Quantifying and Reducing Bias in Maximum Likelihood Estimation of Structured Anomalies

Uthsav Chitra, Kimberly Ding, Jasper C.H. Lee, Benjamin J Raphael
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:1908-1919, 2021.

Abstract

Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an anomaly family. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly; however, it was recently observed that for normally distributed data, the MLE is a biased estimator for some anomaly families. In this work, we demonstrate that in the normal means setting, the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. We also provide empirical evidence that the converse is true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families. Next, we derive a new anomaly estimator using a mixture model, and we prove that our anomaly estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the advantages of our estimator versus the MLE on disease outbreak data and highway traffic data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-chitra21a, title = {Quantifying and Reducing Bias in Maximum Likelihood Estimation of Structured Anomalies}, author = {Chitra, Uthsav and Ding, Kimberly and Lee, Jasper C.H. and Raphael, Benjamin J}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {1908--1919}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/chitra21a/chitra21a.pdf}, url = {https://proceedings.mlr.press/v139/chitra21a.html}, abstract = {Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an anomaly family. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly; however, it was recently observed that for normally distributed data, the MLE is a biased estimator for some anomaly families. In this work, we demonstrate that in the normal means setting, the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. We also provide empirical evidence that the converse is true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families. Next, we derive a new anomaly estimator using a mixture model, and we prove that our anomaly estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the advantages of our estimator versus the MLE on disease outbreak data and highway traffic data.} }
Endnote
%0 Conference Paper %T Quantifying and Reducing Bias in Maximum Likelihood Estimation of Structured Anomalies %A Uthsav Chitra %A Kimberly Ding %A Jasper C.H. Lee %A Benjamin J Raphael %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-chitra21a %I PMLR %P 1908--1919 %U https://proceedings.mlr.press/v139/chitra21a.html %V 139 %X Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an anomaly family. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly; however, it was recently observed that for normally distributed data, the MLE is a biased estimator for some anomaly families. In this work, we demonstrate that in the normal means setting, the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. We also provide empirical evidence that the converse is true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families. Next, we derive a new anomaly estimator using a mixture model, and we prove that our anomaly estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the advantages of our estimator versus the MLE on disease outbreak data and highway traffic data.
APA
Chitra, U., Ding, K., Lee, J.C. & Raphael, B.J.. (2021). Quantifying and Reducing Bias in Maximum Likelihood Estimation of Structured Anomalies. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:1908-1919 Available from https://proceedings.mlr.press/v139/chitra21a.html.

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