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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.

We propose the use of probability models for ranked data as a useful alternative to a quantitative data analysis to investigate the outcome of bioassay experiments, when the preliminary choice of an appropriate normalization method for the raw numerical responses is difficult or subject to criticism. We review standard distance-based and multistage ranking models and in this last context we propose an original generalization of the Plackett-Luce model to account for the order of the ranking elicitation process. The usefulness of the novel model is illustrated with its maximum likelihood estimation for a real data set. Specifically, we address the heterogeneous nature of experimental units via model-based clustering and detail the necessary steps for a successful likelihood maximization through a hybrid version of the Expectation-Maximization algorithm. The performance of the mixture model using the new distribution as mixture components is compared with those relative to alternative mixture models for random rankings. A discussion on the interpretation of the identified clusters and a comparison with more standard quantitative approaches are finally provided., (revised to properly include references)

In the context of capture–recapture experiments heterogeneous capture probabilities are often perceived as one of the most challenging features to be incorporated in statistical models. In this paper we propose within a Bayesian framework a new modeling strategy for inference on the unknown population size in the presence of heterogeneity of subject characteristics. Our approach is attractive in that parameters are easily interpretable. Moreover, no parametric distributional assumptions are imposed on the latent distribution of individual heterogeneous propensities to be captured. Bayesian inference based on marginal likelihood by-passes some common identifiability issues, and a formal default prior distribution can be derived. Alternative default prior choices are considered and compared. Performance of our formal default approach is favorably evaluated with two real data sets and with a small simulation study.

The elicitation of an ordinal judgment on multiple alternatives is often required in many psychological and behavioral experiments to investigate preference/choice orientation of a specific population. The Plackett-Luce model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. The present work introduces a Bayesian finite mixture of Plackett-Luce models to account for unobserved sample heterogeneity of partially ranked data. We describe an efficient way to incorporate the latent group structure in the data augmentation approach and the derivation of existing maximum likelihood procedures as special instances of the proposed Bayesian method. Inference can be conducted with the combination of the Expectation-Maximization algorithm for maximum a posteriori estimation and the Gibbs sampling iterative procedure. We additionally investigate several Bayesian criteria for selecting the optimal mixture configuration and describe diagnostic tools for assessing the fitness of ranking distributions conditionally and unconditionally on the number of ranked items. The utility of the novel Bayesian parametric Plackett-Luce mixture for characterizing sample heterogeneity is illustrated with several applications to simulated and real preference ranked data. We compare our method with the frequentist approach and a Bayesian nonparametric mixture model both assuming the Plackett-Luce model as a mixture component. Our analysis on real datasets reveals the importance of an accurate diagnostic check for an appropriate in-depth understanding of the heterogenous nature of the partial ranking data.

Summary. Motivated by the absolute risk predictions required in medical decision making and patient counseling, we propose an approach for the combined analysis of case-control and prospective studies of disease risk factors. The approach is hierarchical to account for parameter heterogeneity among studies and among sampling units of the same study. It is based on modeling the retrospective distribution of the covariates given the disease outcome, a strategy that greatly simplifies both the combination of prospective and retrospective studies and the computation of Bayesian predictions in the hierarchical casecontrol context. Retrospective modeling differentiates our approach from most current strategies for inference on risk factors, which are based on the assumption of a specific prospective model. To ensure modeling flexibility, we propose using a mixture model for the retrospective distributions of the covariates. This leads to a general nonlinear regression family for the implied prospective likelihood. After introducing and motivating our proposal, we present simple results that highlight its relationship with existing approaches, develop Markov chain Monte Carlo methods for inference and prediction, and present an illustration using ovarian cancer data.

Motivated by a real data set deriving from a study on the genetic determinants of the behavior of Mycobacterium tuberculosis (MTB) hosted in macrophage, we take advantage of the presence of control spots and illustrate modelling issues for background correction and the ensuing empirical findings resulting from a Bayesian hierarchical approach to the problem of detecting differentially expressed genes. We prove the usefulness of a fully integrated approach where background correction and normalization are embedded in a single model-based framework, creating a new tailored model to account for the peculiar features of DNA array data where null expressions are planned by design. We also advocate the use of an alternative normalization device resulting from a suitable reparameterization. The new model is validated by using both simulated and our MTB data. This work suggests that the presence of a substantial fraction of exact null expressions might be the effect of an imperfect background calibration and shows how this can be suitably re-calibrated with the information coming from control spots. The proposed idea can be extended to all experiments in which a subset of genes whose expression levels can be ascribed mainly to background noise is planned by design.