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  1. With an increased focus on incorporating fairness in machine learning models, it becomes imperative not only to assess and mitigate bias at each stage of the machine learning pipeline but also to understand the downstream impacts of bias across stages. Here we consider a general, but realistic, scenario in which a predictive model is learned from (potentially biased) training data, and model predictions are assessed post-hoc for fairness by some auditing method. We provide a theoretical analysis of how a specific form of data bias, differential sampling bias, propagates from the data stage to the prediction stage. Unlike prior work, we evaluate the downstream impacts of data biases quantitatively rather than qualitatively and prove theoretical guarantees for detection. Under reasonable assumptions, we quantify how the amount of bias in the model predictions varies as a function of the amount of differential sampling bias in the data, and at what point this bias becomes provably detectable by the auditor. Through experiments on two criminal justice datasets– the well-known COMPAS dataset and historical data from NYPD’s stop and frisk policy– we demonstrate that the theoretical results hold in practice even when our assumptions are relaxed. 
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  2. We propose a new approach, the calibrated nonparametric scan statistic (CNSS), for more accurate detection of anomalous patterns in large-scale, real-world graphs. Scan statistics identify connected subgraphs that are interesting or unexpected through maximization of a likelihood ratio statistic; in particular, nonparametric scan statistics (NPSSs) identify subgraphs with a higher than expected proportion of individually significant nodes. However, we show that recently proposed NPSS methods are miscalibrated, failing to account for the maximization of the statistic over the multiplicity of subgraphs. This results in both reduced detection power for subtle signals, and low precision of the detected subgraph even for stronger signals. Thus we develop a new statistical approach to recalibrate NPSSs, correctly adjusting for multiple hypothesis testing and taking the underlying graph structure into account. While the recalibration, based on randomization testing, is computationally expensive, we propose both an efficient (approximate) algorithm and new, closed-form lower bounds (on the expected maximum proportion of significant nodes for subgraphs of a given size, under the null hypothesis of no anomalous patterns). These advances, along with the integration of recent core-tree decomposition methods, enable CNSS to scale to large real-world graphs, with substantial improvement in the accuracy of detected subgraphs. Extensive experiments on both semi-synthetic and real-world datasets are demonstrated to validate the effectiveness of our proposed methods, in comparison with state-of-the-art counterparts. 
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  3. We generalize the spatial and subset scan statistics from the single to the multiple subset case. The two main approaches to defining the log-likelihood ratio statistic in the single subset case—the population-based and expectation-based scan statistics—are considered, leading to risk partitioning and multiple cluster detection scan statistics, respectively. We show that, for distributions in a separable exponential family, the risk partitioning scan statistic can be expressed as a scaled f-divergence of the normalized count and baseline vectors, and the multiple cluster detection scan statistic as a sum of scaled Bregman divergences. In either case, however, maximization of the scan statistic by exhaustive search over all partitionings of the data requires exponential time. To make this optimization computationally feasible, we prove sufficient conditions under which the optimal partitioning is guaranteed to be consecutive. This Consecutive Partitions Property generalizes the linear-time subset scanning property from two partitions (the detected subset and the remaining data elements) to the multiple partition case. While the number of consecutive partitionings of n elements into t partitions scales as O(n^(t−1)), making it computationally expensive for large t, we present a dynamic programming approach which identifies the optimal consecutive partitioning in O(n^2 t) time, thus allowing for the exact and efficient solution of large-scale risk partitioning and multiple cluster detection problems. Finally, we demonstrate the detection performance and practical utility of partition scan statistics using simulated and real-world data. Supplementary materials for this article are available online. 
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  4. Advances in medical imaging technology have created opportunities for computer‐aided diagnostic tools to assist human practitioners in identifying relevant patterns in massive, multiscale digital pathology slides. This work presents Hierarchical Linear Time Subset Scanning, a novel statistical method for pattern detection. Hierarchical Linear Time Subset Scanning exploits the hierarchical structure inherent in data produced through virtual microscopy in order to accurately and quickly identify regions of interest for pathologists to review. We take a digital image at various resolution levels, identify the most anomalous regions at a coarse level, and continue to analyze the data at increasingly granular resolutions until we accurately identify its most anomalous subregions. We demonstrate the performance of our novel method in identifying cancerous locations on digital slides of prostate biopsy samples and show that our methods detect regions of cancer in minutes with high accuracy, both as measured by the ROC curve (measuring ability to distinguish between benign and cancerous slides) and by the spatial precision‐recall curve (measuring ability to pick out the malignant areas on a slide which contains cancer). Existing methods need small scale images (small areas of a slide preselected by the pathologist for analysis, eg, 32 × 32 pixels) and may not work effectively on large, raw digitized images of size 100K × 100Kpixels. In this work, we provide a methodology to fill this significant gap by analyzing large digitized images and identifying regions of interest that may be indicative of cancer.

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  5. Summary

    We propose a new ‘fast subset scan’ approach for accurate and computationally efficient event detection in massive data sets. We treat event detection as a search over subsets of data records, finding the subset which maximizes some score function. We prove that many commonly used functions (e.g. Kulldorff’s spatial scan statistic and extensions) satisfy the ‘linear time subset scanning’ property, enabling exact and efficient optimization over subsets. In the spatial setting, we demonstrate that proximity-constrained subset scans substantially improve the timeliness and accuracy of event detection, detecting emerging outbreaks of disease 2 days faster than existing methods.

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