Cross-City Comparison of Crime-Prediction Models: A Scale Confound in Aggregate Poisson Likelihood, with a Four-City Demonstration

Objectives. Aggregate event-weighted Poisson predictive log-likelihood (PLL) is widely reported when comparing count-prediction models across populations of differing density (e.g., crime prediction across cities). This paper asks whether the metric is fit for that purpose and proposes an alternative metric pair for cross-population evaluation. Methods. The paper documents, algebraically and empirically, that aggregate PLL is confounded by zero-rate composition: the per-event log-likelihood ceiling, max_y log p(y | μ) ≈ −(1/2) log(2πμ), becomes more negative as the predicted mean grows, regardless of model quality. The confound is demonstrated on a four-city LightGBM crime-prediction pipeline (Boston, Leeds, Birmingham, London) using publicly available incident data at H3-resolution-9 monthly granularity, with 5-seed ensembling and per-(city, type) honest decomposition. Results. Three different cities take rank-1 under three metrics — Birmingham on aggregate PLL, Boston on nonzero_pll_mean, London on calibration_ratio — and no city holds rank across all three. Denser populations appear systematically worse on aggregate PLL even when their honest per-event predictive likelihood and calibration are comparable or better. Four configuration-specific gradient-space negative results are also reported; the divergence-detection diagnostic against which they were tested measures sparsity rather than predictive pathology. Conclusions. The paper proposes nonzero_pll_mean and calibration_ratio as the primary cross-population evaluation pair, with zero_rate as descriptive context. Implications for cross-city benchmarking of crime-prediction models are discussed.