Poisson Latent Feature Calculus for Generalized Indian Buffet Processes

The purpose of this work is to describe a unified, and indeed simple, mechanism for non-parametric Bayesian analysis, construction and generative sampling of a large class of latent feature models which one can describe as generalized notions of Indian Buffet Processes(IBP). This is done via the Poisson Process Calculus as it now relates to latent feature models. The IBP was ingeniously devised by Griffiths and Ghahramani in (2005) and its generative scheme is cast in terms of customers entering sequentially an Indian Buffet restaurant and selecting previously sampled dishes as well as new dishes. In this metaphor dishes corresponds to latent features, attributes, preferences shared by individuals. The IBP, and its generalizations, represent an exciting class of models well suited to handle high dimensional statistical problems now common in this information age. The IBP is based on the usage of conditionally independent Bernoulli random variables, coupled with completely random measures acting as Bayesian priors, that are used to create sparse binary matrices. This Bayesian non-parametric view was a key insight due to Thibaux and Jordan (2007). One way to think of generalizations is to to use more general random variables. Of note in the current literature are models employing Poisson and Negative-Binomial random variables. However, unlike their closely related counterparts, generalized Chinese restaurant processes, the ability to analyze IBP models in a systematic and general manner is not yet available. The limitations are both in terms of knowledge about the effects of different priors and in terms of models based on a wider choice of random variables. This work will not only provide a thorough description of the properties of existing models but also provide a simple template to devise and analyze new models.Comment: This version provides more details for the multivariate extensions in section 5. We highlight the case of a simple multinomial distribution and showcase a multivariate Levy process prior we call a stable-Beta Dirichlet process. Section 4.1.1 expande

The Spinobulbar System in Lamprey

Locomotor networks in the spinal cord are controlled by descending systems which in turn receive feedback signals from ascending systems about the state of the locomotor networks. In lamprey, the ascending system consists of spinobulbar neurons which convey spinal network signals to the two descending systems, the reticulospinal and vestibulospinal neurons. Previous studies showed that spinobulbar neurons consist of both ipsilaterally and contralaterally projecting cells distributed at all rostrocaudal levels of the spinal cord, though most numerous near the obex. The axons of spinobulbar neurons ascend in the ventrolateral spinal cord and brainstem to the caudal mesencephalon and within the dendritic arbors of reticulospinal and vestibulospinal neurons. Compared to mammals, the ascending system in lampreys is more direct, consisting of excitatory and inhibitory monosynaptic inputs from spinobulbar neurons to reticulospinal neurons. The spinobulbar neurons are rhythmically active during fictive locomotion, representing a wide range of timing relationships with nearby ventral root bursts including those in phase, out of phase, and active during burst transitions between opposite ventral roots. The spinobulbar neurons are not simply relay cells because they can have mutual synaptic interactions with their reticulospinal neuron targets and they can have synaptic outputs to other spinal neurons. Spinobulbar neurons not only receive locomotor inputs but also receive direct inputs from primary mechanosensory neurons. Due to the relative simplicity of the lamprey nervous system and motor control system, the spinobulbar neurons and their interactions with reticulospinal neurons may be advantageous for investigating the general organization of ascending systems in the vertebrate

Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and Beta-Gamma processes

Suppose that P_{\theta}(g) is a linear functional of a Dirichlet process with shape \theta H, where \theta >0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in conjunction with a Gamma identity, we show easily that a generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, Beta-Gamma processes. This represents a generalization of an identity due to Cifarelli and Regazzini, which is also known as the Markov-Krein identity for mean functionals of Dirichlet processes. These results also provide new explanations and interpretations of results in the literature. The identities are analogues to quite useful identities for Beta and Gamma random variables. We give a result which can be used to ascertain specifications on H such that the Dirichlet functional is Beta distributed. This avoids the need for an inversion formula for these cases and points to the special nature of the Dirichlet process, and indeed the functional Beta-Gamma calculus developed in this paper.Comment: Published at http://dx.doi.org/10.1214/009053604000001237 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Compression of Morbidity: In Retrospect and in Prospect

By postponing the age at which chronic infirmity begins,disability and morbidity could be compressed into a shorter period of the average human life span, resulting in a society in which the active and vital years of life would increase in length, the disabilities and frailties of ageing would be postponed,and the total amount of lifetime disability and morbidity would decrease

Review of The Sacrifice of Jesus: Understanding Atonement Biblically

Article reviews the book The Sacrifice of Jesus: Understanding Atonement Biblically, by Christian Eberhart