1. The decision problem

For one homeowner, there are 3 realistic choices:

  1. Remediate now (pay ≈ \$2000, get radon down to 2 pCi/L if it was higher).
  2. Do nothing.
  3. Measure first (pay ≈ \$50 for a good long-term measurement), then—based on the number—either remediate or not.

That’s already a two-stage decision (like the bronchoscopy example): first “measure or not”, then “remediate or not”.

We care about this because radon risk is uncertain before measuring.

2. Why hierarchical Bayes?

Radon levels vary:

  • by house features (basement? basement used?),
  • by county (soil, geology, region),
  • and you may have only a short-term measurement (biased, more variable).

The available national data are of two types:

  • ~5,000 good long-term measurements from 125 counties,
  • ~80,000 noisier / biased short-term measurements from many counties.

So they fit one big hierarchical model to both, to:

  • calibrate short-term to long-term,
  • estimate county-level effects,
  • and get, for every house type in every county, a predictive distribution of radon.

That predictive distribution is then the prior for an individual homeowner.

3. The hierarchical radon model (sketch)

At the house level:

$y_i \sim N(X_i \beta + \alpha_{j(i)}, \sigma_i^2)$

  • $y_i$​ = log radon measurement,
  • $X_i$ = house-level predictors (basement, basement-as-living-space, short-term vs long-term),
  • $\alpha_{j(i)}$ = county effect,
  • $\sigma_i^2$​ bigger for short-term.

At the county level:

$\alpha_j \sim N(W_j \gamma + \delta_{k(j)}, \tau^2)$

  • $W_j$​: county predictors (climate, soil uranium),
  • $\delta_{k(j)}$​: 1 of 19 geology types,
  • $\delta_k \sim N(0, \kappa^2)$.

They fit this by region to allow regional differences.

Outcome: for an unmeasured house i, you get a predictive distribution for

$\theta_i = \log R_i \sim N(M_i, S_i^2)$

That’s your prior for the house’s radon.

Typical numbers:

  • geometric SD (i.e. $e^{S_i}$) ≈ 2.1–2.5 → we’re still uncertain by about a factor of 2
  • geometric mean (i.e. $e^{M_i}$​) usually between 0.6 and 1.6 pCi/L, but can be much higher in bad counties.

4. Decision under certainty

If you somehow knew the true radon level $R$, the question is: at what R should I remediate?

They define:

  • \$2000 ≈ cost of remediation.
  • remediation brings you down to $R_{\text{remed}} = 2$ pCi/L if above, otherwise no change.
  • $D_r$ = dollars per pCi/L per 30 years you’re willing to pay for risk reduction.

Then:

$\text{benefit} = D_r (R_{\text{action}} – R_{\text{remed}})$

Set = \$2000 → solve for action level:

$R_{\text{action}} = \frac{2000}{D_r} + R_{\text{remed}}$

Governments set an $R_{\text{action}}$ (4 pCi/L in the U.S.), which implicitly fixes $D_r$ (and even the implied \$ per microlife).

So: under certainty, the rule is simple: remediate if R > $R_{\text{action}}$.

5. Now add uncertainty (the Bayesian part)

We don’t know R, we only know $\theta = \log R \sim N(M, S^2)$

from the hierarchical model.

We might then measure:

$y \mid \theta \sim N(\theta, \sigma^2)$

(long-term measurement: small $\sigma$; short-term: larger $\sigma$ + bias correction)

Then posterior: $\theta \mid M, y \sim N(\Lambda, V)$

with the usual normal–normal formulas:

$\Lambda = \frac{M/S^2 + y/\sigma^2}{1/S^2 + 1/\sigma^2}, \quad V = \frac{1}{1/S^2 + 1/\sigma^2}$

So: measure → shrink toward prior → smaller variance.

6. Three branches of the decision tree

They compute expected loss (in dollars) for each top-level choice, using

  • \$2000 for remediation
  • \$50 for a long-term measurement
  • the “dollarized” radon exposure using $D_r = 2000/(R_{\text{action}} – R_{\text{remed}})$, so everything’s on the same scale.

Let $R = e^\theta$.

(1) Remediate now

Pay \$2000 now + expected radon exposure after remediation (min(R, 2)):

$L_1 = 2000 + D_r \, E[\min(R, R_{\text{remed}})]$

They write it out in closed form using the lognormal moments.

(2) Do nothing

Just expected radon exposure:

$L_2 = D_r \, E[R] = D_r \, e^{M + \frac12 S^2}$

(3) Measure, then choose

Immediate loss:

  • \$50 for the measurement
  • plus 1/30 of the radon exposure for that year (since you haven’t remediated yet)

Then, conditional on y, you pick the smaller of:

  • “measure + remediate” loss (their 9.10)
  • “measure + don’t remediate” loss (their 9.11)

So the expected loss for branch 3 is $L_3 = E_y \big[ \min(L_{3a}(y), L_{3b}(y)) \big]$

They actually estimate this by simulation over y.

Then you compare $L_1, L_2, L_3$​: pick the smallest.

7. What the policy looks like

If we fix some realistic numbers:

  • measurement σ: log(1.2)
  • prior S: log(2.3) (typical predictive uncertainty)
  • action level $R_{\text{action}} = 4$ pCi/L (U.S. recommendation)

then the optimal rule by prior geometric mean eMe^MeM is:

  • If $e^M < 1.0$ pCi/L → do nothing (about 68% of U.S. houses).
  • If $1.0 \le e^M \le 3.5$ pCi/L → measure (about 27%).
  • If $e^M > 3.5$ pCi/L → remediate now (about 5%).

That’s the content of below image.

So: even if the national action level is 4 pCi/L, most houses shouldn’t even measure, because their prior probability of being above 4 is so small that a \$50 test isn’t worth it.

8. Scaling to the whole U.S.

They then say: suppose everyone follows this optimal rule.

  • ~26–28% of homes would measure
  • ~4–5% would remediate
  • Cost over 30 years ≈ \$7.3B
  • Lives saved ≈ 83,000 (49k smokers + 35k nonsmokers)

Then they compare to simpler but cruder national policies like:

  • “everyone measures short-term, remediate if > 4” (EPA-ish)
  • “everyone measures long-term, remediate if > 4”

Result: the hierarchical, targeted strategy is more efficient:

  • for \$7.3B it saves ~83k lives,
  • the cruder “uncorrected short-term for everybody” would save only ~64k for the same money, or need ~\$12B to save 83k.

So the hierarchical Bayes part isn’t just about nicer estimates—it makes the decision rule more cost-effective when applied at scale.

9. Core takeaways

  1. If you can build a decent hierarchical predictive model, that model itself becomes the prior for individual decisions.
  2. Then you can evaluate “measure or not” as a value-of-information problem, for every house separately.
  3. When many units follow that rule, you can aggregate to get national cost–benefit curves.
  4. And you see that “measure everyone” is worse than “measure only the uncertain ones.”