value for the posterior beta second shape parameter |
a0 |
The first shape parameter for the prior beta distribution |
b0 |
The second shape parameter for the prior beta distribution |
BF10 |
The Bayes factor for the alternative over the null hypothesis |
BF01 |
The Bayes factor for the null over the alternative hypothesis |
null_hypothesis |
The value for the null hypothesis when |
H0lower |
The lower limit of the null hypothesis when |
H0upper |
The upper limit of the null hypothesis when |
dpriorH0 |
The prior probability density for the null point when |
dpostH0 |
The posterior probability density for the null point when |
pH0 |
The prior probability for the null hypothesis when |
pH1 |
The prior probability for the alternative hypothesis when |
postH0 |
The posterior probability for the null hypothesis when |
postH1 |
The posterior probability for the alternative hypothesis when |
References
Chechile, R. A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods. MIT Press.
Dickey, J. M., & Lientz, B. P. (1970). The weighted likelihood ratio, sharp hypotheses about chance, the order of a Markov chain. The Annals of Mathematical Statistics, 41, 214-226.
Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford: Oxford University Press.
Kass, R. E., & Rafftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773-795.
Examples
## Examples with the default uniform prior
dfba_beta_bayes_factor(a_post = 17,
b_post = 5,
method = "interval",
H0 = c(0, .5)
)
dfba_beta_bayes_factor(a_post = 377,
b_post = 123,
method = "point",
H0 = .75)
# An example with the Jeffreys prior
dfba_beta_bayes_factor(a_post = 377.5,
b_post = 123.5,
method = "point",
H0 = .75,
a0 = .5,
b0 = .5
)
dfba_beta_bayes_factor(a_post = 273,
b_post = 278,
method = "interval",
H0 = c(.4975,
.5025)
)