PLRMM (Plackett-Luce Regression Mixture Model)

PLRMM provides a Python implementation of an algorithm for finding clusters within a population of rankers and learning their ranking functions. We call such clusters preference groups. Each ranker is represented by the set of rankings it produces over a set of some items. The items are defined by their features.

PLRMM is a probability model that specifies mixture over observed rankings. Each preference group is identified by its weight vector, which are regression coefficients used to transform a point in the item feature space to its score. The scores are used to induce the ranking over the set of items one wants to rank. The basic intuition is that the higher the score is, the higher the position of the item in the ranking should be. To be precise the scores define the probability distribution (known as Plackett-Luce model) over all possible item permutations and each observed ranking is a sample of this distribution.

If one is interested in a learning to rank model (without the mixture modeling), PLR (Plackett-Luce Regression) is a learning to rank model defined within each preference group of PLRMM and can be modelled as a special case when the expected number of clusters is K = 1.

Keywords: Plackett-Luce Model, Learning to Rank, Probability Model, Expectation Maximization

How to Cite

The detailed description of the algorithms as well as the experiments can be found in the following paper:

Maksim Tkachenko and Hady W. Lauw, Plackett-Luce Regression Mixture Model for Heterogeneous Rankings, CIKM 2016.

If you are using this library in your research, please consider to cite the above paper.

Computer Survey Motivation

One can imagine a scenario where in a survey, M people are asked to rank N computers by their preferences or likelihood of buying the computer. In total we obtain M rankings. In general, there might be more than one rankings produced by the same subject. Each computer is represented with specification attributes including CPU frequency, RAM amount, price, etc.

Based on the subject's background, experience, and requests, the rankings may vary from one person to another. A computer gamer might prefer a computer with a better video card, while an ordinary buyer might be satisfied with the average computer configuration. This observation implies different rankings from the different categories of users. It might be useful to identify these groups for further analysis and to learn their decision making process. And that is exactly what PLRMM does!


python install

The library is implemented for Python 3 and requires a number of additional packages to be installed:

  • NumPy (==1.11.0)
  • SciPy (==0.17.1)
  • Cython (==0.24)

In parenthesis, we specify the versions of the packages that should work fine with the PLRMM.


PLRMM is designed for the use as standalone library that can be manipulated via a number of Python scripts. In case you want to modify or use the model in your code, please refer to the source files.

Training PLRM Model

python [-k <K>] <examples> <model>

-k specifies the number of preference groups (clusters) to be discovered in the dataset.
<examples> is the path to the dataset.
<model> specifies an output file for the trained PLRMM.

The script accepts the number of other optional parameters, such as hyperparameters, number of iterations, etc., for additional information run the script with -h or --help option.

Assigning Group to New Ranker

python <model> <examples> <assignment>

<model> is the path to the trained PLRMM.
<examples> is the path to the dataset file, which is used to infer group assignment for the new rankers given that the trained model is already provided and it should not be modified.
<assignment> specifies an output file where the assignment have to be stored.

Predicting Rankings

python <model> <examples> <assignment> <prediction>

<model> is the path to the trained PLRMM.
<examples> is the path to the dataset file, which is used to predict the ranking, assuming that the group assignments for the rankers are provided.
<assignment> is the path to the group assignment file.
<prediction> specifies an output file for the ranking prediction.

Data Format

Input <examples>

This section specifies the training/test input formats.

The first line of the file contains two integers N and M. N is the number of items available for ranking. M is the number of rankings.

The next N lines specify the items in terms of their features. Each line describes only one item. It contains pairs of integers <index>:<value> separated by spaces. A pair <index>:<value> specifies a feature value: index is non-negative integer, index of the feature; value is a real number, its value. Each of the N items can be referred in the ranking section by a its implicitly assigned index, the first appeared item has index 0, the next 1, and so on.

The next M lines specify the rankings. Each ranking is a set of item indices separated by spaces. The first index identifies item that occupies the first position in the ranking, the second refers to the item that occupies the second position and etc. Each line is for one ranking.

<N> <M>
<index1>:<value1> <index2>:<value2> ... 
<item_index1> <item_index2> ...

Example of three one-hot 'identity' items and two opposite rankings:

3 2
0 1 2
2 1 0

Ranking Output <prediction>

The ranking prediction file (produced by contains only ranking section. It rearranges the item indices with respect to the order implied by the trained model.

Preference Group Assignment Output <assignment>

The group assignment file (produced by is a MATLAB data file (*.MAT) with two fields:

  • 'pz' contains M x K matrix that specifies the posterior probability distribution over the group assignments;
  • 'assignment' contains 1 x M vector that specifies the most probable preference group for each ranking.

NOTE: If one ranker produces several rankings in the dataset, it might be useful to assign all these rankings to the same preference group. This block assignment strategy is not implemented yet at the inference phase, but one may use the probability over possible group assignments ('pz' field of the *.MAT file) to implement, for example, majority vote.

Model <model>

A trained model is stored in MATLAB data file (*.MAT). For the accurate up-to-date information of the stored fields please refer to the source files.

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