# Chapter 02-S02: Normal–Normal Conjugacy for One Mean

#### Kjell Nygren

#### 2026-07-26

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

## 1. The model

### 1.1 Likelihood

Suppose we observe \(n\) measurements \(y_1, \dots, y_n\) drawn independently from a Normal distribution with unknown mean \(\mu\) and known standard deviation \(\sigma\):

\[ Y_i \mid \mu \;\sim\; \mathcal{N}(\mu,\, \sigma^2), \qquad i = 1,\dots,n. \]

The sufficient statistic is the sample mean \(\bar y\), which itself follows

\[ \bar{y} \mid \mu \;\sim\; \mathcal{N}\!\left(\mu,\, \frac{\sigma^2}{n}\right). \]

In practice \(\sigma\) is rarely truly known, but treating it as fixed provides the cleanest template for understanding posterior updating. The full treatment — unknown mean and unknown variance — is handled in Chapter 3 with `lmb()`.

### 1.2 Prior

Since \(\mu\) can take any real value, the Normal distribution is the natural conjugate prior:

\[ \mu \;\sim\; \mathcal{N}(\mu_0,\, \tau_0^2). \]

Here \(\mu_0\) is your prior best guess and \(\tau_0\) encodes your uncertainty about it.

### 1.3 Posterior derivation

The key is to work with precision (reciprocal variance): \(\kappa = 1/\sigma^2\).

Symbol | Formula | Meaning
--- | --- | ---
\(\kappa_0 = 1/\tau_0^2\) | Prior precision | Information in the prior
\(\kappa_L = n/\sigma^2\) | Likelihood precision | Information in the data

Multiplying the Normal likelihood by the Normal prior kernel and completing the square gives the conjugate posterior:

\[ \mu \mid \bar{y} \;\sim\; \mathcal{N}\!\left(\mu_n,\; \frac{1}{\kappa_n}\right), \]

where

\[ \kappa_n \;=\; \kappa_0 + \kappa_L \qquad\text{(precisions add)} \]

\[ \mu_n \;=\; \frac{\kappa_0\,\mu_0 + \kappa_L\,\bar{y}}{\kappa_n} \;=\; \frac{\kappa_0}{\kappa_n}\,\mu_0 \;+\; \frac{\kappa_L}{\kappa_n}\,\bar{y}. \]

The posterior mean \(\mu_n\) is a precision-weighted average of the prior mean and the data mean.

Intuition.

- Precisions add because information is additive: each independent source of knowledge about \(\mu\) contributes its own precision.
- A very diffuse prior (\(\tau_0 \to \infty\), \(\kappa_0 \to 0\)) gives \(\mu_n \approx \bar y\) — the data take over entirely.
- A very small sample (\(n\) small) or noisy data (\(\sigma\) large) keeps \(\kappa_L\) small, so the prior retains more influence.
- As \(n \to \infty\), \(\mu_n \to \bar y\) and the posterior variance \(1/\kappa_n \to 0\) — the Bayesian estimate converges to the classical MLE.

## 2. `bayesrules` illustration

The `bayesrules` package provides `plot_normal_normal()` and `summarize_normal_normal()` for quickly visualizing and summarizing the Normal–Normal update.

Scenario. A sleep researcher’s prior belief is that undergraduates average about 7 hours of sleep per night. They encode this as \(\mu_0 = 7\), \(\tau_0 = 1.5\) — fairly uncertain, allowing for means anywhere from 4 to 10 hours. The population standard deviation is assumed known at \(\sigma = 1.8\) hours (from large-scale studies). A sample of \(n = 22\) students yields a sample mean of \(\bar y = 6.1\) hours.

    prior_mean <- 7;    prior_sd <- 1.5   ## prior: N(7, 1.5^2)
    sigma      <- 1.8                     ## known population SD
    y_bar      <- 6.1;  n_sleep  <- 22L   ## observed sample

    library(bayesrules)
    ## Overlay prior, likelihood, and posterior
    plot_normal_normal(
      mean       = prior_mean,
      sd         = prior_sd,
      sigma      = sigma,
      y_bar      = y_bar,
      n          = n_sleep,
      prior      = TRUE,
      likelihood = TRUE,
      posterior  = TRUE
    )

    ## Tabular summary of prior, likelihood, and posterior parameters
    summarize_normal_normal(
      mean  = prior_mean,
      sd    = prior_sd,
      sigma = sigma,
      y_bar = y_bar,
      n     = n_sleep
    )
    #>       model    mean    mode       var        sd
    #> 1     prior 7.00000 7.00000 2.2500000 1.5000000
    #> 2 posterior 6.15529 6.15529 0.1382253 0.3717866

Reading the output. The `summarize_normal_normal()` table reports the prior mean and SD, the likelihood (data) mean and SE, and the resulting posterior mean and SD. The plot overlays all three: the wide prior, the sharper likelihood centered at 6.1, and the posterior — narrower than both and sitting between them, pulled strongly toward the data because 22 observations outweigh the prior.

Analytic posterior.

    prec_0 <- 1 / prior_sd^2
    prec_L <- n_sleep / sigma^2
    prec_n <- prec_0 + prec_L

    post_mean <- (prec_0 * prior_mean + prec_L * y_bar) / prec_n
    post_sd   <- sqrt(1 / prec_n)

    nn_analytic <- data.frame(
      Example = "Sleep hours (BayesRules)",
      n = n_sleep,
      `y-bar` = y_bar,
      Posterior = sprintf("N(%.4f, %.4f^2)", post_mean, post_sd),
      Mean = post_mean,
      SD = post_sd,
      check.names = FALSE
    )
    knitr::kable(nn_analytic, digits = 4,
      caption = "Conjugate Normal--Normal posterior (known sigma)")

Conjugate Normal–Normal posterior (known sigma)
Example | n | y-bar | Posterior | Mean | SD
--- | --- | --- | --- | --- | ---
Sleep hours (BayesRules) | 22 | 6.1 | N(6.1553, 0.3718^2) | 6.1553 | 0.3718

## 3. Real data: Old Faithful waiting times

The `faithful` dataset (built into R) records waiting times between eruptions of the Old Faithful geyser in Yellowstone National Park. We treat the waiting time \(Y_i\) as approximately Normal with unknown mean \(\mu\) and use a Normal–Normal conjugate model.

