Bayesian analysis of ranking data with the Extended Plackett–Luce model

Abstract

Multistage ranking models, including the popular Plackett–Luce distribution (PL), rely on the assumption that the ranking process is performed sequentially, by assigning the positions from the top to the bottom one (forward order). A recent contribution to the ranking literature relaxed this assumption with the addition of the discrete-valued reference order parameter, yielding the novel Extended Plackett–Luce model (EPL). Inference on the EPL and its generalization into a finite mixture framework was originally addressed from the frequentist perspective. In this work, we propose the Bayesian estimation of the EPL in order to address more directly and efficiently the inference on the additional discrete-valued parameter and the assessment of its estimation uncertainty, possibly uncovering potential idiosyncratic drivers in the formation of preferences. We overcome initial difficulties in employing a standard Gibbs sampling strategy to approximate the posterior distribution of the EPL by combining the data augmentation procedure and the conjugacy of the Gamma prior distribution with a tuned joint Metropolis–Hastings algorithm within Gibbs. The effectiveness and usefulness of the proposal is illustrated with applications to simulated and real datasets.

Appendix

1.1 An example of EPL model

Here is a simple example to clarify the EPL formulation introduced in (1). Without loss of generality, let us suppose that an $\text {EPL}(\dot{\rho },\underline{\dot{p}})$ with parameter values $\dot{\rho }=(4,1,3,2)$ and $\underline{\dot{p}}=(0.4,0.3,0.2,0.1)$ represents the true data generating mechanism. Under this EPL scenario, the positions are assigned according to an alternating preference scheme: the item selected at the first stage corresponds to the least-liked alternative ($\dot{\rho }(1)=4$); at the second stage, the most-liked item is specified ($\dot{\rho }(2)=1$); the item ranked at the third stage is the one receiving the third position ($\dot{\rho }(3)=3$) and, finally, the remaining alternative of the forth stage is placed second in order of preference ($\dot{\rho }(4)=2$). Regarding the support parameters, they reflect a decreasing first-stage choice probability such that ${\dot{p}}_i\propto (K+1)-i$. Hence, the chance of being ranked last reduces when proceeding from alternative 1 up to alternative 4: item 1 is more likely to be chosen at the first step and, thus, to be ranked last, followed in the order by item 2, 3 and 4.

Since the rank assignment order $\rho$ is not restricted to the identity permutation $\rho _{\text {F}}$ as in the PL, a generic ordering $\pi ^{-1}$ does not coincide with the sequence $\eta ^{-1}=\pi ^{-1}\circ \rho$ listing the items selected at each stage of the ranking process. For example, the considered $\text {EPL}(\dot{\rho },\underline{\dot{p}})$ implies that observing the ordering $\pi ^{-1}=(3,1,4,2)$ corresponds to the sequential item selections indicated by the composition below

$$\begin{aligned} \eta ^{-1}=\pi ^{-1}\circ \dot{\rho }&= (\pi ^{-1}(\dot{\rho }(1)),\pi ^{-1}(\dot{\rho }(2)),\pi ^{-1}(\dot{\rho }(3)),\pi ^{-1}(\dot{\rho }(4)))\\&= (\pi ^{-1}(4),\pi ^{-1}(1),\pi ^{-1}(3),\pi ^{-1}(2))=(2,3,4,1), \end{aligned}$$

that is, one chooses item 2 at the first stage, item 3 at the second stage, item 4 at the third stage and item 1 at the last stage.

Equation (1) states that the probability mass associated to $\pi ^{-1}=(3,1,4,2)$ under the specified $\text {EPL}(\dot{\rho },\underline{\dot{p}})$ can be computed from the PL distribution after rearranging the components of $\pi ^{-1}$ according the reference order $\dot{\rho }$:

$$\begin{aligned} {{\,\mathrm{\mathbf {P}}\,}}_{\text {EPL}}(\pi ^{-1}&=(3,1,4,2) |\dot{\rho },\underline{\dot{p}})={{\,\mathrm{\mathbf {P}}\,}}_{\text {PL}}(\eta ^{-1}=(2,3,4,1) |\underline{\dot{p}}) \\ &= \frac{0.3}{1} \cdot \frac{0.2}{0.4+0.2+0.1} \cdot \frac{0.1}{0.4+0.1} \cdot 1 \approx 0.017. \end{aligned}$$