Author: Kjell Nygren
Date: 2026-07-19

# Chapter 02-S03: Beta–Binomial Conjugacy for One Proportion

#### Kjell Nygren

#### 2026-07-19

    library(glmbayes)
    if (requireNamespace("bayesrules", quietly = TRUE)) library(bayesrules)

## 1. The model

### 1.1 Likelihood

Suppose we observe \(n\) independent binary trials — each a success (1) or failure (0) — with a common success probability \(\theta \in (0,1)\). The total number of successes \(y = \sum_i y_i\) follows:

\[ y \mid \theta \;\sim\; \mathrm{Binomial}(n,\, \theta). \]

The likelihood is

\[ p(y \mid \theta) \;=\; \binom{n}{y}\,\theta^y\,(1-\theta)^{n-y}, \]

and everything relevant for updating our beliefs about \(\theta\) is captured by the two sufficient statistics \(y\) (successes) and \(n - y\) (failures).

### 1.2 Prior

The parameter \(\theta\) is a probability, so its support is \([0,1]\). The Beta distribution is the natural conjugate prior:

\[ p(\theta) \;=\; \frac{\theta^{a-1}(1-\theta)^{b-1}}{B(a,b)}, \qquad \theta \in [0,1], \]

with shape parameters \(a > 0\) and \(b > 0\). Its mean is \(a/(a+b)\) and its variance is \(ab/[(a+b)^2(a+b+1)]\).

Interpreting the parameters. Think of \(a - 1\) as a “prior number of successes” and \(b - 1\) as a “prior number of failures” accumulated before the current study. The total “prior sample size” is \(a + b - 2\) and the prior mean is \(a/(a+b)\).

Prior belief | Parameter choice
--- | ---
No information — completely flat | \(a = b = 1\) (uniform on \([0,1]\))
Mild belief near 0.5 | \(a = b = 3\)
Moderate belief near 0.4 | \(a = 4, b = 6\) (mean = 0.4)
Strong belief near 0.2 | \(a = 4, b = 16\) (mean = 0.2)

### 1.3 Posterior derivation

Multiply the likelihood kernel \(\theta^y(1-\theta)^{n-y}\) by the prior kernel \(\theta^{a-1}(1-\theta)^{b-1}\):

\[ p(\theta \mid y) \;\propto\; \theta^{(a+y)-1}(1-\theta)^{(b+n-y)-1}. \]

This is the kernel of a Beta distribution, so the conjugate posterior is:

\[ \boxed{\theta \mid y \;\sim\; \mathrm{Beta}(a + y,\; b + n - y).} \]

The update rule is simple:

- Add the number of successes \(y\) to the first shape parameter.
- Add the number of failures \(n - y\) to the second shape parameter.

### 1.4 The posterior mean as a compromise

\[ \mathbb{E}[\theta \mid y] \;=\; \frac{a + y}{a + b + n} \;=\; \frac{a+b}{a+b+n}\cdot\frac{a}{a+b} \;+\; \frac{n}{a+b+n}\cdot\frac{y}{n}. \]

This is a weighted average of the prior mean \(a/(a+b)\) and the data frequency \(y/n\), with weights proportional to the “prior sample size” \(a+b\) and the actual sample size \(n\). When \(n\) is much larger than \(a + b\) the data dominate; when \(n\) is small the prior pulls the estimate toward its own mean.

## 2. `bayesrules` illustration

The `bayesrules` package provides `plot_beta_binomial()` and `summarize_beta_binomial()` for quick visualization and tabular summaries of the Beta–Binomial update (Johnson et al. 2022).

