Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance

X. NguyenJulien EppsJames Bailey

article2010JMLR2,427 citations

Establishes which information-theoretic clustering comparison measures satisfy metric, normalization, and chance-correction properties, motivating normalized information distance as a principled default.

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Clustering methods are essential for discovering hidden patterns across large datasets, such as biological samples and customer groups. To assess how well these algorithms perform, analysts compare their results against known categories or alternative groupings using comparison scores. While information-based comparison tools are popular due to their ability to detect complex non-linear associations, practitioners have faced ambiguity regarding which specific formula to choose, whether distances satisfy formal geometric properties, and how to prevent random chance from inflating evaluation scores.

The article systematically evaluates, categorizes, and proves the mathematical properties of information-based clustering comparison measures. It aims to establish which measures satisfy essential mathematical criteria, derive exact adjustments for chance agreement, and demonstrate when these adjustments are necessary.

To conduct this evaluation, the authors performed formal mathematical analyses to prove or disprove whether various measures behave as true geometric metrics and whether they remain normalized within a fixed scale. They derived exact statistical adjustments under a permutation framework and established theoretical upper bounds on expected chance agreement. The authors then validated their theoretical findings using numerical simulations and applied them to practical tasks, including eight synthetic and real-world genomic datasets to determine the true number of clusters.

The study established five key findings. First, among the distance measures evaluated, only normalized variation of information and normalized information distance satisfy both the strict definition of a geometric metric and standard scaling between zero and one, with normalized information distance utilizing the available range most effectively. Second, unadjusted measures suffer from significant baseline inflation; as the number of clusters increases, random groupings appear deceptively similar to true classifications. Third, the authors derived an exact adjustment formula that centers the baseline score at zero for purely random clusterings. Fourth, a trade-off exists: applying a chance correction removes the strict geometric metric property. Fifth, mathematical bounds prove that chance adjustment is critical primarily when the ratio of sample items to clusters is small, whereas unadjusted measures naturally approach adjusted values as sample sizes become large relative to the number of clusters (such as when the ratio exceeds 100).

These findings have immediate operational implications for data science workflows, algorithmic design, and research compliance. Unadjusted metrics create substantial risk in small-sample environments, such as rare disease genomics or high-dimensional biomedical studies, where algorithms might falsely favor models with too many clusters simply due to random statistical noise. In contrast, for large datasets, organizations can safely use the unadjusted normalized information distance to preserve true distance geometry, which is beneficial for cluster search and optimization routines.

Based on these results, decision-makers and analytics teams should adopt normalized information distance as their general-purpose metric. Teams must require adjusted-for-chance formulations whenever analyzing datasets with low sample-to-cluster ratios to prevent misleading performance assessments. For large-scale data applications requiring geometric consistency, the unadjusted form should be used.

The authors note limitations, including that adjusted measures lose formal metric properties and that random baseline behavior depends on underlying partition assumptions. Additionally, the exact impact of data-dependent scaling bounds requires careful interpretation during validation. Nevertheless, the theoretical proofs and empirical validations provide high confidence that following these selection guidelines will ensure sound and reproducible clustering evaluations.

CSIRO’s Data61 University of Melbourne University of New South Wales clustering algorithms information theory user information clusterings comparison information theoretic measures mutual information

Nguyen et al (2010).pdf

Paper: Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation, David M. W. Powers (2011). Reading this foundational analysis of chance-corrected performance measures and bias in evaluation metrics provides essential context for understanding the source paper's emphasis on chance correction for clustering comparisons.
Paper: Cluster Ensembles – A Knowledge Reuse Framework for Combining Multiple Partitions, Alexander Strehl et al. (2002). This text establishes mutual information as a core framework for comparing and combining data partitions, directly motivating the information-theoretic measures evaluated in the source paper.

No sufficiently relevant recommendations were found.

Cover for Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance

Abstract

Information theoretic measures form a fundamental class of measures for comparing clusterings, and have recently received increasing interest. Nevertheless, a number of questions concerning their properties and inter-relationships remain unresolved. In this paper, we perform an organized study of information theoretic measures for clustering comparison, including several existing popular measures in the literature, as well as some newly proposed ones. We discuss and prove their important properties, such as the metric property and the normalization property. We then highlight to the clustering community the importance of correcting information theoretic measures for chance, especially when the data size is small compared to the number of clusters present therein. Of the available information theoretic based measures, we advocate the normalized information distance (NID) as a general measure of choice, for it possesses concurrently several important properties, such as being both a metric and a normalized measure, admitting an exact analytical adjusted-for-chance form, and using the nominal [0, 1] range better than other normalized variants.

