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Modelling of the dependence structure across heterogeneous data is crucial for Bayesian inference, since it directly impacts the borrowing of information. Despite extensive advances over the past two decades, most available methods only allow for nonnegative correlations. We derive a new class of dependent nonparametric priors that can induce correlations of any sign, thus introducing a new and more flexible idea of borrowing of information. This is achieved thanks to a novel concept, which we term hyper-tie, and represents a direct and simple measure of dependence. We investigate prior and posterior distributional properties of the model and develop algorithms to perform posterior inference. Illustrative examples on simulated and real data show that the proposed method outperforms alternatives in terms of prediction and clustering. [ABSTRACT FROM AUTHOR]

This is a review paper, beginning with de Finetti's work on partial exchangeability, continuing with his approach to approximate exchangeability, and then his (surprising) approach to assigning informative priors in nonstandard situations. Recent progress on Markov chain Monte Carlo methods for drawing conclusions is supplemented by a review of work by Gerencsér and Ottolini on getting honest bounds for rates of convergence. The paper concludes with a speculative approach to combining classical asymptotics with Monte Carlo. This promises real speed‐ups and makes a nice example of how theory and computation can interact. [ABSTRACT FROM AUTHOR]

Schur-constant vectors are used to model duration phenomena in various areas of economics and statistics. They form a particular class of exchangeable vectors and, as such, rely on a strong property of symmetry. To broaden the field of applications, partially Schur-constant vectors are introduced which correspond to partially exchangeable vectors. First, their copulas of survival, said to be partially Archimedean, are explicitly obtained and analyzed. Next, much attention is devoted to the construction of different partially Schur-constant models with two groups of exchangeable variables. Finally, partial Schur-constancy is briefly extended to the modeling of nested and multi-level dependencies.

Bayesian hierarchical models are powerful tools for learning common latent features across multiple data sources. The Hierarchical Dirichlet Process (HDP) is invoked when the number of latent components is a priori unknown. While there is a rich literature on finite sample properties and performance of hierarchical processes, the analysis of their frequentist posterior asymptotic properties is still at an early stage. Here we establish theoretical guarantees for recovering the true data generating process when the data are modeled as mixtures over the HDP or a generalization of the HDP, which we term boosted because of the faster growth in the number of discovered latent features. By extending Schwartz's theory to partially exchangeable sequences we show that posterior contraction rates are crucially affected by the relationship between the sample sizes corresponding to the different groups. The effect varies according to the smoothness level of the true data distributions. In the supersmooth case, when the generating densities are Gaussian mixtures, we recover the parametric rate up to a logarithmic factor, provided that the sample sizes are related in a polynomial fashion. Under ordinary smoothness assumptions more caution is needed as a polynomial deviation in the sample sizes could drastically deteriorate the convergence to the truth. [ABSTRACT FROM AUTHOR]

A parameter free version of classical models for contingency tables is developed along the lines of de Finetti’s notions of partial exchangeability.

Partial exchangeability is sufficient for the identification of some causal effects of interest. Here we review the use of common graphical tools and the sufficient component cause model in the context of partial exchangeability. We illustrate the utility of single world intervention graphs (SWIGs) in depicting partial exchangeability and provide an illustrative example of when partial exchangeability might hold in the absence of complete exchangeability. [ABSTRACT FROM AUTHOR]

Surrogate endpoints play an important role in drug development when they can be used to measure treatment effect early compared to the final clinical outcome and to predict clinical benefit or harm. Such endpoints are assessed for their predictive value of clinical benefit by investigating the surrogate relationship between treatment effects on the surrogate and final outcomes using meta-analytic methods. When surrogate relationships vary across treatment classes, such validation may fail due to limited data within each treatment class. In this paper, two alternative Bayesian meta-analytic methods are introduced which allow for borrowing of information from other treatment classes when exploring the surrogacy in a particular class. The first approach extends a standard model for the evaluation of surrogate endpoints to a hierarchical meta-analysis model assuming full exchangeability of surrogate relationships across all the treatment classes, thus facilitating borrowing of information across the classes. The second method is able to relax this assumption by allowing for partial exchangeability of surrogate relationships across treatment classes to avoid excessive borrowing of information from distinctly different classes. We carried out a simulation study to assess the proposed methods in nine data scenarios and compared them with subgroup analysis using the standard model within each treatment class. We also applied the methods to an illustrative example in colorectal cancer which led to obtaining the parameters describing the surrogate relationships with higher precision. [ABSTRACT FROM AUTHOR]

