# Evidential machine learning

These days I skim almost every thread on R-devel, which is the email list for discussion about improving base R and its internal functionality. I also occasionally skim selected threads in R-package-devel, which is the email list for discussion about R package development (how to get your package accepted on CRAN).

Since I have studied/lived in France for 4 years, I am usually excited
to read messages from French colleagues on these email lists. In one
thread
I noticed the French name Thierry Denouex, who wrote for help about
his `evclust`

package. I studied clustering during my PHD, and in my
ICML’11 clusterpath
paper, I proposed
a new algorithm for convex clustering. So I was interested to read
about Thierry’s proposed clustering algorithm.

I followed the link in his email signature to his web page, where I found that he is a member of the HEUDIASYC laboratory. Coincidentally that is also the workplace of Yves Grandvalet, who was one of the people who officially reviewed and approved my PHD. Small world!

About clustering, the goal is to group N observations into K clusters, where each cluster should contain a subset of observations which are similar in some sense. Classical algorithms include hierarchical/agglomerative clustering, K-means, and Gaussian mixture models (Expectation-Maximization). The evclust package vignette begins by reviewing these and other algorithms (fuzzy k-means, sparse k-means, etc). K-means is an example of a “hard” clustering algorithm, because each observation is assigned to exactly one cluster (an integer value from 1 to K). In contrast, “soft” clustering algorithms like Gaussian mixtures give each observation a vector of K real numbers — one for each cluster, larger values mean that cluster is more likely for this observation. Typically these numbers are constrained to be on the probability simplex (non-negative, sum to one). When the sum to one constraint is removed, we obtain “possibilistic clustering.” When each observation is assigned a set of clusters, we obtain “rough clustering.”

The “evidential clustering” approach uses mass functions and focal sets, and generalizes most of these other methods. It is based on the the Dempster-Shafer theory that assumes a question has one answer among a finite set of possibilities. The mass function takes a subset of possibilities, and returns a value in [0,1]. Each subset with mass function value greater than 0 is called a focal set, and summing over all subsets must yield one. To apply this formalism to clustering, the set of K clusters is used as the finite set of possibilities, and each observation has a corresponding mass function (the N-tuple of mass functions is called a credal/evidential partition). Some special cases are

- When mass functions are Bayesian (focal sets are singletons, meaning only one element in each set) we get a fuzzy/soft partition (vector of real numbers, one for each cluster).
- When mass functions are logical (only one focal set) we get a rough partition (each observation assigned a set of clusters).
- When mass functions are certain (both Bayesian and logical, meaning the only focal set is a singleton containing the cluster ID for that observation) we get a hard partition (like K-means).
- When mass functions are consonant (focal sets are nested, meaning for any two sets, one must be a subset of the other) we get possibilistic clustering algorithms (vector of real numbers with no sum to one constraint).

How interesting! I wonder why this alternative to / generalization of probability is not more commonly discussed in the statistical machine learning literature? Thierry’s Belief Functions and Machine Learning page lists dozens of publications on the subject. He implemented classification algorithms based on Belief functions theory in the evclass package, which can handle “the case where there is an unknown class, not represented in the learning set.”