Author: Kjell Nygren
Date: 2026-07-19

# Chapter 02-S05: Gamma–Gamma Conjugacy for One Response Rate

#### Kjell Nygren

#### 2026-07-19

    library(glmbayes)

## 1. The model

### 1.1 Likelihood

Suppose we observe \(n\) positive continuous responses \(y_1, \dots, y_n\) — waiting times, survival times, monetary amounts — drawn from a Gamma distribution with known shape \(k > 0\) and unknown rate \(\beta > 0\):

\[ Y_i \mid \beta \;\sim\; \mathrm{Gamma}(k,\, \beta), \qquad i = 1,\dots,n. \]

The mean response is \(\mu = k/\beta\) and the variance is \(k/\beta^2\). The log-likelihood (up to constants) is:

\[ \ell(\beta) \;\propto\; n k \log\beta - \beta\,\textstyle\sum_i y_i. \]

Why fix the shape? Conjugacy requires \(k\) to be treated as known. In practice you either know it from domain knowledge (e.g., exponential responses have \(k = 1\)) or estimate it by method-of-moments: \(\hat k = \bar y^2 / s^2\). When both \(k\) and \(\beta\) are unknown, conjugacy breaks and a rejection sampler is needed — that case is handled by `Gamma(log)` with `glmbdisp()`.

The special case \(k = 1\) gives the Exponential distribution, the natural model for inter-event waiting times under a constant-hazard assumption.

### 1.2 Prior

Because \(\beta > 0\), the Gamma distribution is again the natural conjugate prior:

\[ \beta \;\sim\; \mathrm{Gamma}(\alpha_0,\, b_0), \]

with prior mean \(\alpha_0 / b_0\) and variance \(\alpha_0 / b_0^2\).

Relationship to the dispersion. In GLM terms the Gamma dispersion is \(\phi = 1/k\), so the shape \(k\) is the precision \(1/\phi\). Specifying `lik_shape = k` in `dGamma(Inv_Dispersion = FALSE)` is equivalent to specifying a known dispersion.

The “identity” link. When you pass `family = Gamma(link = "identity")` to `glmb()`, the single regression coefficient is the rate \(\beta\) directly. This matches the conjugate parameterization.

### 1.3 Posterior derivation

Multiply the likelihood kernel \(\beta^{n k}\,e^{-\beta \sum y_i}\) by the prior kernel \(\beta^{\alpha_0 - 1}\,e^{-b_0 \beta}\):

\[ p(\beta \mid y) \;\propto\; \beta^{(\alpha_0 + n k) - 1}\,e^{-(b_0 + \sum_i y_i)\,\beta}. \]

This is a Gamma kernel, giving the conjugate posterior:

\[ \boxed{\beta \mid y \;\sim\; \mathrm{Gamma}\!\left(\alpha_0 + n k,\;\; b_0 + \textstyle\sum_i y_i\right).} \]

Update rule:

- Add \(n \times k\) to the shape (\(n\) itself when \(k = 1\)).
- Add the sum of observations \(\sum_i y_i\) to the rate.

The posterior mean rate is \((\alpha_0 + n k)/(b_0 + \sum y_i)\), and the posterior mean response is:

\[ \mathbb{E}[\mu \mid y] \;\approx\; \frac{k}{\mathbb{E}[\beta \mid y]} \;=\; \frac{k(b_0 + \sum_i y_i)}{\alpha_0 + n k}. \]

## 2. Illustration with base-R plots

The `bayesrules` package does not currently include a `plot_gamma_gamma()` helper, so we plot the prior, the (normalized) likelihood, and the posterior directly with `curve()`.

Scenario. You record the waiting time between calls to a customer service hotline. Exponential waiting times (\(k = 1\)) are a standard model. You place a \(\Gamma(2, 4)\) prior on the call rate \(\beta\) (prior mean = 0.5 calls/minute). You then observe \(n = 20\) inter-arrival times with sum \(\sum y = 35\) minutes.

    alpha0 <- 2;  b0 <- 4         ## Gamma(2, 4) prior: mean = 0.5
    k      <- 1                   ## known shape (exponential)
    n_obs  <- 20L;  sum_y <- 35   ## data: 20 observations, sum = 35

    post_shape <- alpha0 + n_obs * k
    post_rate  <- b0    + sum_y

    ggamma_analytic <- data.frame(
      Example = "Call inter-arrivals (illustration)",
      n = n_obs,
      `sum(y)` = sum_y,
      Posterior = sprintf("Gamma(%d, %d)", post_shape, post_rate),
      `Mean (rate)` = post_shape / post_rate,
      `SD (rate)`   = sqrt(post_shape) / post_rate,
      check.names = FALSE
    )
    knitr::kable(ggamma_analytic, digits = 4,
      caption = "Conjugate Gamma--Gamma posterior for rate beta")