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence weaker than exchangeability, but equivalent to it in the presence of stationarity. We extend such notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, which is shown to be equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies at the foundations of Bayesian statistics. Natural examples of partially c.i.d. constructions are interacting randomly reinforced processes satisfying certain conditions on the reinforcement. [ABSTRACT FROM AUTHOR]

The prediction of future outcomes of a random phenomenon is typically based on a certain number of “analogous” observations from the past. When observations are generated by multiple samples, a natural notion of analogy is partial exchangeability and the problem of prediction can be effectively addressed in a Bayesian nonparametric setting. Instead of confining ourselves to the prediction of a single future experimental outcome, as in most treatments of the subject, we aim at predicting features of an unobserved additional sample of any size. We first provide a structural property of prediction rules induced by partially exchangeable arrays, without assuming any specific nonparametric prior. Then we focus on a general class of hierarchical random probability measures and devise a simulation algorithm to forecast the outcome of m future observations, for any m ≥ 1 . The theoretical result and the algorithm are illustrated by means of a real dataset, which also highlights the “borrowing strength” behavior across samples induced by the hierarchical specification. [ABSTRACT FROM AUTHOR]

A new class of dependent random measures which we call compound random measures is proposed and the use of normalized versions of these random measures as priors in Bayesian non-parametric mixture models is considered. Their tractability allows the properties of both compound random measures and normalized compound random measures to be derived. In particular, we show how compound random measures can be constructed with gamma, σ-stable and generalized gamma process marginals. We also derive several forms of the Laplace exponent and characterize dependence through both the Lévy copula and the correlation function. An augmented Pólya urn scheme sampler and a slice sampler are described for posterior inference when a normalized compound random measure is used as the mixing measure in a non-parametric mixture model and a data example is discussed. [ABSTRACT FROM AUTHOR]

This article is concerned with binary random matrices that are exchangeable with the probability of any finite submatrix only depending on its row- and column sums. We describe basic representations of such matrices both in the case of full row- and column exchangeability and the case of weak exchangeability. Finally the results are interpreted in terms of random graphs with exchangeable labels and with a view towards their potential application to social network analysis.

We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partially exchangeable property is the vertex reinforced jump process, under some technical conditions (Theorem 4). This result gives a counterpart to the characterization of edge reinforced random walk given by Rolles (Probab. Theory Related Fields 126 (2003) 243-260). [ABSTRACT FROM AUTHOR]

Efforts to personalize medicine in oncology have been limited by reductive characterizations of the intrinsically complex underlying biological phenomena. Future advances in personalized medicine will rely on molecular signatures that derive from synthesis of multifarious interdependent molecular quantities requiring robust quantitative methods. However, highly parameterized statistical models when applied in these settings often require a prohibitively large database and are sensitive to proper characterizations of the treatment-by-covariate interactions, which in practice are difficult to specify and may be limited by generalized linear models. In this article, we present a Bayesian predictive framework that enables the integration of a high-dimensional set of genomic features with clinical responses and treatment histories of historical patients, providing a probabilistic basis for using the clinical and molecular information to personalize therapy for future patients. Our work represents one of the first attempts to define personalized treatment assignment rules based on large-scale genomic data. We use actual gene expression data acquired from The Cancer Genome Atlas in the settings of leukemia and glioma to explore the statistical properties of our proposed Bayesian approach for personalizing treatment selection. The method is shown to yield considerable improvements in predictive accuracy when compared to penalized regression approaches. [ABSTRACT FROM AUTHOR]

Partially exchangeable sequences representable as mixtures of Markov chains are completely specified by de Finetti's mixing measure. The paper characterizes, in terms of a subclass of hidden Markov models, the partially exchangeable sequences with mixing measure concentrated on a countable set, for sequences of random variables both on a discrete space and on a Polish space. [ABSTRACT FROM AUTHOR]

We provide representation theorems for both finite and countable sequences of finite-valued random variables that are considered to be partially exchangeable. In their most general form, our results are presented in terms of sets of desirable gambles, a very general framework for modelling uncertainty. Its key advantages are that it allows for imprecision, is more expressive than almost every other imprecise-probabilistic framework and makes conditioning on events with (lower) probability zero non-problematic. We translate our results to more conventional, although less general frameworks as well: lower previsions, linear previsions and probability measures. The usual, precise-probabilistic representation theorems for partially exchangeable random variables are obtained as special cases. [ABSTRACT FROM AUTHOR]

