# Difference between Weighted Average Entropy and Adjusted Mutual Information (for evaluating Clustering)

I was advised by my team leader to use this weighted average entropy to evaluating the performance of my dbscan clustering algorithm, and its mathematical formulation is: Scikit provides what many would consider standard metrics for clustering performance evaluation such as Adjusted Mutual Information, Homogeneity, completeness and V-measure (which are all based on the calculation of entropy.

I wonder the difference and relationship between two metics, aka Weighted Average Entropy and Adjusted Mutual Information? And are they both good for clustering evaluation? Thanks!

• None is a good measure for evaluating clusterings. Evaluation of clusterings is a black art, and needs lots of manual labor. Looking at some number is not good. – Has QUIT--Anony-Mousse Sep 25 '14 at 8:48
• If the ground truth labels are not known, evaluation must be performed using the model itself. The Silhouette Coefficient (sklearn.metrics.silhouette_score) is an example of such an evaluation scikit-learn.org/stable/modules/clustering.html – KLDavenport May 31 '15 at 16:24

## 1 Answer

It looks like that $\mbox{WAE}$ is just a conditional entropy. If $D$ is your class (ground truth clustering) and $C$ is the clustering solution you generated: $$\mbox{WAE} = H(D|C) = - \sum_{j=1}^c \frac{|c_j|}{n} \sum_{i=1}^cP_{i|j} \log_2{P_{i|j}}$$ (There is one minus missing in your formula)

• $H(D|C) = 0$ when your clustering $C$ is identical to the ground truth $D$
• $H(D|C) = H(D)$ when the clustering solution $C$ is "bad", which is when the random variables corresponding to $D$ and $C$ are independent.

The Adjusted Mutual Information ($\mbox{AMI}$) employs the mutual information ($\mbox{MI}$) to compare clusterings: $$\mbox{MI} = H(D) - H(D|C) = H(D) - \mbox{WAE}$$ $\mbox{MI}$ uses $\mbox{WAE}$ to obtain a similarity measure which has to be maximized: the bigger the better. $$\mbox{AMI} = \frac{\mbox{MI} - E[\mbox{MI}]}{\max{\mbox{MI}} - E[\mbox{MI}]}$$

$\mbox{AMI}$ is a further rescaling of $\mbox{MI}$ to obtain:

• A maximum value of $1$ when clusterings are identical;
• $0$ value when $\mbox{MI}$ is equal to its expected value obtained by random chance agreement between $C$ and $D$.

Given that there are no explicit random variables involved, the baseline value of $0$ for $\mbox{AMI}$ makes more sense.