Universal Approximation of Markov Kernels by Shallow Stochastic Feedforward Networks

We establish upper bounds for the minimal number of hidden units for which a binary stochastic feedforward network with sigmoid activation probabilities and a single hidden layer is a universal approximator of Markov kernels. We show that each possible probabilistic assignment of the states of $n$ output units, given the states of $k\geq1$ input units, can be approximated arbitrarily well by a network with $2^{k-1}(2^{n-1}-1)$ hidden units.Comment: 13 pages, 3 figure

Hierarchical Models as Marginals of Hierarchical Models

We investigate the representation of hierarchical models in terms of marginals of other hierarchical models with smaller interactions. We focus on binary variables and marginals of pairwise interaction models whose hidden variables are conditionally independent given the visible variables. In this case the problem is equivalent to the representation of linear subspaces of polynomials by feedforward neural networks with soft-plus computational units. We show that every hidden variable can freely model multiple interactions among the visible variables, which allows us to generalize and improve previous results. In particular, we show that a restricted Boltzmann machine with less than $[ 2(\log(v)+1) / (v+1) ] 2^v-1$ hidden binary variables can approximate every distribution of $v$ visible binary variables arbitrarily well, compared to $2^{v-1}-1$ from the best previously known result.Comment: 18 pages, 4 figures, 2 tables, WUPES'1

Mixtures and products in two graphical models

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.Comment: 18 pages, 7 figure

Refinements of Universal Approximation Results for Deep Belief Networks and Restricted Boltzmann Machines

We improve recently published results about resources of Restricted Boltzmann Machines (RBM) and Deep Belief Networks (DBN) required to make them Universal Approximators. We show that any distribution p on the set of binary vectors of length n can be arbitrarily well approximated by an RBM with k-1 hidden units, where k is the minimal number of pairs of binary vectors differing in only one entry such that their union contains the support set of p. In important cases this number is half of the cardinality of the support set of p. We construct a DBN with 2^n/2(n-b), b ~ log(n), hidden layers of width n that is capable of approximating any distribution on {0,1}^n arbitrarily well. This confirms a conjecture presented by Le Roux and Bengio 2010

When Does a Mixture of Products Contain a Product of Mixtures?

We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table

Geometry and Expressive Power of Conditional Restricted Boltzmann Machines

Conditional restricted Boltzmann machines are undirected stochastic neural networks with a layer of input and output units connected bipartitely to a layer of hidden units. These networks define models of conditional probability distributions on the states of the output units given the states of the input units, parametrized by interaction weights and biases. We address the representational power of these models, proving results their ability to represent conditional Markov random fields and conditional distributions with restricted supports, the minimal size of universal approximators, the maximal model approximation errors, and on the dimension of the set of representable conditional distributions. We contribute new tools for investigating conditional probability models, which allow us to improve the results that can be derived from existing work on restricted Boltzmann machine probability models.Comment: 30 pages, 5 figures, 1 algorith

Finding a Common Ground between Theology and Women’s Reproductive Rights: Assessing the societal levels of influence of religion on the sexual and reproductive health of women

The principle aim of this study is to explicate and elucidate the intersection between religious beliefs and practices and Sexual and Reproductive Health throughout distinct levels of society in the developing world. A literature review identified relevant peer-reviewed and grey literature on religious beliefs held on sexuality and procreation, the landscape of influence of religion on laws and policies at a national and international level, the effects of religion on individual sexual behavior, and modern interventions aiming to be culturally and religiously sensitive. The intricacies and nuances of three Abrahamic faiths were assessed to highlight the dogma of sacred texts and practices, which highlighted the intrinsic benevolence of these religions. Four semi-structured interviews were conducted with experts in the field of Sexual and Reproductive Health with religion being an impactful factor in their work. The influence of intrinsic religious beliefs was evaluated in three different realms surrounding S&RH, such as: 1) Influences of Religious Beliefs on International and Governmental Entities, Policies, and Programs, 2) Individual Behavior, Lifestyle Choices, and Perceptions impacted by Religious Beliefs, and 3) The “Point of Intersection” at the Community Level: How the Reproductive Health and Sexual Health Agenda can be advanced through Religious Leaders and Faith-based Organizations. This study emphasizes the prevalence of religious beliefs in the individual, the community, and the nation, while seeking to express the importance of religious beliefs in progressing the agenda of S&RH by utilizing religion as a “vehicle for change”