We introduce the branching Ewens–Pitman sampling model for dependent species sequences. The model defines random probability measures having marginally two-parameter Poisson–Dirichlet process distributions. These random measures are associated with the nodes of a binary tree which describes the strength of dependence of the resulting random partitions. We discuss Bayesian analysis under the introduced model and provide algorithms for posterior inference. The model is applied to evaluate similarities across microbial populations in the human esophagus. [ABSTRACT FROM AUTHOR]

In the big data era there is a growing need to model the main features of large and non-trivial data sets. This paper proposes a Bayesian nonparametric prior for modelling situations where data are divided into different units with different densities, allowing information pooling across the groups. Leisen and Lijoi [(2011), ‘Vectors of Poisson–Dirichlet processes’,J. Multivariate Anal., 102, 482–495] introduced a bivariate vector of random probability measures with Poisson–Dirichlet marginals where the dependence is induced through a Lévy's Copula. In this paper the same approach is used for generalising such a vector to the multivariate setting. A first important contribution is the derivation of the Laplace functional transform which is non-trivial in the multivariate setting. The Laplace transform is the basis to derive the exchangeable partition probability function (EPPF) and, as a second contribution, we provide an expression of the EPPF for the multivariate setting. Finally, a novel Markov Chain Monte Carlo algorithm for evaluating the EPPF is introduced and tested. In particular, numerical illustrations of the clustering behaviour of the new prior are provided. [ABSTRACT FROM AUTHOR]

Mixture models for hazard rate functions are widely used tools for addressing the statistical analysis of survival data subject to a censoring mechanism. The present article introduced a new class of vectors of random hazard rate functions that are expressed as kernel mixtures of dependent completely random measures. This leads to define dependent nonparametric prior processes that are suitably tailored to draw inferences in the presence of heterogenous observations. Besides its flexibility, an important appealing feature of our proposal is analytical tractability: we are, indeed, able to determine some relevant distributional properties and a posterior characterization that is also the key for devising an efficient Markov chain Monte Carlo sampler. For illustrative purposes, we specialize our general results to a class of dependent extended gamma processes. We finally display a few numerical examples, including both simulated and real two-sample datasets: these allow us to identify the effect of a borrowing strength phenomenon and provide evidence of the effectiveness of the prior to deal with datasets for which the proportional hazards assumption does not hold true. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Intervention studies in school systems are sometimes aimed not at changing curriculum or classroom technique, but rather at changing the way that teachers, teaching coaches, and administrators in schools work with one another—in short, changing the professional social networks of educators. Current methods of social network analysis are ill-suited to modeling the multiple partially exchangeable networks that arise in randomized field trials and observational studies in which multiple classrooms, schools, or districts are involved, and to detecting the effect of an intervention on the social network itself. To address these needs, we introduce a new modeling framework, the Hierarchical Network Models (HNM) framework. The HNM framework can be used to extend single-network statistical network models to multiple networks, using a hierarchical modeling approach. We show how to generalize the latent space model for a single network to the HNM/multiple-network setting, and illustrate our approach with real and simulated social network data among education professionals. [ABSTRACT FROM AUTHOR]

This paper proposes a new method for constructing a sequence of infinitely exchangeable uniform random variables on the unit interval. For constructing the sequence, we utilize a Pólya urn partially. The resulting exchangeable sequence depends on the initial numbers of balls of the Pólya urn. We also derive the de Finetti measure for the exchangeable sequence. For an arbitrarily given one-dimensional distribution function, we generate sequences of exchangeable random variables with the one-dimensional marginal distribution by transforming the exchangeable uniform sequences with the inverse function of the distribution function. Among them we mainly investigate sequences of exchangeable discrete random variables. They differ from the well-known exchangeable sequence generated only by the Pólya urn scheme. Some examples are also given as applications of the results to exact distributions of some statistics based on sequences of exchangeable trials. Further, from the above exchangeable uniform sequence we construct partial or Markov exchangeable sequences. We also provide numerical examples of statistical inference based on the exchangeable and Markov exchangeable sequences. [ABSTRACT FROM AUTHOR]

We propose a class of dependent processes in which density shape is regressed on one or more predictors through conditional tail-free probabilities by using transformed Gaussian processes. A particular linear version of the process is developed in detail. The resulting process is flexible and easy to fit using standard algorithms for generalized linear models. The method is applied to growth curve analysis, evolving univariate random effects distributions in generalized linear mixed models, and median survival modelling with censored data and covariate-dependent errors. [ABSTRACT FROM PUBLISHER]

Let $${\{ X_{i}\} _{i\geq 1}}$$ be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in $${\{ X_{i}\} _{i\geq1}}$$ . Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models. [ABSTRACT FROM AUTHOR]

