Probability Seminar
Abstract: One very widely used criterion in the theory of Markov chains states that if a Markov operator has the strong Feller property and is topologically irreducible, then it can have at most one invariant measure. While this criterion is very useful in finite-dimensional situations, it fails for many infinite-dimensional problems. In this talk, we will present two different generalizations of the strong Feller property that can be applied to a much larger class of problems. These include semi-linear parabolic stochastic PDEs, stochastic delay equations, and diffusions driven by coloured noise.
Brown University Center for Statistical Sciences Seminar
Abstract: This Seminar addresses the problems that occur in the construction of likelihood-based confidence sets for the parameters of a finite mixture model. The parameters in a standard mixture problem where the number of components is fixed are technically not identifiable, as the labels on the parameters are not identified. The good news is that there is a form of asymptotic identifiability which one can appeal to enable inference on labeled parameters. But we show that when the sample size is not large relative to the separation of the components, the likelihood regions cannot be used to construct confidence sets for labeled parameters, clearly signifying failure of the asymptotic theory. Unfortunately, there is little in the way of published guidance as to how to assess the adequacy of asymptotic theory for finite samples. In this presentation, we provide a simulation-based method to construct labeled confidence sets along with a set of diagnostic tools for ascertaining whether the level of empirical identifiability is sufficient to support the use of labeled confidence regions. If there is sufficient empirical identifiability, our proposed method enables us to describe multiple complex confidence regions without any difficult optimization and to assess simultaneously the potential breakdown of the asymptotic results.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract:
Logic and probability (aka. 'symbols and statistics', 'structured representation and graded inference') are key themes of cognitive science that have long had an uneasy coexistence. I will describe a Probabilistic Language of Thought approach that brings them together into compositional representations with probabilistic meaning. This provides a view of cognition in which mental representations describe causal ``working models'' of the world, that can be used for reasoning and learning by probabilistic inference. Using this framework I will investigate human concept learning, beginning with simple categorization tasks and extending to acquisition of abstract concepts. I will describe a model of learning the meanings of number words that predicts the staged progression exhibited by children (including the striking conceptual reorganization associated with the ``cardinal principle'' transition). I will then briefly describe how a probabilistic language of thought explains phenomena of human reasoning, including both classic syllogistic reasoning tasks and more complex reasoning involving causality and agency.
[pizza will be provided]
Center for Computational Molecular Biology Seminar
Hosted by: Charles (Chip) Lawrence Refreshments will be served at 3:45pm |
Abstract: Hidden Markov models and score-maximizing dynamic programming algorithms are employed for the evaluation of sequential data in a variety of scientific fields, including linguistics, vision, and computational biology. Given a hidden Markov model, efficient "Viterbi" and "forward" algorithms are used to evaluate the probability that the model would generate a given sequence of observations, and similar approaches are employed in the dynamic programming algorithms where the focus is on finding high scores instead of high probabilities. Here we present modifications to the "forward" algorithm that allow additional computations. We can efficiently estimate statistical significance: what is the probability that a randomly generated sequence will score at least as high as the observed sequence does? (We've computed answers down to 1e-4000.) We can also compute how typical a sequence is: for every whole number d, what is the probability that a sequence generated by the hidden Markov model will have exactly d differences from the observed sequence?
PDE Seminar
Abstract: In this talk, I will discuss the quadruple junction solutions in the entire three dimensional space to a vector‐valued Allen‐ Cahn equation which models multiple phase separation. The solution is the basic profile of the local structure near a quadruple junction in three dimensional crystalline material under the generalized Allen‐Cahn model, and is the three dimensional counterpart of triple junction solution which is two dimensional. I will start with one dimensional heteroclinic solutions, and describe how we can construct higher dimensional solutions from the lower dimensional ones, and explain the complications and difficulties in constructing such a solution in three dimensions.