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I modelli mistura bayesiani sono onnipresenti in statistica per la loro semplicità e flessibilità e possono essere facilmente impiegati in un'ampia varietà di contesti. In questa tesi, miriamo a fornire alcuni contributi agli attuali metodi bayesiani di analisi dei dati, spesso motivati ​​da domande di ricerca provenienti da applicazioni biologiche. In particolare, ci concentriamo sullo sviluppo di nuovi modelli mistura bayesiani, tipicamente in un ambiente non parametrico, per migliorare ed estendere aree di ricerca che coinvolgono dati caratterizzati da grande dimensioni: la modellazione di dati nested, test di ipotesi simultaneo e la riduzione della dimensionalità. \\ Pertanto, il nostro obiettivo è duplice: sviluppare metodi statistici robusti motivati da un solido background teorico e proporre algoritmi efficienti, scalabili e trattabili per le loro applicazioni. \\ La tesi è organizzata come segue. Nel capitolo 1 esamineremo brevemente il background metodologico e discuteremo i concetti necessari che appartengono alle diverse aree a cui contribuiremo con questa tesi. \\ Nel capitolo 2 proponiamo un modello di atomi comuni (CAM) per nested data, che supera le limitazioni del processo del nested Dirichlet Process, come discusso in \ citep {Camerlenghi2018}. Deriviamo le sue proprietà teoriche e sviluppiamo uno slice sampler per dati nested al fine di ottenere un algoritmo efficiente per la simulazione della posterior. Abbiamo poi incorporato il modello in un framework di Rounded mixture of Gaussian Kernels, così da applicare il nostro metodo a una abundance table derivante da uno studio di microbioma. \\ Nel capitolo \ref {BNPT} sviluppiamo una versione BNP del two-group model, modellando sia $ f_0 $ che $ f_1 $ con Pitman-Yor mixtures models. Proponiamo di fissare i due parametri $ \sigma_0 $ e $ \sigma_1 $ in modo che $ \sigma_0> \sigma_1 $, in base alla logica secondo cui il PY che modella la distribuzione nulla dovrebbe essere più vicino alla sua mi, Bayesian mixture models are ubiquitous in statistics due to their simplicity and flexibility and can be easily employed in a wide variety of contexts. In this dissertation, we aim at providing a few contributions to current Bayesian data analysis methods, often motivated by research questions from biological applications. In particular, we focus on the development of novel Bayesian mixture models, typically in a nonparametric setting, to improve and extend active research areas that involve large-scale data: the modeling of nested data, multiple hypothesis testing, and dimensionality reduction.\\ Therefore, our goal is twofold: to develop robust statistical methods motivated by a solid theoretical background, and to propose efficient, scalable and tractable algorithms for their applications.\\ The thesis is organized as follows. In Chapter \ref{intro} we shortly review the methodological background and discuss the necessary concepts that belong to the different areas that we will contribute to with this dissertation. \\ In Chapter \ref{CAM} we propose a Common Atoms model (CAM) for nested datasets, which overcomes the limitations of the nested Dirichlet Process, as discussed in \citep{Camerlenghi2018}. We derive its theoretical properties and develop a slice sampler for nested data to obtain an efficient algorithm for posterior simulation. We then embed the model in a Rounded Mixture of Gaussian kernels framework to apply our method to an abundance table from a microbiome study.\\ In Chapter \ref{BNPT} we develop a BNP version of the two-group model \citep{Efron2004}, modeling both the null density $f_0$ and the alternative density $f_1$ with Pitman-Yor process mixture models. We propose to fix the two discount parameters $\sigma_0$ and $\sigma_1$ so that $\sigma_0>\sigma_1$, according to the rationale that the null PY should be closer to its base measure (appropriately chosen to be a standard Gaussian base measure), while the alternative PY should have fewer constr

This paper is meant to introduce a significant extension of the flexible Dirichlet (FD) distribution, which is a quite tractable special mixture model for compositional data, i.e. data representing vectors of proportions of a whole. The FD model displays several theoretical properties which make it suitable for inference, and fairly easy to handle from a computational viewpoint. However, the rigid type of mixture structure implied by the FD makes it unsuitable to describe many compositional datasets. Furthermore, the FD only allows for negative correlations. The new extended model, by considerably relaxing the strict constraints among clusters entailed by the FD, allows for a more general dependence structure (including positive correlations) and greatly expands its applicative potential. At the same time, it retains, to a large extent, its good properties. EM-type estimation procedures can be developed for this more complex model, including ad hoc reliable initialization methods, which permit to keep the computational issues at a rather uncomplicated level. Accurate evaluation of standard error estimates can be provided as well.

