Normalized Mutual Information
Also known as: NMI, mutual information, information criterion
Normalized Mutual Information (NMI), popularized by Danon et al. in 2005, is an external clustering evaluation metric based on information theory. It measures the amount of information shared between a predicted clustering and ground truth labels, normalized to a scale between 0 and 1. A value of 1 indicates perfect agreement, while 0 indicates independence.
Key highlights
- Grounded in information theory; captures the concept of shared information
- Ranges from 0 to 1, making values intuitive and comparable
- Symmetric: NMI(C, L) = NMI(L, C)
- Less sensitive to class imbalance compared to some other metrics
- Works with any number of clusters in both partitions
Intuition
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How it works
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When to use it
Use Normalized Mutual Information when you want an information-theoretic measure of agreement between clustering and ground truth. It is particularly useful when comparing clustering results across different datasets or when cluster sizes vary significantly. NMI is sensitive to the amount of information shared and works well for comparing different algorithms on the same problem.
Strengths & limitations
- Grounded in information theory; captures the concept of shared information
- Ranges from 0 to 1, making values intuitive and comparable
- Symmetric: NMI(C, L) = NMI(L, C)
- Less sensitive to class imbalance compared to some other metrics
- Works with any number of clusters in both partitions
- Requires ground truth labels; not suitable for purely unsupervised evaluation
- Can be computationally expensive for large datasets with many clusters
- Interpretation requires understanding of entropy and information theory
- Different normalization schemes (arithmetic vs. geometric) can yield different values
Common pitfalls
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Applications
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Frequently asked
What is the difference between arithmetic and geometric normalization of NMI?
Arithmetic normalization uses (H(C) + H(L))/2 and is more commonly used; geometric normalization uses the square root of the product. Both range from 0 to 1 but have different sensitivity to entropy differences. Arithmetic normalization penalizes disagreement more symmetrically.
When should I use NMI instead of ARI?
NMI is based on information theory and measures shared information; ARI is based on pair counting and corrects for chance. Use NMI when you value information-theoretic interpretations, and ARI when you prefer statistical adjustments for chance agreement. Both are valid; choose based on your framework preference.
Can NMI be used for comparing two unsupervised clusterings without ground truth?
Yes; NMI can measure agreement between any two partitions, even without ground truth labels. Treat one clustering as 'predicted' and the other as 'reference' to measure their mutual information. This usage compares stability or similarity between different clustering results.
Is NMI affected by the number of clusters?
Partially. The entropy of each partition depends on the number and size distribution of clusters. More balanced partitions have higher entropy. NMI reflects these differences, which can make comparisons across different numbers of clusters complex. Normalization helps but does not fully eliminate this effect.
Sources
- 1.Danon, L., Diaz-Guilera, A., Duch, J., & Arenas, A. (2005). Comparing community structure identification. Journal of Statistical Mechanics: Theory and Experiment, 2005(09), P09008.
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Cite this page
ScholarGate. (2026, June 3). Normalized Mutual Information. ScholarGate. https://scholargate.app/model-evaluation/normalized-mutual-information