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CPS2203 Thierry D. et al.
restrict the considered set of partitions. A similar approach has been
developped in the Gaussian setting in Devijver and Gallopin (2018).
Once the subset of considered partitions has been built we select the best
partition using a classical model selection approach based on a penalized
likelihood criteria.
The estimator of s* is then defined as the maximun likelihood estimator
dened by the chosen data-driven partition.
We implemented our method and applied it to real data. We used the
MovieLens dataset made of ratings of 137000 users and consider the top 1000
most rated movies.
This paper is organized as follows. After introducing the notations used
throughout the paper, we present our three step method : data-driven pre-
selection of the set of partitions of interest, partition selection using a
penalized likelihood approach and calibration of the penalty using the classical
slope heuristic. The performance of the method is studied using synthetic data
in Section 4.1 and using the MovieLens dataset in Section 4.2.

2. Covariates partitions
2.1 Basic notations
Let p ∈ N be a the number of covariates. Consider the index set {1,...,}. In
the following, we denote by () the ℎ component of a vector and by
() = () ; ∈ ) the group of variables from a cluster B ⊆ {1,...,p}.
Throughout the article, m = {B1,B2,...,BK} will denote a partition of the

covariates into K disjoint clusters B1,B2,...,BK with ∪ = {1, … , }.

=1
Denote by k = |k| the number of variables in the cluster .
Denote by the set of all possible partitions of variables. The set is
large: its size corresponds to the Bell’s number which exponentially growth
with .

2.2 Model collection associated with a partition
Let ∈ be a covariates partition. We associate with a set of
p
probability densities with respect with the uniform measure on {0,1} defined
by

= {() = ∏ ( ( ) )}

=1
where, for any ∈ {1, . . . , }, is a probability density on {0,1} .

3. The Method
Suppose that we observe some data , , . . . , ∈ {0,1} considered as an
p
1
2

p
i.i.d. realizations of an unknown probability distribution on {0,1} . On the
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