There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We propose an integrated sampling, rarefaction, and extrapolation methodology to compare species richness of a set of communities based on samples of equal completeness (as measured by sample coverage) instead of equal size. Traditional rarefaction or extrapolation to equal-sized samples can misrepresent the relationships between the richnesses of the communities being compared because a sample of a given size may be sufficient to fully characterize the lower diversity community, but insufficient to characterize the richer community. Thus, the traditional method systematically biases the degree of differences between community richnesses. We derived a new analytic method for seamless coverage-based rarefaction and extrapolation. We show that this method yields less biased comparisons of richness between communities, and manages this with less total sampling effort. When this approach is integrated with an adaptive coverage-based stopping rule during sampling, samples may be compared directly without rarefaction, so no extra data is taken and none is thrown away. Even if this stopping rule is not used during data collection, coverage-based rarefaction throws away less data than traditional size-based rarefaction, and more efficiently finds the correct ranking of communities according to their true richnesses. Several hypothetical and real examples demonstrate these advantages.

https://esajournals.onlinelibrary.wiley.com/doi/epdf/10.1890/11-1952.1

Coverage‐based rarefaction and extrapolation: standardizing samples by completeness rather than size

The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propagated by the chain rule through the graph are low variance unbiased estimators of the gradients of the expected loss. While many continuous random variables have such reparameterizations, discrete random variables lack useful reparameterizations due to the discontinuous nature of discrete states. In this work we introduce Concrete random variables---continuous relaxations of discrete random variables. The Concrete distribution is a new family of distributions with closed form densities and a simple reparameterization. Whenever a discrete stochastic node of a computation graph can be refactored into a one-hot bit representation that is treated continuously, Concrete stochastic nodes can be used with automatic differentiation to produce low-variance biased gradients of objectives (including objectives that depend on the log-probability of latent stochastic nodes) on the corresponding discrete graph. We demonstrate the effectiveness of Concrete relaxations on density estimation and structured prediction tasks using neural networks.

The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables

Background: Expressed sequence tags (ESTs) analyses are a fundamental tool for gene identification in organisms. Given a preliminary EST sample from a certain library, several statistical prediction problems arise. In particular, it is of interest to estimate how many new genes can be detected in a future EST sample of given size and also to determine the gene discovery rate: these estimates represent the basis for deciding whether to proceed sequencing the library and, in case of a positive decision, a guideline for selecting the size of the new sample. Such information is also useful for establishing sequencing efficiency in experimental design and for measuring the degree of redundancy of an EST library. Results: In this work we propose a Bayesian nonparametric approach for tackling statistical problems related to EST surveys. In particular, we provide estimates for: a) the coverage, defined as the proportion of unique genes in the library represented in the given sample of reads; b) the number of new unique genes to be observed in a future sample; c) the discovery rate of new genes as a function of the future sample size. The Bayesian nonparametric model we adopt conveys, in a statistically rigorous way, the available information into prediction. Our proposal has appealing properties over frequentist nonparametric methods, which become unstable when prediction is required for large future samples. EST libraries, previously studied with frequentist methods, are analyzed in detail. Conclusion: The Bayesian nonparametric approach we undertake yields valuable tools for gene capture and prediction in EST libraries. The estimators we obtain do not feature the kind of drawbacks associated with frequentist estimators and are reliable for any size of the additional sample.

/pdf/a-bayesian-nonparametric-method-for-prediction-in-est-1pbxczi8jp.pdf

A Bayesian nonparametric method for prediction in EST analysis

Dependent mixture models

Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as “sharing of information.” In this paper, we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman–Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.

Distribution theory for hierarchical processes

SummaryFor the most popular discrete nonparametric models, beyond the Dirichlet process, the prior guess at the shape of the data-generating distribution, also known as the base measure, is assumed to be diffuse. Such a specification greatly simplifies the derivation of analytical results, allowing for a straightforward implementation of Bayesian nonparametric inferential procedures. However, in several applied problems the available prior information leads naturally to the incorporation of an atom into the base measure, and then the Dirichlet process is essentially the only tractable choice for the prior. In this paper we fill this gap by considering the Pitman–Yor process with an atom in its base measure. We derive computable expressions for the distribution of the induced random partitions and for the predictive distributions. These findings allow us to devise an effective generalized Polya urn Gibbs sampler. Applications to density estimation, clustering and curve estimation, with both simulated and real data, serve as an illustration of our results and allow comparisons with existing methodology. In particular, we tackle a functional data analysis problem concerning basal body temperature curves.

/pdf/on-the-pitman-yor-process-with-spike-and-slab-base-measure-3gimsqe69h.pdf

On the Pitman–Yor process with spike and slab base measure

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalizing to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop a Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by-product. The results and their inferential implications are showcased on synthetic and real data.

/pdf/latent-nested-nonparametric-priors-with-discussion-pkonqndnp0.pdf

Igor Prünster

Papers

A Bayesian nonparametric method for prediction in EST analysis

Dependent mixture models

Distribution theory for hierarchical processes

On the Pitman–Yor process with spike and slab base measure

Latent Nested Nonparametric Priors (with Discussion)