Prior. Based on decades of visitor records, a geologist believes the mean waiting time is around 72 minutes, with considerable uncertainty (\(\tau_0 = 15\) minutes). We estimate the population SD from the data and treat it as fixed at \(\hat\sigma\).

    y_faith <- faithful$waiting
    n_faith <- length(y_faith)

    ybar_faith  <- mean(y_faith)
    sigma_faith <- sd(y_faith)           ## treat as known

    prior_mean_f <- 72;  prior_sd_f <- 15   ## N(72, 15^2) prior

    cat(sprintf("n = %d,  y-bar = %.2f,  sigma = %.2f\n",
                n_faith, ybar_faith, sigma_faith))
    #> n = 272,  y-bar = 70.90,  sigma = 13.59

Analytic posterior.

    prec_0f <- 1 / prior_sd_f^2
    prec_Lf <- n_faith / sigma_faith^2
    prec_nf <- prec_0f + prec_Lf

    post_mean_f <- (prec_0f * prior_mean_f + prec_Lf * ybar_faith) / prec_nf
    post_sd_f   <- sqrt(1 / prec_nf)

    faith_analytic <- data.frame(
      Dataset = "Old Faithful waiting",
      n = n_faith,
      `y-bar` = ybar_faith,
      Posterior = sprintf("N(%.4f, %.4f^2)", post_mean_f, post_sd_f),
      Mean = post_mean_f,
      SD = post_sd_f,
      check.names = FALSE
    )
    knitr::kable(faith_analytic, digits = 4,
      caption = "Conjugate Normal--Normal posterior (known sigma)")

Conjugate Normal–Normal posterior (known sigma)
Dataset | n | y-bar | Posterior | Mean | SD
--- | --- | --- | --- | --- | ---
Old Faithful waiting | 272 | 70.8971 | N(70.9004, 0.8231^2) | 70.9004 | 0.8231

Reading the output. With \(n = 272\) observations, the likelihood precision dominates the prior, so the posterior mean is almost identical to \(\bar y\). The posterior SD is small — the data overwhelm the prior.

### 3.1 Fitting with `lmb()`

In glmbayes, the intercept-only model uses `lmb()` with `dNormal()` and fixed dispersion \(\sigma^2\):

    df_faith <- data.frame(y = y_faith)

    mu_f    <- matrix(prior_mean_f, nrow = 1, ncol = 1,
                      dimnames = list(NULL, "(Intercept)"))
    Sigma_f <- matrix(prior_sd_f^2, nrow = 1, ncol = 1,
                      dimnames = list("(Intercept)", "(Intercept)"))

    pf_faith <- dNormal(mu = mu_f, Sigma = Sigma_f,
                        dispersion = sigma_faith^2)

    set.seed(2026)
    fit_faith <- lmb(
      n       = 20000,
      y ~ 1,
      data    = df_faith,
      pfamily = pf_faith
    )
    print(fit_faith)
    #>
    #> Call:
    #> lmb(formula = y ~ 1, n = 20000, data = df_faith)
    #>
    #> Posterior Mean Coefficients:
    #> (Intercept)
    #>       70.91

    faith_compare <- data.frame(
      Dataset = "Old Faithful waiting",
      Posterior = faith_analytic$Posterior,
      `Analytic Mean` = faith_analytic$Mean,
      `Analytic SD`   = faith_analytic$SD,
      `lmb Post.Mean` = fit_faith$coef.means["(Intercept)"],
      `lmb Post.Sd`   = sd(fit_faith$coefficients[, "(Intercept)", drop = TRUE]),
      check.names = FALSE
    )
    knitr::kable(faith_compare, digits = 4,
      caption = "Analytic vs. lmb() posterior mean and SD")

Analytic vs. lmb() posterior mean and SD
| Dataset | Posterior | Analytic Mean | Analytic SD | lmb Post.Mean | lmb Post.Sd
--- | --- | --- | --- | --- | --- | ---
(Intercept) | Old Faithful waiting | N(70.9004, 0.8231^2) | 70.9004 | 0.8231 | 70.9091 | 0.8261

The `lmb Post.Mean` and `lmb Post.Sd` columns should match the analytic Mean and SD to Monte Carlo error.

## 4. Connection to glmbayes

This scalar case is the template for Gaussian linear models with `lmb()` (Chapter 3). In regression, the single unknown \(\mu\) is replaced by a coefficient vector \(\beta \in \mathbb{R}^p\), the scalar prior variance \(\tau_0^2\) becomes a \(p \times p\) prior covariance matrix \(\Sigma_0\), and the precision-weighted average becomes the matrix formula \(\mu_n = (\Sigma_0^{-1} + X^\top X / \sigma^2)^{-1}(\Sigma_0^{-1}\mu_0 + X^\top y / \sigma^2)\) — but the logic is identical.

## See also

- Chapter-02-S01 — introduction to Bayesian inference and conjugacy.
- Chapter-02-S03 — Beta–Binomial conjugacy for one proportion.
- Chapter-02-S04 — Gamma–Poisson conjugacy for one count rate.
- Chapter-02-S05 — Gamma–Gamma conjugacy for a Gamma response rate.
- Chapter 03 — `lmb()` for multivariate Gaussian linear models.