Scenario. A clinical researcher believes, based on earlier trials, that about 40% of patients respond to a new analgesic. They encode this as a Beta(4, 6) prior (mean = 0.4, effective prior sample size = 10). In a new pilot study they observe \(y = 14\) responders out of \(n = 30\) patients.

    a0 <- 4;  b0 <- 6          ## Beta(4, 6) prior: mean = 0.4
    y  <- 14; n  <- 30          ## data: 14/30 successes

    ## Analytic posterior
    post_a <- a0 + y            ## 4 + 14 = 18
    post_b <- b0 + (n - y)      ## 6 + 16 = 22

    library(bayesrules)
    ## Overlay prior, likelihood (scaled), and posterior densities
    plot_beta_binomial(
      alpha      = a0,
      beta       = b0,
      y          = y,
      n          = n,
      prior      = TRUE,
      likelihood = TRUE,
      posterior  = TRUE
    )

    ## Tabular summary of prior, likelihood, and posterior
    summarize_beta_binomial(alpha = a0, beta = b0, y = y, n = n)
    #>       model alpha beta mean      mode         var         sd
    #> 1     prior     4    6 0.40 0.3750000 0.021818182 0.14770979
    #> 2 posterior    18   22 0.45 0.4473684 0.006036585 0.07769547

Reading the output. The posterior parameters are Beta(18, 22), giving a posterior mean of \(18/40 = 0.45\) — pulled from the prior mean (0.40) toward the data frequency (\(14/30 \approx 0.47\)) in proportion to their relative weights. The posterior is noticeably narrower than the prior, reflecting the information gained from 30 observations.

Analytic posterior.

    bb_analytic <- data.frame(
      Example = "Analgesic trial (BayesRules)",
      n = n,
      y = y,
      Posterior = sprintf("Beta(%d, %d)", post_a, post_b),
      Mean = post_a / (post_a + post_b),
      SD = sqrt(post_a * post_b / ((post_a + post_b)^2 * (post_a + post_b + 1))),
      check.names = FALSE
    )
    knitr::kable(bb_analytic, digits = 4,
      caption = "Conjugate Beta--Binomial posterior")

Conjugate Beta–Binomial posterior
Example | n | y | Posterior | Mean | SD
--- | --- | --- | --- | --- | ---
Analgesic trial (BayesRules) | 30 | 14 | Beta(18, 22) | 0.45 | 0.0777

## 3. Real data: the Bechdel test

The Bechdel test asks whether a film features at least two women who talk to each other about something other than a man. The `bayesrules::bechdel` dataset records the test result for 1794 films (Johnson et al. 2022).

Research question. What fraction of films pass the Bechdel test?

Prior. Based on earlier analyses suggesting roughly 45% of films pass, we specify a Beta(9, 11) prior (mean = 0.45, effective sample size = 20 — moderately informative).

    library(bayesrules)

    ## Binary outcome: 1 = PASS, 0 = FAIL
    pass   <- as.integer(bechdel[["binary"]] == "PASS")
    n_bech <- length(pass)
    y_bech <- sum(pass)

    cat(sprintf("n = %d,  y (PASS) = %d,  proportion = %.3f\n",
                n_bech, y_bech, y_bech / n_bech))
    #> n = 1794,  y (PASS) = 803,  proportion = 0.448

    a0_b <- 9;  b0_b <- 11    ## Beta(9, 11) prior: mean = 0.45

    post_a_b <- a0_b + y_bech
    post_b_b <- b0_b + (n_bech - y_bech)

    c(prior_mean  = a0_b / (a0_b + b0_b),
      data_freq   = round(y_bech / n_bech, 4),
      post_mean   = round(post_a_b / (post_a_b + post_b_b), 4))
    #> prior_mean  data_freq  post_mean
    #>     0.4500     0.4476     0.4476

    ## 95% credible interval
    round(qbeta(c(0.025, 0.975), post_a_b, post_b_b), 4)
    #> [1] 0.4248 0.4706

With over 1700 films the data dominate the prior; the posterior mean is nearly identical to the observed proportion.

    analytic_mean_b <- post_a_b / (post_a_b + post_b_b)
    analytic_sd_b   <- sqrt(post_a_b * post_b_b /
                            ((post_a_b + post_b_b)^2 * (post_a_b + post_b_b + 1)))

    bech_analytic <- data.frame(
      Dataset = "Bechdel test",
      n = n_bech,
      y = y_bech,
      Posterior = sprintf("Beta(%d, %d)", post_a_b, post_b_b),
      Mean = analytic_mean_b,
      SD = analytic_sd_b,
      check.names = FALSE
    )
    knitr::kable(bech_analytic, digits = 4,
      caption = "Conjugate Beta--Binomial posterior")

