Individualized Posterior Taste Weight Recovery via Bayes' Rule
Random utility mixed logit models estimate population-level preference distributions, but evaluating individual-level treatment effect heterogeneity requires estimating each specific decision-maker's utility weights. Standard plug-in estimates introduce classical measurement error that attenuates second-stage causal estimates.
Picture this
Imagine estimating a diner's specific love for spice not just from general customer trends, but by combining population trends with the exact dishes that diner ordered when given full menu choice. Bayes' rule uses their specific selections to refine the population average into a personalized preference score.
What the evidence says
Individual posterior weights ($\hat{\beta}_i^S$) ranged from near 0 to above 3.0 across applicants. Because measurement error ($\beta_i^S - \hat{\beta}_i^S$) is orthogonal to the posterior expectation by construction, using $\hat{\beta}_i^S$ as an instrumental variable interaction avoids classical measurement error attenuation bias.
- Who
- N = 2,591 human students in randomized lottery admission priority groups within Charlotte-Mecklenburg Schools, North Carolina.
- How
- Empirical Bayes posterior parameter calculation evaluating individual choice sets and choices using 1,000 Monte Carlo draws from estimated mixed logit population parameter distributions.
What to do
Calculate empirical Bayes posterior expectations of individual random coefficients to create un-attenuated interaction terms for secondary causal regressions.
From the source
"$\hat{\beta}_{i}^{S}$ is a posterior estimate of the weight each parent placed on school tests scores, calculated from our demand model using Bayes' rule... coefficient estimates for terms involving $\hat{\beta}_{i}^{S}$ are not attenuated by the usual measurement error bias because the measurement error $(\beta_{i}^{S}-\hat{\beta}_{i}^{S})$ is uncorrelated with the posterior."
Heterogeneous Preferences and the Efficacy of Public School Choice