Topics

clustering algorithms information theory user information

Keywords

clusterings comparison information theoretic measures mutual information Rand index consensus clustering ensemble clustering information distance chance agreement cluster analysis

1. Introduction
2. A Brief Review of Measures for Comparing Clusterings
3. Information Theoretic Based Measures - Variants and Properties
4. Adjustment for Chance
4.1 The Proposed Adjusted Measures
4.2 Properties of the Adjusted Measures
4.3 An Example Application
5. Related Work
6. Conclusion
Acknowledgments
Appendix A. Proofs
References

Knowls

Knowl 1 — Exact Expected Mutual Information Between Clusterings Under the Permutation Model
equation
Under the generalized hypergeometric model of randomness (the permutation model), two partitional clusterings $U = \{U_1, \dots, U_R\}$ and $V = \{V_1, \dots, V_C\}$ of a set of $N$ data items are generated independently subject to fixed cluster sizes $a_i = |U_i|$ for $i=1,\dots,R$ and $b_j = |V_j|$ for $j=1,\dots,C$ (such that $\sum_{i=1}^R a_i = \sum_{j=1}^C b_j = N$ ). The exact analytical expectation of the mutual information $I(U,V)$ between $U$ and $V$ is given by:

$E\{I(U,V)\} = \sum_{i=1}^R \sum_{j=1}^C \sum_{n_{ij}=\max(a_i+b_j-N, 0)}^{\min(a_i,b_j)} \frac{n_{ij}}{N} \log\left(\frac{N n_{ij}}{a_i b_j}\right) \frac{a_i! b_j! (N - a_i)! (N - b_j)!}{N! n_{ij}! (a_i - n_{ij})! (b_j - n_{ij})! (N - a_i - b_j + n_{ij})!}$

where $n_{ij} = |U_i \cap V_j|$ denotes the number of data points common to cluster $U_i$ and cluster $V_j$ . The innermost summation runs over all feasible overlap counts $n_{ij}$ under the hypergeometric distribution constraints for cell $(i,j)$ of the $R \times C$ contingency table.
Knowl 2 — Adjusted Mutual Information and Adjusted Distance Measures
definition
To provide clustering comparison measures with a constant baseline value of zero under random chance, the adjusted mutual information (AMI) applies the chance-adjustment framework to normalized mutual information variants:

$\text{AMI}(U,V) = \frac{I(U,V) - E\{I(U,V)\}}{\text{Bound}(H(U), H(V)) - E\{I(U,V)\}}$

where $U = \{U_1, \dots, U_R\}$ and $V = \{V_1, \dots, V_C\}$ are partitions of $N$ data items, $H(U) = -\sum_{i=1}^R \frac{|U_i|}{N} \log \frac{|U_i|}{N}$ is the entropy of $U$ , $I(U,V) = \sum_{i=1}^R \sum_{j=1}^C \frac{|U_i \cap V_j|}{N} \log \frac{N |U_i \cap V_j|}{|U_i||V_j|}$ is the mutual information, and $E\{I(U,V)\}$ is the expected mutual information under the permutation model.

Depending on the upper bound chosen for normalization, four variants of adjusted mutual information are defined:

$\text{AMI}_{\max}(U,V) = \frac{I(U,V) - E\{I(U,V)\}}{\max\{H(U), H(V)\} - E\{I(U,V)\}}$

$\text{AMI}_{\text{sum}}(U,V) = \frac{I(U,V) - E\{I(U,V)\}}{\frac{1}{2}[H(U) + H(V)] - E\{I(U,V)\}}$

$\text{AMI}_{\text{sqrt}}(U,V) = \frac{I(U,V) - E\{I(U,V)\}}{\sqrt{H(U) H(V)} - E\{I(U,V)\}}$

$\text{AMI}_{\min}(U,V) = \frac{I(U,V) - E\{I(U,V)\}}{\min\{H(U), H(V)\} - E\{I(U,V)\}}$

The corresponding adjusted distance measures are their unit complements, defined as $\text{Ad}(U,V) = 1 - \text{AMI}(U,V)$ . These measures equal $1$ (for similarity) or $0$ (for distance) when $U$ and $V$ are identical, and equal $0$ (similarity) or $1$ (distance) when the mutual information equals its expected value under chance agreement.
Knowl 3 — Metric Property of Normalized Information Distance and Normalized Variation of Information
theoretical result
Let $U$ and $V$ be two partitional clusterings of a finite dataset with $N$ items, with marginal entropies $H(U)$ and $H(V)$ , joint entropy $H(U,V)$ , and mutual information $I(U,V)$ .