Conjugate Beta–Binomial posterior
Dataset | n | y | Posterior | Mean | SD
--- | --- | --- | --- | --- | ---
Bechdel test | 1794 | 803 | Beta(812, 1002) | 0.4476 | 0.0117

### 3.1 Fitting with `glmb()` using `dBeta()`

The scalar Beta–Binomial model generalizes to `glmb()` with `family = binomial(link = "identity")` and the `dBeta()` prior. In this intercept-only setting the single regression coefficient is \(\theta\) directly on the probability scale.

    df_bech <- data.frame(y = pass)

    bech_beta <- matrix(a0_b / (a0_b + b0_b), nrow = 1, ncol = 1,
                        dimnames = list(NULL, "(Intercept)"))

    bech_pf <- dBeta(shape1 = a0_b, shape2 = b0_b, beta = bech_beta)

    set.seed(2026)
    fit_bech <- glmb(
      n       = 20000,
      y ~ 1,
      data    = df_bech,
      weights = rep(1L, n_bech),
      family  = binomial(link = "identity"),
      pfamily = bech_pf
    )
    print(fit_bech)
    #>
    #> Call:  glmb(formula = y ~ 1, family = binomial(link = "identity"), pfamily = bech_pf,
    #>     n = 20000, data = df_bech, weights = rep(1L, n_bech))
    #>
    #> Posterior Mean Coefficients:
    #> (Intercept)
    #>      0.4477
    #>
    #> Effective Number of Parameters: 0.9969965
    #> Expected Residual Deviance: 2468.272
    #> DIC: 2469.269

    bech_compare <- data.frame(
      Dataset = "Bechdel test",
      Posterior = bech_analytic$Posterior,
      `Analytic Mean` = bech_analytic$Mean,
      `Analytic SD`   = bech_analytic$SD,
      `glmb Post.Mean` = fit_bech$coef.means["(Intercept)"],
      `glmb Post.Sd`   = sd(fit_bech$coefficients[, "(Intercept)", drop = TRUE]),
      check.names = FALSE
    )
    knitr::kable(bech_compare, digits = 4,
      caption = "Analytic vs. glmb() posterior mean and SD")

Analytic vs. glmb() posterior mean and SD
| Dataset | Posterior | Analytic Mean | Analytic SD | glmb Post.Mean | glmb Post.Sd
--- | --- | --- | --- | --- | --- | ---
(Intercept) | Bechdel test | Beta(812, 1002) | 0.4476 | 0.0117 | 0.4477 | 0.0117

The `glmb Post.Mean` and `glmb Post.Sd` columns should match the analytic Mean and SD to Monte Carlo error because `dBeta()` implements the exact conjugate update.

## 4. Connection to glmbayes

This scalar prototype is the intuition behind Binomial `glmb()` (Chapters 7–9), where instead of a single \(\theta\) you model \(\mathrm{logit}(\theta_i) = x_i^\top \beta\) and place a multivariate Normal prior on the coefficient vector \(\beta\). The conjugate `dBeta()` path is specific to intercept-only models with the identity link; once covariates enter, the conjugacy is lost and the general envelope sampler takes over.

## See also

- Chapter-02-S01 — introduction to Bayesian inference and conjugacy.
- Chapter-02-S02 — Normal–Normal conjugacy for one mean.
- Chapter-02-S04 — Gamma–Poisson conjugacy for one count rate.
- Chapter-02-S05 — Gamma–Gamma conjugacy for a Gamma response rate.
- Chapters 07–09 — Binomial regression with `glmb()`.

# References

Johnson, Alicia A., Miles Q. Ott, and Mine Dogucu. 2022. Bayes Rules! An Introduction to Applied Bayesian Modeling. CRC Press. https://www.bayesrulesbook.com.