< List of probability distributions > *Mallows distribution*

## What is the Mallows distribution?

The **Mallows distribution**, or *Mallows model*, is used to model the order of rankings. It is an exponential probability distribution over permutations, based on distances. In the context of the Mallows distribution, *distances *— usually measured with Kendall’s Tau, are measures of similarity between two rankings. Introduced in the mid-1950s by Mallows [1], it is perhaps the most popular member of the distance-based ranking models, which also include the Thurstone, Pearson and Plackett-Luce models.

The Mallows distribution can be viewed as being like a normal distribution for permutations: given a center permutation with the highest probability value (π_{0}) and a parameter for spread (θ,) the model defines a distribution for all permutations [2].

## Mallows distribution properties

The Mallows distribution is parameterized by a parameter *θ*, which controls the amount of “disagreement” in the rankings. If θ = 0, then the Mallows distribution is the uniform distribution, which means that all rankings are equally likely. As θ increases, the Mallows distribution becomes more concentrated around the “most popular” ranking.

Formally, the **Mallows distribution **is a probability distribution that satisfies, for all rankings π ∈ 𝕊_{n}:

where

- θ defines the concentration around the distribution’s peak π
_{0}, - Rankings π
_{0}and θ ≥ 0 are the model parameters, - φ (0) is a normalizing constant [3],
*d*is the distance in 𝕊_{n}, which can be one of many distances including those defined by Kendall, Cayley and Spearman [4].

The most popular method for defining distance is **Kendall’s Tau** which is the number of concordant pairs minus the number of discordant pairs. A concordant pair is a pair of items that are ranked the same way in both rankings, while a discordant pair is a pair of items that are ranked in opposite ways in the two rankings. The **Cayley** **distance **is the number of transpositions (swaps) needed to transform one permutation into another. The distance measure defined by **Spearman **is the *Spearman footrule distance*.

If θ = 0, a uniform distribution is obtained; for θ = +∞, Mallows model assigns probabilities equal to 1 to π_{0} and zero for the remaining permutations [5].

Key challenges in using the Mallows distribution are the exponential support size and the absence of a closed-form expression for choice probabilities [6].

## Other distance-based ranking models

While the most popular distance-based ranking model is Mallows — due to its simplicity — there are other models including Plackett-Luce, Bradley-Terry, Thurstone, and Spearman. These models estimate rankings using item distances based on various factors, such as similarity, voter preferences, or item values.

- The Plackett-Luce model and the Bradley-Terry model are based on pairwise comparisons. Both are popular but more complex.
- The Thurstone model focuses on item similarities and offers flexibility.
- The Spearman model, based on Spearman’s footrule distance, is a non-probabilistic model based on item values and serves as a benchmark.

## Mallows Cp vs Mallows distribution

There is a relationship between the Mallows distribution and Mallows Cp. Mallows Cp serves as a criterion for model selection, and it is based on the Mallows distribution.

Mallows Cp is defined as the expected value of the residual sum of squares (RSS) for a model within the Mallows distribution. The RSS measures the fit of a model to the data, with a smaller RSS indicating a better fit.

In simple terms, Mallows Cp quantifies the expected fit of a model to the data, assuming that the true ranking follows the Mallows distribution. The lower the value of Mallows Cp, the greater the likelihood that the model is a good fit to the data.

## References

[1] Mallows CL (1957) Non-null ranking models. Biometrika 44(1): 114–130.

[2] Chierichetti et. al. Mallows Models for Top-k Lists

[3] Recent Trends in Applied Artificial Intelligence. 26th International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, IEA/AIE 2013, Amsterdam, The Netherlands, June 17-21, 2013, Proceedings. p. 104.

[4] Fligner & Verducci. Distance based ranking models. Journal of the Royal Statistical Society 48(3), 359-369 (1986).

[5] Symbolic and Quantitative Approaches to Reasoning with Uncertainty. 15th European Conference, ECSQARU 2019, Belgrade, Serbia, September 18-20, 2019, Proceedings. p. 353.

[6] Desir et. al. Mallows-Smoothed Distribution over Rankings Approach for Modeling Choice. OPERATIONS RESEARCH. Vol. 69, No. 4, July–August 2021 pp. 1206–1227.