Recursive inversion models for permutations

Abstract

We develop a new exponential family model for permutations that can capture hierarchical structure in preferences, and that has the well known Mallows models as a subclass. The Recursive Inversions Model (RIM), unlike most distributions over permutations of n items, has a flexible structure, represented by a binary tree. We describe how to compute marginals in the RIM, including the partition function, in closed form. Further we introduce methods for the Maximum Likelihood estimation of parameters and structure search for this model. We demonstrate that this added flexibility both improves predictive performance and enables a deeper understanding of collections of permutations.

Supplementary Figures

Figure 8 shows an example of the type of structure used and found in Sect. 6.1, chosen from $N=1000$ as an example containing model mismatch.

The left model represents the true model and parameters from which a sample was drawn, with the marginal ${\bar{Q}}(\tau (\mathbf {\theta }))$. The right model represents the found structure and parameters using the same search parameters as the experiments, along with the sample ${\hat{Q}}$ that was used to fit the model. The Frobenius norm of the difference between the true ${\bar{Q}}(\tau (\mathbf {\theta }))$ and ${\hat{Q}}$ is $\Vert {\bar{Q}}(\tau (\mathbf {\theta }))-{\hat{Q}}\Vert _F=0.0828$, while the norm between ${\hat{Q}}$ and the marginal ${\bar{Q}}({\hat{\tau }}(\hat{\mathbf {\theta }}))$ (not shown) is only 0.0322, and $\Vert {\bar{Q}}(\tau (\mathbf {\theta }))-{\bar{Q}}({\hat{\tau }}(\hat{\mathbf {\theta }}))\Vert _F$ is 0.0785. This lack of ability to identify the true model from the sample is likely a consequence of two adjacent values of $\theta _i$ being very (0.6924 and 0.762). This can be seen playing out in the misplacement of item $e_6$. Figure 9 shows the training log-likelihoods, in an otherwise identical format to the plots found in Fig. 4. We can see that the alternate initialization has little influence on the final log-likelihood, with many instances finding an identical model to the runs using a more nuanced starting ranking. Similarly, and unsurprisingly, we see the RIM model as a clear winner when comparing testing likelihoods, largely due to the improved flexibility over other models. It is likely that the Rank Inversion model would have fitted the training data better than the $\mathtt HG$ model we see here, which only uses the structure of the found Rank Inversion models but constrains them to the exponential inversion model of shuffle likelihoods.

Fig. 9

These plots show the training log-likelihoods compared against the training log likelihood of the best selected model