Bayes factor under the hypergeometric urn model

Computes the Bayes factor for a working theory \(H_1\) against a single rival \(H_R\), given observed counts y_W of evidence favorable to \(H_1\) and y_R of evidence favorable to \(H_R\), under Formulation C of the urn construction. The Working Theory Favorable (WTF) sub-model has urn composition \((y_W + 1, \max(1, y_R))\); the Rival Theory Favorable (RTF) sub-model has urn composition \((y_W, y_W + 1)\). Both constructions tilt in favor of the rival, so the Bayes factor returned is a lower bound on the evidence for \(H_1\).

Usage

bf_urn(y_W, y_R, omega = 1)

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).

Value

A length-1 numeric: the Bayes factor in favor of \(H_1\). NA_real_ if the RTF urn is too small for the sample; Inf if the RTF probability of the observed pattern is numerically zero.

Details

This model fits research designs where the evidence base is bounded and fixed — a closed historical archive, a fixed roster of documents. For open-ended evidence collection, see bf_binomial().

Observation bias omega enters as the odds ratio in Fisher's non-central hypergeometric distribution (via BiasedUrn::dFNCHypergeo()): omega = 1 is unbiased; omega > 1 makes pro-\(H_1\) items likelier to be drawn; omega < 1 makes them less likely.

For weighted analyses, sum the weights first and pass the sums: bf_urn(sum(w_W), sum(w_R)).

When y_R > y_W + 1, the RTF urn cannot supply a sample of size \(n = y_W + y_R\), the construction is undefined, and the function returns NA_real_.

See also

bf_binomial() for the open-ended-evidence case; sens_urn() for sensitivity to omega.

Examples

bf_urn(7, 3)
#> [1] 39
bf_urn(7, 3, omega = 0.5)
#> [1] 281.6842