Relabelling in Bayesian mixture models by pivotal units

A simple procedure based on relabelling to deal with label switching when exploring complex posterior distributions by MCMC algorithms is proposed. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and low computational burden. A possible area where it proves to be useful is when deriving a sample for the posterior distribution arising from finite mixture models when no simple or rational ordering between the components is available

Developments in Bayesian Hierarchical Models and Prior Specification with Application to Analysis of Soccer Data

In the recent years the challenge for new prior specifications and for complex hierarchical models became even more relevant in Bayesian inference. The advent of the Markov Chain Monte Carlo techniques, along with new probabilistic programming languages and new algorithms, extended the boundaries of the field, both in theoretical and applied directions. In the present thesis, we address theoretical and applied tasks. In the first part we propose a new class of prior distributions which might depend on the data and specified as a mixture between a noninformative and an informative prior. The generic prior belonging to this class provides less information than an informative prior and is more likely to not dominate the inference when the data size is small or moderate. Such a distribution is well suited for robustness tasks, especially in case of informative prior misspecification. Simulation studies within the conjugate models show that this proposal may be convenient for reducing the mean squared errors and improving the frequentist coverage. Furthermore, under mild conditions this class of distributions yields some other nice theoretical properties. In the second part of the thesis we use hierarchical Bayesian models for predicting some soccer quantities and we extend the usual match goals’ modeling strategy by including the bookmakers’ information directly in the model. Posterior predictive checks on in-sample and out-of sample data show an excellent model fit, a good model calibration and, ultimately, the possibility for building efficient betting strategies

Relabelling in Bayesian mixture models by pivotal units

4noLabel switching is a well-known and fundamental problem in Bayesian estimation of finite mixture models. It arises when exploring complex posterior distributions by Markov Chain Monte Carlo (MCMC) algorithms, because the likelihood of the model is invariant to the relabelling of mixture components. If the MCMC sampler randomly switches labels, then it is unsuitable for exploring the posterior distributions for component-related parameters. In this paper, a new procedure based on the post-MCMC relabelling of the chains is proposed. The main idea of the method is to perform a clustering technique on the similarity matrix, obtained through the MCMC sample, whose elements are the probabilities that any two units in the observed sample are drawn from the same component. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and very low computational burden.partially_openembargoed_20180828Egidi, Leonardo; Pappadà, Roberta; Pauli, Francesco; Torelli, NicolaEgidi, Leonardo; Pappada', Roberta; Pauli, Francesco; Torelli, Nicol