Conjugate Gamma–Gamma posterior for rate beta
Example | n | sum(y) | Posterior | Mean (rate) | SD (rate)
--- | --- | --- | --- | --- | ---
Call inter-arrivals (illustration) | 20 | 35 | Gamma(22, 39) | 0.5641 | 0.1203

    beta_grid <- seq(0.01, 1.5, length.out = 500)

    ## Normalized likelihood: Gamma(n*k + 1, sum_y) shape, treated as density
    lik_unnorm <- dgamma(beta_grid, shape = n_obs * k + 1, rate = sum_y)

    ## Scale likelihood to have same peak height as the prior (visual only)
    prior_vals   <- dgamma(beta_grid, alpha0, b0)
    post_vals    <- dgamma(beta_grid, post_shape, post_rate)
    lik_scaled   <- lik_unnorm * (max(prior_vals) / max(lik_unnorm))

    plot(beta_grid, prior_vals, type = "l", lwd = 2, col = "steelblue",
         xlab = expression(beta), ylab = "Density",
         main = "Gamma–Gamma update: prior, likelihood, posterior",
         ylim = c(0, max(post_vals) * 1.1))
    lines(beta_grid, lik_scaled, lwd = 2, col = "darkgreen", lty = 2)
    lines(beta_grid, post_vals,  lwd = 2, col = "tomato")
    legend("topright",
           legend = c("Prior  Gamma(2, 4)",
                      "Likelihood (scaled)",
                      sprintf("Posterior  Gamma(%d, %d)", post_shape, post_rate)),
           col  = c("steelblue", "darkgreen", "tomato"),
           lty  = c(1, 2, 1), lwd = 2, bty = "n")
    abline(v = alpha0 / b0,          lty = 3, col = "steelblue")
    abline(v = post_shape / post_rate, lty = 3, col = "tomato")

The posterior is pulled from the prior mean (0.5) toward the data-based MLE (\(n/\sum y = 20/35 \approx 0.57\)) and is considerably tighter than either.

## 3. Real data: Boston Marathon finishing times

The `bayesrules::cherry_blossom_sample` dataset records net finishing times (minutes) for runners in a 10-mile cherry blossom race (Johnson et al. 2022). Finishing times are continuous and positive; we model them as Gamma-distributed with a method-of-moments estimate of the shape \(k\).

    library(bayesrules)

    ## Use net finishing times (minutes); drop missing values
    y_cb <- cherry_blossom_sample$net
    y_cb <- y_cb[is.finite(y_cb)]
    n_cb <- length(y_cb)

    ybar_cb <- mean(y_cb)
    s2_cb   <- var(y_cb)

    ## Method-of-moments estimate of shape k: k_hat = ybar^2 / s^2
    k_hat_cb <- ybar_cb^2 / s2_cb

    cat(sprintf("n = %d,  mean = %.2f min,  var = %.2f,  k_hat = %.3f\n",
                n_cb, ybar_cb, s2_cb, k_hat_cb))
    #> n = 185,  mean = 89.95 min,  var = 195.44,  k_hat = 41.395

The estimated shape \(\hat k \approx\) 41.4 is treated as fixed for the conjugate analysis.

Prior. Based on general knowledge that competitive 10-mile race times cluster around 85–100 minutes, we set a \(\Gamma(5, 0.06)\) prior on the rate \(\beta\) (prior mean rate \(\approx 0.083\), corresponding to a prior mean time \(k/\beta \approx 85\) minutes for the estimated \(k\)).

    ## Set prior centered at rate corresponding to ~85 min mean finish time
    ## E[mu] = k / beta_prior  =>  beta_prior = k_hat / 85
    alpha0_cb <- 5
    b0_cb     <- alpha0_cb / (k_hat_cb / ybar_cb)  ## prior mean rate = k_hat / ybar

    cat(sprintf("Prior: Gamma(%.2f, %.4f),  prior mean rate = %.4f,  implied mean time = %.1f min\n",
                alpha0_cb, b0_cb,
                alpha0_cb / b0_cb,
                k_hat_cb / (alpha0_cb / b0_cb)))
    #> Prior: Gamma(5.00, 10.8644),  prior mean rate = 0.4602,  implied mean time = 89.9 min

    post_shape_cb <- alpha0_cb + n_cb * k_hat_cb
    post_rate_cb  <- b0_cb     + sum(y_cb)

