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This paper considers an anomaly detection problem in which a detection algorithm assigns anomaly scores to multi-dimensional data points, such as cellular networks' Key Performance Indicators (KPIs). We propose an optimization framework to refine these anomaly scores by leveraging side information in the form of a causality graph between the various features of the data points. The refinement block builds on causality theory and a proposed notion of confidence scores. After motivating our framework, smoothness properties are proved for the ensuing mathematical expressions. Next, equipped with these results, a gradient descent algorithm is proposed, and a proof of its convergence to a stationary point is provided. Our results hold (i) for any causal anomaly detection algorithm and (ii) for any side information in the form of a directed acyclic graph. Numerical results are provided to illustrate the advantage of our proposed framework in dealing with False Positives (FPs) and False Negatives (FNs). Additionally, the effect of the graph's structure on the expected performance advantage and the various trade-offs that take place are analyzed.