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A recent wave of research has attempted to define fairness quantitatively. In particular, this work has explored what fairness might mean in the context of decisions based on the predictions of statistical and machine learning models. The rapid growth of this new field has led to wildly inconsistent motivations, terminology, and notation, presenting a serious challenge for cataloging and comparing definitions. This article attempts to bring much-needed order. First, we explicate the various choices and assumptions made—often implicitly—to justify the use of prediction-based decision-making. Next, we show how such choices and assumptions can raise fairness concerns and we present a notationally consistent catalog of fairness definitions from the literature. In doing so, we offer a concise reference for thinking through the choices, assumptions, and fairness considerations of prediction-based decision-making.

This prognostic study validates a machine learning (random forest) prediction model of elevated blood lead levels by comparing with a parsimonious logistic regression among children in a Women, Infants, and Children cohort., Key Points Question How does a machine learning model compare with logistic regression in predicting childhood lead poisoning during infancy? Findings This prognostic study of 6812 children in a Women, Infants, and Children cohort in Chicago, Illinois, used blood lead level surveillance data, housing characteristics, and demographic characteristics to predict lead poisoning risk. Using predictors from a range of spatiotemporal scales, a random forest model significantly outperformed a parsimonious logistic regression. Meaning These findings suggest that machine learning can be used to more precisely predict risk of elevated blood lead levels to guide allocation of prevention resources to children most at risk., Importance Childhood lead poisoning causes irreversible neurobehavioral deficits, but current practice is secondary prevention. Objective To validate a machine learning (random forest) prediction model of elevated blood lead levels (EBLLs) by comparison with a parsimonious logistic regression. Design, Setting, and Participants This prognostic study for temporal validation of multivariable prediction models used data from the Women, Infants, and Children (WIC) program of the Chicago Department of Public Health. Participants included a development cohort of children born from January 1, 2007, to December 31, 2012, and a validation WIC cohort born from January 1 to December 31, 2013. Blood lead levels were measured until December 31, 2018. Data were analyzed from January 1 to October 31, 2019. Exposures Blood lead level test results; lead investigation findings; housing characteristics, permits, and violations; and demographic variables. Main Outcomes and Measures Incident EBLL (≥6 μg/dL). Models were assessed using the area under the receiver operating characteristic curve (AUC) and confusion matrix metrics (positive predictive value, sensitivity, and specificity) at various thresholds. Results Among 6812 children in the WIC validation cohort, 3451 (50.7%) were female, 3057 (44.9%) were Hispanic, 2804 (41.2%) were non-Hispanic Black, 458 (6.7%) were non-Hispanic White, and 442 (6.5%) were Asian (mean [SD] age, 5.5 [0.3] years). The median year of housing construction was 1919 (interquartile range, 1903-1948). Random forest AUC was 0.69 compared with 0.64 for logistic regression (difference, 0.05; 95% CI, 0.02-0.08). When predicting the 5% of children at highest risk to have EBLLs, random forest and logistic regression models had positive predictive values of 15.5% and 7.8%, respectively (difference, 7.7%; 95% CI, 3.7%-11.3%), sensitivity of 16.2% and 8.1%, respectively (difference, 8.1%; 95% CI, 3.9%-11.7%), and specificity of 95.5% and 95.1% (difference, 0.4%; 95% CI, 0.0%-0.7%). Conclusions and Relevance The machine learning model outperformed regression in predicting childhood lead poisoning, especially in identifying children at highest risk. Such a model could be used to target the allocation of lead poisoning prevention resources to these children.

This paper studies the evaluation of methods for targeting the allocation of limited resources to a high-risk subpopulation. We consider a randomized controlled trial to measure the difference in efficiency between two targeting methods and show that it is biased. An alternative, survey-based design is shown to be unbiased. Both designs are simulated for the evaluation of a policy to target lead hazard investigations using a predictive model. Based on our findings, we advised the Chicago Department of Public Health to use the survey design for their field trial. Our work anticipates further developments in economics that will be important as predictive modeling becomes an increasingly common policy tool.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of K\"ahler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer [math] as the sum of [math] triangular numbers is asymptotically equivalent to the modified divisor function [math] .