Abstract: The definition of vectors of dependent random probability measures is a topic of interest in applications to Bayesian statistics. They represent dependent nonparametric prior distributions that are useful for modelling observables for which specific covariate values are known. In this paper we propose a vector of two-parameter Poisson–Dirichlet processes. It is well-known that each component can be obtained by resorting to a change of measure of a -stable process. Thus dependence is achieved by applying a Lévy copula to the marginal intensities. In a two-sample problem, we determine the corresponding partition probability function which turns out to be partially exchangeable. Moreover, we evaluate predictive and posterior distributions. [ABSTRACT FROM AUTHOR]

A parameter free version of classical models for contingency tables is developed along the lines of de Finetti's notions of partial exchangeability. [ABSTRACT FROM AUTHOR]

We discuss the analysis of data from single-nucleotide polymorphism arrays comparing tumour and normal tissues. The data consist of sequences of indicators for loss of heterozygosity (LOH) and involve three nested levels of repetition: chromosomes for a given patient, regions within chromosomes and single-nucleotide polymorphisms nested within regions. We propose to analyse these data by using a semiparametric model for multilevel repeated binary data. At the top level of the hierarchy we assume a sampling model for the observed binary LOH sequences that arises from a partial exchangeability argument. This implies a mixture of Markov chains model. The mixture is defined with respect to the Markov transition probabilities. We assume a non-parametric prior for the random-mixing measure. The resulting model takes the form of a semiparametric random-effects model with the matrix of transition probabilities being the random effects. The model includes appropriate dependence assumptions for the two remaining levels of the hierarchy, i.e. for regions within chromosomes and for chromosomes within patient. We use the model to identify regions of increased LOH in a data set coming from a study of treatment-related leukaemia in children with an initial cancer diagnostic. The model successfully identifies the desired regions and performs well compared with other available alternatives. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a novel approach to constructing bivariate prior distributions. The idea is based on the notion of partial exchangeability. In particular, in a simple extension of the exchangeable sequence, we create two dependent exchangeable sequences via a branching mechanism. This implies the existence of a bivariate prior distribution. [Copyright &y& Elsevier]

Abstract: We define a reinforced stochastic process of random variables indexed by the vertices of a k-tree and with values in a Polish space. The work presents a natural extension from an exchangeable to a partially exchangeable setting of previous work done by the authors. [Copyright &y& Elsevier]

This article discusses inference on the order of dependence in binary sequences. The proposed approach is based on the notion of partial exchangeability of order k. A partially exchangeable binary sequence of order k can be represented as a mixture of Markov chains. The mixture is with respect to the unknown transition probability matrix θ. We use this defining property to construct a semiparametric model for binary sequences by assuming a nonparametric prior on the transition matrix θ. This enables us to consider inference on the order of dependence without constraint to a particular parametric model. Implementing posterior simulation in the proposed model is complicated by the fact that the dimension of θ changes with the order of dependence k. We discuss appropriate posterior simulation schemes based on a pseudo prior approach. We extend the model to include covariates by considering an alternative parameterization as an autologistic regression which allows for a straightforward introduction of covariates. The regression on covariates raises the additional inference problem of variable selection. We discuss appropriate posterior simulation schemes, focusing on inference about the order of dependence. We discuss and develop the model with covariates only to the extent needed for such inference. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general Bayesian model which allows multiple grouping of parameters, where the components within a subgroup are exchangeable. The general idea is then illustrated for the normal means estimation problem under priors which are scale mixture of normals. We discuss also implementation of the Bayes procedure via Markov chain Monte Carlo integration techniques. We illustrate the proposed methods with a numerical example. [Copyright &y& Elsevier]

A Bayesian analysis for factorial experiments is presented, using finite mixture distributions to model the main effects and interactions. This allows both estimation and an analogue of hypothesis testing in a posterior analysis using a single prior specification. A detailed formulation based on this approach is provided for the case of the two-way model with replication, allowing interactions. Issues in formulating a suitable prior are discussed in detail, and, in the context of two illustrative applications, we discuss implementation, presentation of posterior distributions, sensitivity and performance of the Markov chain Monte Carlo methods that are used. [ABSTRACT FROM PUBLISHER]

This paper deals with a central limit problem for columnwise exchangeable arrays of a Hilbert space valued random variables satisfying the condition of conditional uniform asymptotic negligibility. It is proved that rowwise sums may converge in distribution only to an exchangeable sequence of random variables whose distribution is a mixture of in nitely divisible distributions. Conditions for convergence to a speci c distribution and, in particular, to a mixture of Gaussian distributions, are also given. [ABSTRACT FROM AUTHOR]