Brown Analysis Seminar
Brown University Center for Statistical Sciences Seminar
Abstract: Sufficient cause interactions concern cases in which a particular causal mechanism for some outcome will operate only if two or more specific causes are present. Empirical conditions have been derived to test for sufficient cause interactions. However, when regression outcome models are used to control for confounding variables in tests for sufficient cause interactions, the outcome models impose restrictions on the relationship between the confounding variables and certain unidentified background causes within the sufficient cause framework; often these assumptions are implausible. By using marginal structural models, rather than outcome regression models, to test for sufficient cause interactions, assumptions are instead made on the relationship between the causes of interest and the confounding variables; these assumptions will often be more plausible. The use of marginal structural models also allows for testing for sufficient cause interactions in the presence of time-dependent confounding. Such time-dependent confounding may arise in cases in which a genetic factor of interest affects both an environmental factor of interest and the outcome. It is furthermore shown that marginal structural models can be used not only to test for sufficient cause interactions but to give lower bounds on the prevalence of such sufficient cause interactions. The methods are illustrated with an application to sufficient cause interactions between the effects of well-arsenic exposure and smoking on the development of skin lesions using data from a study in Bangladesh.
Lefschetz Center for Dynamical Systems Seminar
Abstract: The model of a periodic potential with a defect is a common one in many areas of physics and applied mathematics. We consider a model with a dislocation type defect, where there is an asymptotic phase shift in the periodic potential between plus and minus infinity. Using Evan's function techniques and the Hamiltonian structure of the problem we prove an index theorem that counts the number of point eigenvalues created in the gaps in the continuous spectrum, and show that in the large energy limit the number of defect eigenvalues in a particular gap depends on the solvability of a certain Diophantine approximation problem.
Probability Seminar
Abstract: Typical contingent claims such as options are written on two or more underlying assets. Each of the underlying assets can be chosen as a numeraire for the purposes of pricing and hedging as long as the price of such asset is positive. This leads to at least two alternative formulations of the pricing problem, depending on the number of available reference assets with a positive price that enter a given contract. We show that the prices when expressed under different numeraires are connected by a functional relationship known as perspective mapping. This technique of computing prices under different reference assets is more general than simply computing the prices as expected discounted payoffs under the martingale measure associated with a given numeraire since it works also in situations when the reference asset does not have a corresponding martingale measure. For instance, an asset that represents the maximum price in the payoff of lookback options does not have a martin! gale measure, but the price of the contract with respect to the maximum can still be expressed using perspective mapping. This method applies for a general evolution of the price process. We give examples of the relationship of the pricing measures in the binomial model, the diffusion model, and the L\'evy jump model. We give two formulations of the pricing problem for European and American options, and three formulations of the problem for exotic options such as quantos, lookbacks, or Asians. In diffusion models, we obtain partial differential equations that correspond to the pricing problem.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: A probabilistic grammar for the grouping and labeling of parts and objects, when taken together with pose and part-dependent appearance models, constitutes a generative scene model and a Bayesian framework for image analysis. To the extent that the generative model generates features, as opposed to pixel intensities, the posterior distribution (i.e. the conditional distribution on part and object labels given the image) is based on incomplete information; feature vectors are generally insufficient to recover the original intensities. I will propose a way to learn pixel-level models for the appearances of parts. I will demonstrate the utility of the models with some experiments in Bayesian image classification.
Department of Mathematics Colloquium
Scientific Computing Seminar
Abstract:
Predictive simulation of complex engineering systems increasingly rests on the interplay of experimental observations with computational models. Key inputs, parameters, or structural aspects of models may be incomplete or unknown, and must be developed from indirect and limited observations. At the same time, quantified uncertainties are needed to qualify computational predictions in the support of design and decision-making. In this context, Bayesian statistics provides a foundation for inference from noisy and limited data. Computationally intensive forward models, however, can render a Bayesian approach prohibitive.
Polynomial chaos expansions, typically used in the forward propagation of uncertainty, are an extremely useful tool in the inverse context as well. We introduce a stochastic spectral formulation that accelerates the Bayesian solution of inverse problems via rapid evaluation of a surrogate posterior distribution. The posterior is constructed by either stochastic collocation or stochastic Galerkin methods. Theoretical convergence results are verified with several numerical examples---in particular, parameter estimation in transport equations and in chemical kinetic systems. We also extend this approach to the inference of spatially distributed quantities in a hierarchical Bayesian setting, achieving dimensionality reduction via Karhunen-Loeve representations of Gaussian process priors.
Finally, we discuss the utility of polynomial chaos expansions in optimal experimental design---choosing experimental conditions to maximize information gain in parameters or outputs of interest. A Bayesian formulation of the design problem fully accounts for uncertainty in the parameters and relevant observables.
PDE Seminar
Abstract: Block copolymers are macromolecules that can form variety of microstructures as a result of incomplete phase separation. For this reason, they are natural candidates for controllednanoscale self-assembly and possess novel material properties. This talk focuses on the mathematical issues surrounding density functional models of dilute diblock copolymer mixtures and their related gradient flows. Isolated structures emerge in the subcritical regime that resemble amphiphilic bilayers and micelles. Existence and stability properties of these solutions are discussed, including a surprising secondary bifurcation that results in self-replication phenomena. Well above the subcritical regime, there is a Ostwald-ripening-type process which is inhibited by long range interactions. The dynamics can be reduced by a series of approximations to interacting particle systems and coarse-grained statistical descriptions which characterize the large scale behavior.