  Anomaly detection (AD), also referred to as outlier detection, is a
statistical process aimed at identifying observations within a dataset that
significantly deviate from the expected pattern of the majority of the data.
Such a process finds wide application in various fields, such as finance and
healthcare. While the primary objective of AD is to yield high detection
accuracy, the requirements of explainability and privacy are also paramount.
The first ensures the transparency of the AD process, while the second
guarantees that no sensitive information is leaked to untrusted parties. In
this work, we exploit the trade-off of applying Explainable AI (XAI) through
SHapley Additive exPlanations (SHAP) and differential privacy (DP). We perform
AD with different models and on various datasets, and we thoroughly evaluate
the cost of privacy in terms of decreased accuracy and explainability. Our
results show that the enforcement of privacy through DP has a significant
impact on detection accuracy and explainability, which depends on both the
dataset and the considered AD model. We further show that the visual
interpretation of explanations is also influenced by the choice of the AD
algorithm.
