Normal Distribution with Known Variance#

Part 1 · Stage 2 · 📍 Single-Parameter Models & Priors · Lesson 015 of 144 · beginner

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Important

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The second workhorse#

After the Beta–Binomial, the normal model with known variance is the most useful closed-form case, and the template for nearly every hierarchical and regression result later in the course. Take \(n\) observations \(y_i \sim \mathrm{N}(\theta, \sigma^2)\) with \(\sigma^2\) known, and a conjugate prior \(\theta \sim \mathrm{N}(\mu_0, \tau_0^2)\).

Precision adds#

The algebra is cleanest in terms of precision — the reciprocal of variance, \(1/\sigma^2\) — because precisions simply add. The posterior is normal,

\[\theta \mid y \;\sim\; \mathrm{N}(\mu_n, \tau_n^2), \qquad \frac{1}{\tau_n^2} = \frac{1}{\tau_0^2} + \frac{n}{\sigma^2},\]

and its mean is the precision-weighted average of the prior mean and the sample mean:

\[\mu_n = \frac{\dfrac{1}{\tau_0^2}\,\mu_0 + \dfrac{n}{\sigma^2}\,\bar{y}} {\dfrac{1}{\tau_0^2} + \dfrac{n}{\sigma^2}} .\]

This is the compromise property again, now with precision as the weight. Data contribute precision \(n / \sigma^2\); the prior contributes \(1/\tau_0^2\), equivalent to \(\sigma^2 / \tau_0^2\) observations. Each new observation increases posterior precision, so the posterior variance can only shrink.

In code#

import numpy as np
from scipy import stats

mu0, tau0, sigma = 0.0, 2.0, 1.0        # prior N(0, 2^2); known sd = 1
y = np.array([1.2, 0.8, 1.5, 0.9])
n, ybar = len(y), y.mean()

prec = 1 / tau0**2 + n / sigma**2
mun = (mu0 / tau0**2 + n * ybar / sigma**2) / prec
post = stats.norm(mun, np.sqrt(1 / prec))
post.mean(), post.interval(0.95)         # ≈ 1.04, (0.61, 1.47)

Where it leads#

Two consequences echo through the rest of the course. As \(n \to \infty\) the prior’s precision becomes negligible and \(\mu_n \to \bar{y}\) — data win, as always. And a flat prior (\(\tau_0 \to \infty\)) gives exactly \(\theta \mid y \sim \mathrm{N}(\bar{y}, \sigma^2/n)\), the familiar sampling-theory result reappearing as a Bayesian posterior. That coincidence, and its limits, is the subject of Stage 4.

Hint