The normalized variation of information (NVI):

$d_{\text{joint}}(U,V) = 1 - \frac{I(U,V)}{H(U,V)} = \frac{H(U|V) + H(V|U)}{H(U,V)}$

and the normalized information distance (NID):

$d_{\max}(U,V) = 1 - \frac{I(U,V)}{\max\{H(U), H(V)\}} = \frac{\max\{H(U|V), H(V|U)\}}{\max\{H(U), H(V)\}}$

are both true metrics on the space of clusterings. They satisfy non-negativity, identity of indiscernibles ( $d(U,V)=0 \iff U=V$ ), symmetry ( $d(U,V)=d(V,U)$ ), and the triangle inequality ( $d(U,W) \le d(U,V) + d(V,W)$ for all clusterings $U, V, W$ ).

Both metrics are bounded within $[0, 1]$ . Because $\max\{H(U), H(V)\} \le H(U,V)$ , $d_{\max}$ provides a tighter normalization bound than $d_{\text{joint}}$ , enabling it to use the full nominal range $[0, 1]$ more effectively.
Knowl 4 — Non-Metric Property of Min, Sum, and Sqrt Distance Variants
theoretical result
Among unnormalized and normalized information-theoretic clustering distance measures, the min, sum, and geometric mean variants violate the triangle inequality and are therefore not metrics:
1. The unnormalized distance measures:
$D_{\min}(U,V) = \min\{H(U), H(V)\} - I(U,V)$

$D_{\text{sqrt}}(U,V) = \sqrt{H(U)H(V)} - I(U,V)$

are not metrics. Counterexample: let $X$ and $Y$ be independent binary variables with $P(X=1)=P(X=0)=P(Y=1)=P(Y=0)=1/2$ , and $Z = (X,Y)$ . Then $D_{\min}(X,Y) = 1 > D_{\min}(X,Z) + D_{\min}(Y,Z) = 0$ .
1. The normalized distance measures:
$d_{\min}(U,V) = 1 - \frac{I(U,V)}{\min\{H(U), H(V)\}}$

$d_{\text{sum}}(U,V) = 1 - \frac{2I(U,V)}{H(U) + H(V)}$

$d_{\text{sqrt}}(U,V) = 1 - \frac{I(U,V)}{\sqrt{H(U)H(V)}}$

are not metrics, each failing the triangle inequality on the same counterexample.
Knowl 5 — Upper Bounds on Expected Mutual Information and Asymptotic Convergence
theoretical result
For two random partitional clusterings $U$ (with $R$ clusters of sizes $a_1, \dots, a_R$ ) and $V$ (with $C$ clusters of sizes $b_1, \dots, b_C$ ) on a dataset of $N$ data items under the generalized hypergeometric distribution model of randomness, the expected mutual information $E\{I(U,V)\}$ satisfies:

$E\{I(U,V)\} \le \sum_{i=1}^R \sum_{j=1}^C \frac{a_i b_j}{N^2} \log\left(\frac{N(a_i - 1)(b_j - 1)}{(N - 1)a_i b_j} + \frac{N}{a_i b_j}\right) \le \log\left(\frac{N + RC - R - C}{N - 1}\right)$

Consequently, for fixed cluster counts $R$ and $C$ , the expected mutual information vanishes asymptotically as sample size increases:

$\lim_{N \to \infty} E\{I(U,V)\} = 0$

As $N \to \infty$ , all adjusted mutual information measures $\text{AMI}$ converge to their unadjusted normalized counterparts $\text{NMI}$ . When the ratio of sample size to number of clusters is large (e.g., $N/K \ge 100$ ), $E\{I(U,V)\}$ is close to zero and chance correction is unnecessary.
Knowl 6 — Non-Metric Property of Adjusted Information Distance Measures
theoretical result
All chance-adjusted information-theoretic distance measures:

$\text{Ad}_{\max}(U,V) = 1 - \text{AMI}_{\max}(U,V)$

$\text{Ad}_{\text{sum}}(U,V) = 1 - \text{AMI}_{\text{sum}}(U,V)$

$\text{Ad}_{\text{sqrt}}(U,V) = 1 - \text{AMI}_{\text{sqrt}}(U,V)$

$\text{Ad}_{\min}(U,V) = 1 - \text{AMI}_{\min}(U,V)$

fail to satisfy the triangle inequality and therefore are not true metrics on the space of clusterings.

