Getting started with DrWrinch

library(DrWrinch)

The problem

A researcher classifying within-case evidence for a working theory ( $H_1$ ) against a single rival ( $H_R$ ) wants to summarize the weight of evidence with a Bayes factor. The challenge is that the per-observation probabilities a Bayes factor requires are hard to motivate without overcommitting. DrWrinch supplies two fully specified generative models that produce those probabilities, both tilted against $H_1$ by construction, so the reported Bayes factor is a lower bound on the evidence in favor of the working theory.

This vignette works the running example from Lopez, Bowers, and Gajardo Cooper (2026): nine pieces of evidence favoring $H_1$ and three favoring $H_R$ .

Two models

Binomial model. Each observation is an independent Bernoulli draw from an infinite evidence universe with success probability $\theta$ . $H_1$ corresponds to $\theta > 1/2$ . Under a uniform prior on $\theta$ , the Bayes factor is the ratio of posterior masses on the two sides of $1/2$ . This model fits research designs where the evidence universe is open-ended: ongoing interviews, an expanding archive.

bf_binomial(y_W = 9, y_R = 3)
#> [1] 20.67196

Hypergeometric urn model (Formulation C). Evidence is drawn without replacement from a finite urn. 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 $(y_W, y_W + 1)$ . The construction adds one unobserved pro- $H_1$ item to the working theory’s urn and tilts the rival’s urn toward $H_R$ by exactly one item. This model fits research designs where the evidence base is bounded: a closed historical archive, a fixed roster of documents.

bf_urn(y_W = 9, y_R = 3)
#> [1] 323

The hypergeometric BF of 323 has a closed form: $(10/13) / (120/50388) = 323$ .

When does each model apply?

Design	Model
Ongoing interviews, growing archive, open-ended fieldwork	`bf_binomial()`
Closed historical archive, fixed roster of documents	`bf_urn()`

When in doubt, compute both. They report different but compatible quantities: the binomial conditions on the evidence universe being infinite; the urn conditions on it being bounded.

A property worth knowing

Under symmetric evidence ( $y_W = y_R$ ), both Bayes factors are exactly $1$ – no evidence in either direction.

bf_binomial(5, 5)
#> [1] 1
bf_urn(5, 5)
#> [1] 1

For bf_urn() this is the central property distinguishing Formulation C from naive constructions: the urn model returns 1 rather than $\infty$ or $0$ at the equipoise point.

Sensitivity to observation bias

A reviewer might ask: how robust is the conclusion to the possibility that pro- $H_1$ items were more likely to be observed than pro- $H_R$ items? sens_urn() reports the smallest observation-bias factor $\omega > 1$ at which the Bayes factor first drops below a decision threshold (the default threshold is $20$ , the conventional cutoff for “strong” evidence).

s_urn <- sens_urn(9, 3, threshold = 20)
s_urn$bf
#> [1] 323
s_urn$omega_star
#> [1] 2.433984

The reading: pro- $H_1$ items would have had to be roughly 2.4 times more likely to be noticed than pro- $H_R$ items before the urn conclusion reverses.

Sensitivity to coding error

A peer who cannot re-read every document can still ask how many of the pro- $H_1$ observations would have to be re-coded as pro- $H_R$ before the conclusion changes. sens_coding() reports that tipping point for either model: re-coding $x$ observations moves the counts from $(y_W, y_R)$ to $(y_W - x, y_R + x)$ and recomputes the same Bayes factor.

sens_coding(9, 3, model = "binomial", threshold = 20)$x_star
#> [1] 1
sens_coding(9, 3, model = "urn", threshold = 20)$x_star
#> [1] 2

One re-coding overturns the binomial conclusion and two overturn the hypergeometric. The binomial tolerates less because its Bayes factor (20.67) starts only just above the threshold; the urn starts at 323 and so absorbs one more re-coding before crossing.

Sensitivity to the prior on $\theta$

For the binomial model, sens_binomial() additionally reports the smallest rival-tilted prior at which the conclusion reverses. The prior $\text{Beta}(1, M + 1)$ posits $M$ pseudo-observations all favoring the rival; M_star is the smallest such $M$ that drives the Bayes factor below threshold.

s_binom <- sens_binomial(9, 3, threshold = 20)
s_binom$bf
#> [1] 20.67196
s_binom$omega_star
#> [1] 1.00981
s_binom$M_star
#> [1] 1

At the running example the binomial Bayes factor only just clears the threshold, so M_star is 1: a single rival-favoring pseudo-observation in the prior drops it below 20. A larger evidence base would tolerate more.

Weighted evidence

A researcher who believes one observation carries more probative weight than the others can sum integer weights and pass the totals:

# Eight ordinary pro-H1 items at weight 1 and one "smoking gun" at
# weight 10, matching the paper's weighted running example: W = 8 + 10 = 18.
w_W <- c(10, rep(1, 8))
w_R <- rep(1, 3)
bf_binomial(sum(w_W), sum(w_R))
#> [1] 2336.962
bf_urn(sum(w_W), sum(w_R))
#> [1] 11475735

The “+1” rival-favoring construction goes through unchanged. See the paper’s discussion of probative weight for the rationale.