Anomaly detection under coordinate transformations

There is a growing need for machine-learning-based anomaly detection strategies to broaden the search for beyond-the-Standard-Model physics at the Large Hadron Collider (LHC) and elsewhere. The first step of any anomaly detection approach is to specify observables and then use them to decide on a set of anomalous events. One common choice is to select events that have low probability density. It is a well-known fact that probability densities are not invariant under coordinate transformations, so the sensitivity can depend on the initial choice of coordinates. The broader machine learning community has recently connected coordinate sensitivity with anomaly detection and our goal is to bring awareness of this issue to the growing high-energy physics literature on anomaly detection. In addition to analytical explanations, we provide numerical examples from simple random variables and from the LHC Olympics dataset that show how using probability density as an anomaly score can lead to events being classified as anomalous or not depending on the coordinate frame.