Bayesian Hierarchical Modeling of Serious Crime Rates (LAPD 2024–2025)

Abstract

This project investigates spatial heterogeneity in serious crime rates across the 21 geographic areas of the Los Angeles Police Department (LAPD). Using data from January 2024 to October 2025, we modeled the count of serious offenses (homicide, robbery, aggravated assault) as a Binomial outcome relative to the total reported crimes in each area. A Bayesian hierarchical Beta-Binomial model was implemented to borrow strength across districts, stabilizing estimates for areas with smaller sample sizes. Posterior inference performed via Gibbs sampling revealed significant geographic disparities, with the 77th Street area exhibiting the highest risk (~15%) and West LA the lowest (~2%).

Data

Source: LAPD “Crime Data from 2020 to Present” (Data.gov).
Study Period: January 1, 2024 – October 31, 2025.
Variables:

• Unit of Analysis: 21 LAPD Geographic Areas.
• Serious Crime: Incidents with Primary Crime Code 1 < 300 (e.g., homicide, rape, robbery, aggravated assault).
• Exposure: Total number of crime incidents reported in the area ($N_i$).
• Response: Count of serious crimes ($y_i$).

Model

• Likelihood: The number of serious crimes $y_i$ in area $i$ follows a Binomial distribution:
  $y_i | p_i, N_i \sim \text{Binomial}(N_i, p_i)$.
• Prior: Area-specific rates $p_i$ share a conjugate Beta prior:
  $p_i \sim \text{Beta}(\alpha, \beta)$.
• Hyperprior: The shape parameters $\alpha$ and $\beta$ are assigned independent weakly informative Gamma priors:
  $\alpha, \beta \sim \text{Gamma}(2, 0.5)$.
• Computation: Implemented in JAGS using Gibbs sampling. We ran 3 parallel chains with 100,000 iterations each (after 1,500 burn-in) to approximate the posterior distribution.

Methodology

1. Data Wrangling: Filtered raw incident logs for the 2024–2025 period and aggregated counts by police division to calculate empirical proportions.
2. MCMC Simulation: Executed the JAGS sampler to generate posterior draws for $\alpha$, $\beta$, and all area-specific rates $p_i$.
3. Diagnostics: Confirmed convergence using Trace Plots and the Gelman-Rubin statistic (PSRF ≈ 1.0). Validated model fit via Posterior Predictive Checks (Bayesian p-value ≈ 0.52).
4. Sensitivity Analysis: Compared results against diffuse priors ($\text{Gamma}(1, 0.25)$) to ensure robustness of the posterior estimates.

Downloads

• 📄 Report: STAT431 Final Project Report