Quantifying and Reducing Bias in Maximum Likelihood Estimation of Structured Anomalies

TL;DR

This paper investigates the bias in maximum likelihood estimators for structured anomalies, demonstrating how the size of the anomaly family affects bias and proposing a new, unbiased estimator using a mixture model.

Contribution

It provides a theoretical analysis of MLE bias depending on anomaly family size and introduces a novel unbiased estimator applicable to various anomaly structures.

Findings

MLE bias depends on the size of the anomaly family

MLE is asymptotically unbiased if the number of sets is sub-exponential

Proposed mixture model estimator is asymptotically unbiased regardless of family size

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 $\textit{anomaly family}$. 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 $\textit{biased}$ 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 and highway traffic data.