Calibrating Discrete Entropy Prediction Floors for Urban Crime Counts: A Multicity Evaluation

Urban forecasting systems increasingly draw on mobility and other dynamic data, yet improved test accuracy alone cannot show whether an added source lowers an information-constrained error bound or merely benefits a particular model. We develop a finite-sample evaluation framework based on the Discrete Entropy Lower Bound (DELB) for integer-valued prediction. DELB maps conditional Shannon entropy to a mean squared error (MSE) lower bound through the integer-lattice maximum-entropy envelope. The empirical analysis separately examines sparse-state estimation bias, evidence from prespecified randomization schemes, and held-out performance. We apply the framework to 503,128 crime events and 86,269,976 flow-eligible bicycle trips from Washington, DC, New York City, and Vancouver during 2020–2022, aggregated into 1,241,768 1 km grid–day observations. In the primary high-dimensional specification, the observed conditional mutual information did not exceed the temporal, spatial, circular-shift, or baseline-state-matched reference distributions. The positive estimated DELB reductions are therefore finite-sample point estimates, not calibrated evidence of source-specific information: they were small over the complete crime-supported populations and 7.9%, 17.3%, and 18.0% within bicycle-covered areas. A lower-cardinality sensitivity detected a small calibrated difference only in New York City (0.00105 bits above the matched-reference mean; Benjamini–Hochberg (BH)-adjusted $q=0.003$), showing that empirical resolution depends on state complexity. In the scale–sample reference experiment, semi-synthetic null rejection was near nominal, whereas sparse-state settings were conservative (Type I error 0.011–0.014) and power was limited for small effects; held-out gains varied across cities and models. The results support a staged evaluation of new urban data sources: the estimated bound change, randomization-based evidence, and realized forecasting gain should be reported separately.