A Bayesian Hierarchical Generalization of Empirical Bayes for Crash Rate Estimation with Missing Traffic Volume

The Empirical Bayes (EB) procedure of Hauer et al. (2002) is the workhorse of highway safety analysis: it combines a Safety Performance Function with observed crash counts to produce shrinkage estimates of segment-level crash rates. EB delivers practicality by holding several quantities fixed at calibration: SPF coefficients, per-type overdispersion, observed ADT, and a fixed exposure exponent. These assumptions strain when ADT is missing on a majority of segments. We present a fully Bayesian hierarchical model that generalizes EB by relaxing each of these assumptions in a single joint inference. Fit on Ohio's road inventory (408,304 segments, 2.9 million crashes, 2013-2025), the model jointly imputes missing ADT and estimates per-segment crash rates with uncertainty. Posterior predictive checks of an initial fixed-exposure model expose a tail misfit; relaxing the exposure structure to a per-functional-class exposure exponent and an estimated length exponent, in place of a single scalar and a fixed offset, resolves it and improves out-of-sample predictive accuracy (PSIS-LOO $\Delta\mathrm{elpd}$ = 9,394, SE 238). Crash count is sublinear in traffic in every class (exposure exponents 0.49-0.70, all $<1$, the safety-in-numbers effect) and sublinear in segment length ($\beta_{\mathrm{len}} = 0.69$). Partial pooling substantially improves out-of-sample predictive accuracy over complete pooling (PSIS-LOO $\Delta\mathrm{elpd}$ = 4,780, SE 225). On equal features, the Bayesian ADT submodel attains $R^2_{\log} = 0.756$ versus $0.653$ for LightGBM. The output is a posterior crash rate distribution per segment, replacing the median-by-type point estimates used in our prior risk-aware routing framework.