Computes the Bayes factor for a working theory \(H_1: \theta >
\theta_{\text{cut}}\) against a rival \(H_R: \theta \le
\theta_{\text{cut}}\), given y_W pieces of evidence favorable to
\(H_1\) and y_R pieces favorable to \(H_R\), treated as
\(n = y_W + y_R\) independent Bernoulli draws from an infinite
evidence universe with success probability theta.
Arguments
- y_W
Non-negative integer. Count (or summed integer weight) of evidence favorable to the working theory.
- y_R
Non-negative integer. Count (or summed integer weight) of evidence favorable to the rival.
- omega
Positive numeric. Observation-bias odds ratio. Default
1(unbiased).- prior_a, prior_b
Positive numerics. Beta prior shape parameters on
theta. Default uniform prior,1and1.- theta_cut
Numeric in (0, 1). Cutpoint separating \(H_1\) from \(H_R\). Default
0.5.
Value
A length-1 numeric: the Bayes factor in favor of \(H_1\).
Returns NA_real_ if the integrated likelihood is numerically zero.
Details
This model fits research designs where the evidence universe is
open-ended — ongoing interviews, an expanding archive, a growing
set of cases. For bounded archives, see bf_urn().
Under a Beta(prior_a, prior_b) prior on theta and observation
bias omega = 1 (unbiased), the posterior is
Beta(prior_a + y_W, prior_b + y_R) and the Bayes factor is the
ratio of posterior masses on the two sides of theta_cut. Under
observation bias omega != 1, the probability that an item supports
\(H_1\) given it was observed is `q = omega * theta / (1 + (omega -
theta)
(Fisher's odds-ratio transformation); the Bayes factor is then computed by numerical integration.omega > 1makes pro- \eqn{H_1} items likelier to be observed than they are in the universe;omega < 1` makes them less likely.
For weighted analyses, sum the weights first and pass the sums:
bf_binomial(sum(w_W), sum(w_R)). Integer weights act as effective
replication; see the paper for the rationale.
See also
bf_urn() for the bounded-archive case;
sens_binomial() for sensitivity to omega and prior.