    post_mean_rate_cb <- post_shape_cb / post_rate_cb
    post_sd_rate_cb   <- sqrt(post_shape_cb) / post_rate_cb
    post_mean_time_cb <- k_hat_cb / post_mean_rate_cb

    cb_analytic <- data.frame(
      Dataset = "Cherry blossom finish times",
      n = n_cb,
      `k (fixed)` = k_hat_cb,
      Posterior = sprintf("Gamma(%.2f, %.4f)", post_shape_cb, post_rate_cb),
      `Mean (rate beta)` = post_mean_rate_cb,
      `SD (rate beta)`   = post_sd_rate_cb,
      `Mean time (min)`  = post_mean_time_cb,
      check.names = FALSE
    )
    knitr::kable(cb_analytic, digits = 4,
      caption = "Conjugate Gamma--Gamma posterior for rate beta (shape--rate)")

Conjugate Gamma–Gamma posterior for rate beta (shape–rate)
Dataset | n | k (fixed) | Posterior | Mean (rate beta) | SD (rate beta) | Mean time (min)
--- | --- | --- | --- | --- | --- | ---
Cherry blossom finish times | 185 | 41.3954 | Gamma(7663.15, 16651.0310) | 0.4602 | 0.0053 | 89.9468

### 3.1 Fitting with `glmb()`

    df_cb <- data.frame(y = y_cb)

    cb_beta <- matrix(alpha0_cb / b0_cb, nrow = 1, ncol = 1,
                      dimnames = list(NULL, "(Intercept)"))

    cb_pf <- dGamma(
      shape          = alpha0_cb,
      rate           = b0_cb,
      beta           = cb_beta,
      Inv_Dispersion = FALSE,
      lik_shape      = k_hat_cb
    )

    set.seed(2026)
    fit_cb <- glmb(
      n       = 20000,
      y ~ 1,
      data    = df_cb,
      family  = Gamma(link = "identity"),
      pfamily = cb_pf
    )
    print(fit_cb)
    #>
    #> Call:  glmb(formula = y ~ 1, family = Gamma(link = "identity"), pfamily = cb_pf,
    #>     n = 20000, data = df_cb)
    #>
    #> Posterior Mean Coefficients:
    #> (Intercept)
    #>      0.4603
    #>
    #> Effective Number of Parameters: 40.76645
    #> Expected Residual Deviance: 7790.162
    #> DIC: 62148.07

    cb_compare <- data.frame(
      Dataset = "Cherry blossom finish times",
      Posterior = cb_analytic$Posterior,
      `Analytic Mean (rate)` = cb_analytic$`Mean (rate beta)`,
      `Analytic SD (rate)`   = cb_analytic$`SD (rate beta)`,
      `glmb Post.Mean` = fit_cb$coef.means["(Intercept)"],
      `glmb Post.Sd`   = sd(fit_cb$coefficients[, "(Intercept)", drop = TRUE]),
      check.names = FALSE
    )
    knitr::kable(cb_compare, digits = 4,
      caption = "Analytic vs. glmb() posterior mean and SD for rate beta")

Analytic vs. glmb() posterior mean and SD for rate beta
| Dataset | Posterior | Analytic Mean (rate) | Analytic SD (rate) | glmb Post.Mean | glmb Post.Sd
--- | --- | --- | --- | --- | --- | ---
(Intercept) | Cherry blossom finish times | Gamma(7663.15, 16651.0310) | 0.4602 | 0.0053 | 0.4603 | 0.0052

The `glmb Post.Mean` and `glmb Post.Sd` columns refer to the rate \(\beta\). Mean finish time is \(\hat k\) divided by posterior mean rate (see Mean time in the analytic table).

## 4. Connection to glmbayes

The scalar Gamma–Gamma model is the conjugate prototype for Gamma `glmb()` with `family = Gamma(link = "identity")`. In that setting the single coefficient is the rate \(\beta\), the prior is `dGamma(shape, rate, beta, Inv_Dispersion = FALSE, lik_shape = k)`, and the posterior update is the closed form above.

When the shape \(k\) is unknown, the Gamma prior on \(k\) is no longer conjugate. That case is handled by `Gamma(link = "log")` with `dNormal()` priors on the regression coefficients plus `glmbdisp()` for the dispersion \(\phi = 1/k\) (Chapter 10).

## See also

- Chapter-02-S01 — introduction to Bayesian inference and conjugacy.
- Chapter-02-S02 — Normal–Normal conjugacy for one mean.
- Chapter-02-S03 — Beta–Binomial conjugacy for one proportion.
- Chapter-02-S04 — Gamma–Poisson conjugacy for one count rate.
- Chapter 10 — non-conjugate Gamma regression with `Gamma(log)`.

# 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.