### Understanding KL Divergence: A Comprehensive Guide

Kullback-Leibler (KL) divergence, also known as relative entropy, is a fundamental concept in information theory. It quantifies the difference between two probability distributions, making it a popular yet occasionally misunderstood metric. This guide explores the math, intuition, and practical applications of KL divergence, particularly its use in drift monitoring.

#### Calculating KL Divergence

KL divergence is a non-symmetric measure of the relative entropy between two distributions, reflecting the information difference between them. In model monitoring, practitioners often use the discrete form of KL divergence, which involves binning data to create discrete distributions. While the continuous and discrete forms of KL divergence converge as sample sizes and bin numbers increase, practical applications typically involve fewer bins and specific techniques to handle bins with zero samples.

#### Using KL Divergence in Model Monitoring

In production environments, KL divergence monitors feature and prediction data to detect significant deviations from a baseline. This baseline could be derived from a training production window or a validation dataset. Drift monitoring is particularly valuable for teams receiving delayed ground truth, as it allows them to use changes in prediction and feature distributions as performance proxies.

KL divergence is applied to individual features, highlighting how each one diverges from baseline values. Each bin contributes to the total KL divergence, providing a comprehensive view of distributional changes.

Note: While perfecting the mathematics of data changes is a common goal, it’s crucial to remember that real data in production evolves constantly. The aim is to maintain a robust and practical metric for effective troubleshooting.

#### Asymmetry in KL Divergence

KL divergence is inherently asymmetric, meaning swapping the baseline and sample distributions results in different values. This asymmetry can complicate comparisons in troubleshooting workflows. To address this, tools like Arize often use the Population Stability Index (PSI), a symmetric variant of KL divergence, for monitoring distribution changes.

#### Differences Between Numeric and Categorical Features

KL divergence can measure differences in both numeric and categorical distributions:

• Numeric Features: Data is binned based on cutoff points, bin sizes, and widths. The chosen binning strategy significantly impacts KL divergence.
• Categorical Features: For categorical data, KL divergence tracks major distributional shifts. High cardinality can reduce the metric’s usefulness, ideally limited to 50–100 unique values.

For high cardinality features, traditional statistical distances might not be effective. Alternative approaches include:

• Embeddings: Monitoring embedding drift for features like User ID or Content ID.
• Pure High Cardinality Categorical: Monitoring the top 50–100 items and grouping others as “other” can be effective.

Additionally, specific monitors can track metrics like the percentage of new values or bins within a period.

#### Conclusion

When considering KL divergence for drift measurement, it’s essential to:

1. Prefer the discrete form, using binning for practical distribution creation.
2. Understand the underlying math and intuition but remain open to other metrics like PSI or alternative approaches based on the use case.

By keeping these principles in mind, practitioners can effectively use KL divergence to monitor and troubleshoot model performance in dynamic production environments.