Mixture of Species Sampling Models

We introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric priors recently introduced to provide sparsity. Furthermore, we show how mSSS arise while considering hierarchical species sampling random probabilities (e.g., the hierarchical Dirichlet process). Extending previous results, we prove that mSSS are obtained by assigning the values of an exchangeable sequence to the classes of a latent exchangeable random partition. Using this representation, we give an explicit expression of the Exchangeable Partition Probability Function of the partition generated by an mSSS. Some special cases are discussed in detail—in particular, species sampling sequences with general base measures and a mixture of species sampling sequences with Gibbs-type latent partition. Finally, we give explicit expressions of the predictive distributions of an mSSS.

Decomposition of Random Sequences into Mixtures of Simpler Ones and Its Application in Network Analysis

A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti’s Theorem. In its original form, it applies to infinite 0–1 valued sequences with the special property that the distribution is invariant to permutations (called an exchangeable sequence). Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more detail than usual, with the intent of making it also accessible to readers who may not be closely familiar with the subject.

The deFinetti representation of generalised Marshall–Olkin sequences

We show that each infinite exchangeable sequence τ1, τ2, . . . of random variables of the generalised Marshall–Olkin kind can be uniquely linked to an additive subordinator via its deFinetti representation. This is useful for simulation, model estimation, and model building.

A note on the Galambos copula and its associated Bernstein function

There is an infinite exchangeable sequence of random variables {Xk}k∈ℕ such that each finitedimensional distribution follows a min-stable multivariate exponential law with Galambos survival copula, named after [7]. A recent result of [15] implies the existence of a unique Bernstein function Ψ associated with {Xk}k∈ℕ via the relation Ψ(d) = exponential rate of the minimum of d members of {Xk}k∈ℕ. The present note provides the Lévy–Khinchin representation for this Bernstein function and explores some of its properties.

Bayesian Forecasting of Economic Time Series

A model is suggested to forecast economic time series. This model incorporates some innovative ideas of Harrison and Stevens [20] for building into the forecasting process important external shocks to the systems. Thus the occurrence of possibly significant real-world events may cause a fundamental change in the time series in question. The Jeffreys-Savage (JS) Bayesian theory of hypothesis testing is used to test the hypothesis that a particular event has been such as to free the series from its immediate past behavior. When the event frees the series in this way, then we model the sequence of observations following such an event (until the next such event) as an exchangeable sequence. In the simplest case of 0–1 valued data, such as in recording the ups and downs of the value of a particular commodity or stock, our alternative hypothesis is a Pólya process, and the null hypothesis is a simple random walk (unit roots model) with p = .50. Any exchangeable sequence is strictly stationary, and the observations in the Polya process are positively correlated, which can give rise to “explosive” behavior of the series at isolated time points. We then use the JS theory to predict future observations by taking a weighted average of the optimal predictions for each model, with weights given by the posterior probabilities of the hypotheses. Results of simulation studies are presented which compare the predictive performance of the fully Bayesian method based upon the JS theory with those based upon the “p-value” or pre-test method. The de Finetti method for scoring predictions is used to assess their empirical performance. A theoretical methodology, which extends the “evaluation game” of Hill [28,37], is developed for comparing predictors.

Archimedean copulas, exchangeability, and max-stability

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

Archimedean copulas, exchangeability, and max-stability

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.