Informative Prior Distribution for Cancer Rates#

Part 1 · Stage 2 · 📍 Single-Parameter Models & Priors · Lesson 017 of 144 · beginner

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Important

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The map that lies#

Colour a map of US counties by kidney-cancer rate and something strange appears: the highest-rate counties are mostly small, rural ones. Colour it by the lowest rates and — the same counties appear again. No environmental story explains both. The pattern is a statistical artifact, and it is the classic argument for informative priors.

Small denominators, wild rates#

Kidney cancer is rare. A county of 60,000 people might record 4 cases (about 6.6 per 100,000); a county of 17,000 might record 7 (about 41 per 100,000). One case fewer in the small county would drop its rate by roughly a sixth. The raw rate \(y_j / n_j\) is an unbiased but hopelessly noisy estimate when \(n_j\) is tiny, so the extremes of the ranking are populated not by the most dangerous places, but by the smallest ones.

The Poisson–Gamma remedy#

Model the counts as \(y_j \sim \mathrm{Poisson}(\theta_j n_j)\) with a \(\mathrm{Gamma}(\alpha, \beta)\) prior on the county rate, and the conjugate posterior mean is a weighted average of the county’s own rate and the prior (national) rate:

\[\mathrm{E}[\theta_j \mid y_j] = \frac{\alpha + y_j}{\beta + n_j} = \frac{\beta}{\beta + n_j} \cdot \underbrace{\frac{\alpha}{\beta}}_{\text{prior rate}} \;+\; \frac{n_j}{\beta + n_j} \cdot \underbrace{\frac{y_j}{n_j}}_{\text{raw rate}} .\]

Small counties (\(n_j \ll \beta\)) are pulled hard toward the national rate; large counties keep their own. The prior does not distort — it stabilises.

from scipy import stats
alpha, beta = 20, 1_000_000        # prior: ~20 cases per million person-years
for y, n in [(7, 17_000), (4, 60_000), (250, 1_200_000)]:
    raw = 1e5 * y / n
    shrunk = 1e5 * (alpha + y) / (beta + n)
    print(f"raw {raw:6.1f}   shrunk {shrunk:6.1f}   (n={n:,})")
# small counties move a lot; the large county barely moves

Shrinkage, honestly#

The estimates are shrunk toward a common centre by an amount governed by how much data each county supplies. This buys enormous stability at the price of a small bias toward the mean — an excellent trade when the alternative is ranking noise. And note what this analysis is quietly reaching for: the prior rate \(\alpha/\beta\) should really be estimated from the counties themselves. That is a hierarchical model, and it arrives in Stage 5.

Hint