Alcune classi di dati sono definite su spazi matematici diversi dal classico spazio reale a D dimensioni. Ad esempio, i dati composizionali sono contenuti nel simplesso, cioè lo spazio dei vettori positivi con componenti che sommano all’unità. Questi dati si possono trovare in molti campi (geologia, medicina, economia, psicologia, etc.) e quindi la definizione di una qualche metodologia atta alla loro analisi è necessaria. La distribuzione più famosa definita sul simplesso è la Dirichlet. Nonostante essa abbia numerose proprietà matematiche, in molte applicazioni reali essa non si adatta molto bene ai dati a causa del rigido schema di indipendenze supposto dal modello e/o dalla rigidità imposta sulla parametrizzazione della matrice di varianze e covarianze. Inoltre, la Dirichlet permette di avere al più una moda finita. Lo scopo di questa tesi è quello di confrontare alcune distribuzioni definite in letteratura per superare gli svantaggi derivanti dalla Dirichlet. Nello specifico, sono stati richiamati gli aspetti principali della Additive Logistic-Normal (ALN, proposta da Aitchison nel 1986) e della Flexible Dirichlet (FD, proposta da Ongaro e Migliorati nel 2013). La FD può essere definita come un modello mistura con componenti distribuite secondo una Dirichlet e che quindi consente una maggiore flessibilità (ad esempio nella struttura della matrice di varianze e covarianze e nel permettere più mode finite). Le covarianze tra elementi distinti di un vettore distribuito secondo una FD sono negative. Nonostante questo sia coerente con il vincolo di somma a 1 imposto dal simplesso, spesso nei dati reali le covarianze di dati composizionali possono essere positive. Per questa ragione, in alcune conferenze è stata proposta una nuova distribuzione che generalizza la FD (e, di conseguenza, la Dirichlet): l’Extended Flexible Dirichlet (EFD). Questa distribuzione può essere ottenuta normalizzando una particolare base Y = (Y_1, ..., Y_D), dove Y_r = W_r + X_r, Some kind of data are defined on unusual mathematical spaces instead of classical ones as the D-dimensional real space. For instance, compositional data belong to the D-dimensional simplex, defined as the space of positive vectors subject to a unit-sum constraint (i.e. proportions). Note that compositional data are prevalent in many disciplines (e.g. geology, medicine, economics, psychology, environmetrics, etc.), therefore their proper treatment is a relevant issue. The Dirichlet is one of the most known distribution defined on the simplex. Although it has several mathematical properties, in many real applications it does not fit the data well, due to its extreme forms of simplicial independence or stiffness in modeling cluster structure and the covariance matrix. Moreover, the Dirichlet distribution allows only one finite mode. The purpose of this thesis is to compare some distributions proposed in the literature in order to overcome these drawbacks. In particular, the main aspects of the Additive-Logistic Normal (proposed by Aitchison in his complete methodology in 1986) and the Flexibile Dirichlet (FD, proposed by Ongaro and Migliorati in 2013) distributions are recalled. The FD has a particular finite mixture structure (with Dirichlet components) that allows for multimodality and a more flexibile modelization of the covariance matrix. In particular, the covariance between distinct elements of a FD-distributed vector are negative; this is coherent with the unit-sum constraint imposed by the simplex, but in some applications such a covariance may be positive. For this reason, a new generalization of both the Dirichlet and the FD distributions has been proposed in some conferences: the Extended Flexible Dirichlet. This distribution can be obtained normalizing a particular basis Y = (Y_1, ..., Y_D), where Y_r = W_r + X_r * U_r, r = 1,... , D. W_r ~ Gamma(alpha_r, beta) are independent random variables, U_r ~ Gamma(tau_r, beta) are independent of each other and inde

The flexible Dirichlet (FD) distribution (Ongaro and Migliorati in J. Multvar. Anal. 114: 412–426, 2013) makes it possible to preserve many theoretical properties of the Dirichlet one, without inheriting its lack of flexibility in modeling the various independence concepts appropriate for compositional data, i.e. data representing vectors of proportions. In this paper we tackle the potential of the FD from an inferential and applicative viewpoint. In this regard, the key feature appears to be the special structure defining its Dirichlet mixture representation. This structure determines a simple and clearly interpretable differentiation among mixture components which can capture the main features of a large variety of data sets. Furthermore, it allows a substantially greater flexibility than the Dirichlet, including both unimodality and a varying number of modes. Very importantly, this increased flexibility is obtained without sharing many of the inferential difficulties typical of general mixtures. Indeed, the FD displays the identifiability and likelihood behavior proper to common (non-mixture) models. Moreover, thanks to a novel non random initialization based on the special FD mixture structure, an efficient and sound estimation procedure can be devised which suitably combines EM-types algorithms. Reliable complete-data likelihood-based estimators for standard errors can be provided as well.