For $N = 5$ data items, the three clusterings:

$U = \{U_3, U_1, U_1, U_1, U_2\}, \quad V = \{V_2, V_2, V_3, V_1, V_2\}, \quad X = \{X_2, X_1, X_1, X_1, X_2\}$

(where the $k$ -th entry specifies the cluster assignment of data point $k$ ) violate the triangle inequality for these adjusted measures. For any $N = 5 + d$ ( $d \in \mathbb{N}^+$ ), appending $d$ singleton items to $U, V, X$ yields a corresponding counterexample.
Knowl 7 — Metric Property of the Unnormalized Maximum Information Distance
theoretical result
The unnormalized maximum information distance between two partitional clusterings $U$ and $V$ of a set of $N$ items is defined as:

$D_{\max}(U,V) = \max\{H(U), H(V)\} - I(U,V) = \max\{H(U|V), H(V|U)\}$

where $H(U)$ and $H(V)$ are marginal entropies, $I(U,V)$ is mutual information, and $H(U|V)$ and $H(V|U)$ are conditional entropies.

$D_{\max}(U,V)$ is a true metric on the space of clusterings, satisfying non-negativity ( $D_{\max}(U,V) \ge 0$ ), identity of indiscernibles ( $D_{\max}(U,V) = 0 \iff U = V$ ), symmetry ( $D_{\max}(U,V) = D_{\max}(V,U)$ ), and the triangle inequality:

$D_{\max}(U,W) \le D_{\max}(U,V) + D_{\max}(V,W)$

for any clusterings $U, V, W$ . The range of $D_{\max}(U,V)$ is $[0, \log N]$ .
Knowl 8 — Information-Theoretic Similarity and Distance Normalization Bounds
definition
For two partitional clusterings $U = \{U_1, \dots, U_R\}$ and $V = \{V_1, \dots, V_C\}$ of $N$ data points, the mutual information $I(U,V)$ satisfies the bounding inequalities:

$I(U,V) \le \min\{H(U), H(V)\} \le \sqrt{H(U)H(V)} \le \frac{1}{2}(H(U) + H(V)) \le \max\{H(U), H(V)\} \le H(U,V)$

Dividing $I(U,V)$ by each upper bound yields five normalized mutual information (NMI) similarity measures taking values in $[0,1]$ :
- $\text{NMI}_{\text{joint}}(U,V) = \frac{I(U,V)}{H(U,V)}$
- $\text{NMI}_{\max}(U,V) = \frac{I(U,V)}{\max\{H(U), H(V)\}}$
- $\text{NMI}_{\text{sum}}(U,V) = \frac{2I(U,V)}{H(U) + H(V)}$
- $\text{NMI}_{\text{sqrt}}(U,V) = \frac{I(U,V)}{\sqrt{H(U)H(V)}}$
- $\text{NMI}_{\min}(U,V) = \frac{I(U,V)}{\min\{H(U), H(V)\}}$
Subtracting each normalized similarity measure from $1$ gives the corresponding normalized distance measures $d_{\text{joint}}, d_{\max}, d_{\text{sum}}, d_{\text{sqrt}}, d_{\min} \in [0,1]$ .
Knowl 9 — Consensus Index for Determining the Number of Clusters
model/method
The consensus index (CI) estimates the true number of clusters $K^*$ in a dataset by measuring the internal agreement within an ensemble of clusterings generated under perturbations.