In this paper a new non-central beta distribution is defined. Several properties are derived (including various representations and moments expressions) both for the new and the standard non-central beta distribution, showing a greater tractability and interpretability of the former.

The Dirichlet is the most well known distribution for data representing vectors of proportions (i.e. compositions). The flexible Dirichlet distribution (FD, Ongaro, Migliorati (2013)) generalizes the Dirichlet one allowing to preserve many of its mathematical and compositional properties without inheriting its lack of flexibility in modeling the data dependence structure. The Extended FD model (EFD, Ongaro, Migliorati (2014)) is obtained by generalizing the basis of positive random variables generating the FD by normalization. The new model exhibits a more sophisticated mixture representation and it accommodates for the existence of dependence between composition (the normalized basis) and size (the sum of the variables forming the basis). The present paper investigates the inferential aspects of the EFD, as a model for compositional data. In particular, an analysis of the flexibility of the cluster structure and of the dependence pattern implied by the model reveals its relevance for applications. Furthermore, appropriate estimation procedures are devised based on E--M algorithm. Specifically, an ad hoc initialization strategy is proposed to address the crucial choice of starting values for the E--M. Finally, the potential of the model is illustrated by means of applications to real data sets

In this paper a new non-central beta distribution is defined. Several properties are derived (including various representations and moments expressions) both for the new and the standard non-central beta distribution, showing a greater tractability and interpretability of the former.

In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support

Nel presente articolo viene introdotta una nuova distribuzione Beta non centrale. Vengono derivate varie proprietà (in particolare varie rappresentazioni ed espressioni dei momenti) sia per la nuova Beta non centrale sia per quella standard, mostrando come sotto molti aspetti la prima appaia maggiormente trattabile., A new Non-central Beta distribution is defined. Several properties are derived (including various representations and moments expressions) both for the new and the standard Non-central Beta distribution, showing in many respects a greater tractability of the former.

Nella presente tesi vengono discussi alcuni miscugli di distribuzioni Beta e di Dirichlet. In particolare, l’attenzione è rivolta alle distribuzioni Beta e di Dirichlet non centrali, di cui viene definita una variante maggiormente trattabile. Vengono studiate le principali proprietà e caratteristiche sia delle nuove distribuzioni Beta e di Dirichlet non centrali sia di quelle standard. Più precisamente, vengono proposte varie rappresentazioni, vengono analizzate le differenti forme assunte dalle densità, vengono ricavate approssimazioni delle stesse, vengono derivate formule generali per i momenti e, in ambito bayesiano, viene studiata la coniugatezza ai modelli Binomiale e Multinomiale. Infine, in ambito bayesiano non parametrico, vengono introdotte due estensioni non centrali del processo di Dirichlet. Di tali processi vengono fornite le caratteristiche marginali finito-dimensionali e la rappresentazione delle traiettorie discrete

Often we are confronted with heterogeneous multivariate data, i.e., data coming from several categories, and the interest may center on the differential structure of stochastic dependence among the variables between the groups. The focus in this work is on the two groups problem and is faced modeling the system through a Gaussian directed acyclic graph (DAG) couple linked in a fashion to obtain a joint estimation in order to exploit, whenever they exist, similarities between the graphs. The model can be viewed as a set of separate regressions and the proposal consists in assigning a non-local prior to the regression coefficients with the objective of enforcing stronger sparsity constraints on model selection. The model selection is based on Moment Fractional Bayes Factor, and is performed through a stochastic search algorithm over the space of DAG models.

This paper considers species sampling models using constructions that arise from Bayesian nonparametric prior distributions. A discrete random measure, used to generate a species sampling model, can have either a countable infinite number of atoms, which has been the emphasis in the recent literature, or a finite number of atoms K, while allowing K to be assigned a prior probability distribution on the positive integers. It is the latter class of model we consider here, due to the interpretation of K as the number of species. We demonstrate the consistency of the posterior distribution of K as the sample size increases