For a candidate number of clusters $K \in \{2, \dots, K_{\max}\}$ , an ensemble $\mathcal{U}_K = \{U_1, U_2, \dots, U_B\}$ containing $B$ partitions is generated by applying a clustering algorithm (such as $K$ -means) under perturbations (e.g., random projections or data subsampling). The consensus index is the average pairwise agreement:

$\text{CI}(\mathcal{U}_K) = \frac{\sum_{1 \le i < j \le B} \text{AM}(U_i, U_j)}{B(B-1)/2}$

where $\text{AM}(U_i, U_j)$ is a normalized clustering similarity measure, such as $\text{AMI}_{\max}$ or the adjusted Rand index (ARI). The estimated number of clusters is:

$K^* = \arg\max_{K=2, \dots, K_{\max}} \text{CI}(\mathcal{U}_K)$

When unadjusted measures (such as MI, NMI, or VI) are used for $\text{AM}$ , random chance agreement inflates the baseline similarity as $K$ increases, biasing $K^*$ toward excessively large cluster counts. Adjusted-for-chance measures (AMI, ARI) maintain a zero baseline regardless of $K$ , avoiding this bias.
Knowl 10 — Sensitivity of Clustering Comparison Measures to Skewed Class Distributions
theoretical result
A clustering comparison measure is defective with respect to class distribution if it assigns higher similarity to a clustering with balanced cluster sizes than to a clustering whose cluster size distribution closely matches a skewed ground-truth clustering.

For a ground-truth clustering $U$ with highly skewed cluster sizes (e.g., class sizes $[30, 2, 6, 10, 2]$ ) compared against:
1. A balanced clustering $V$ with cluster sizes $[10, 10, 10, 10, 10]$
2. A skewed clustering $V'$ with cluster sizes $[29, 2, 6, 11, 2]$ matching the structure of $U$
the unadjusted, unnormalized mutual information is defective because it assigns $I(U,V) > I(U,V')$ .

Among normalized and adjusted variants, only the min-normalized measures— $\text{NMI}_{\min}$ , $D_{\min}$ , $d_{\min}$ , and $\text{Ad}_{\min}$ —are defective in this sense. All other normalized and adjusted variants (including $\text{NMI}_{\max}$ , $\text{NMI}_{\text{sum}}$ , $\text{NMI}_{\text{sqrt}}$ , $\text{NMI}_{\text{joint}}$ , $d_{\max}$ , $d_{\text{joint}}$ , and their adjusted forms) correctly assign higher similarity / lower distance to $V'$ over $V$ .

Coverage note — Specific numerical experimental curves on individual microarray datasets are omitted in favor of the general consensus index method and theoretical properties of the measures.

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Citation

MLA

Vinh, N. X., et al. “Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance”. Journal of Machine Learning Research, vol. 11, no. 95, 2010, pp. 2837–54, https://jmlr.org/papers/v11/vinh10a.html.

APA

Vinh, N. X., Epps, J., & Bailey, J. (2010). Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance. Journal of Machine Learning Research, 11(95), 2837–2854. https://jmlr.org/papers/v11/vinh10a.html

Chicago

Vinh, N. X., J. Epps, and J. Bailey. 2010. “Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance”. Journal of Machine Learning Research 11 (95): 2837–54. https://jmlr.org/papers/v11/vinh10a.html.

Harvard

Vinh, N.X., Epps, J. and Bailey, J. (2010) “Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance”, Journal of Machine Learning Research, 11(95), pp. 2837–2854. Available at: https://jmlr.org/papers/v11/vinh10a.html.

Vancouver

1. Vinh NX, Epps J, Bailey J (2010) Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance. Journal of Machine Learning Research 11:2837–2854

BibTeX

@article{vinh2010information,
  title = {Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance},
  author = {Vinh, Nguyen Xuan and Epps, Julien and Bailey, James},
  year = {2010},
  journal = {Journal of Machine Learning Research},
  volume = {11},
  number = {95},
  pages = {2837-2854},
  url = {https://jmlr.org/papers/v11/vinh10a.html}
}

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Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Exact Expected Mutual Information Between Clusterings Under the Permutation Model

Knowl 2 — Adjusted Mutual Information and Adjusted Distance Measures

Knowl 3 — Metric Property of Normalized Information Distance and Normalized Variation of Information

Knowl 4 — Non-Metric Property of Min, Sum, and Sqrt Distance Variants

Knowl 5 — Upper Bounds on Expected Mutual Information and Asymptotic Convergence

Knowl 6 — Non-Metric Property of Adjusted Information Distance Measures

Knowl 7 — Metric Property of the Unnormalized Maximum Information Distance

Knowl 8 — Information-Theoretic Similarity and Distance Normalization Bounds

Knowl 9 — Consensus Index for Determining the Number of Clusters

Knowl 10 — Sensitivity of Clustering Comparison Measures to Skewed Class Distributions

References

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