Recent & Upcoming Talks

Dan Roy: Admissibility is Bayes Optimality with Infinitesimals

We give an exact characterization of admissibility in statistical decision problems in terms of Bayes optimality in a so-called nonstandard extension of the original decision problem, as introduced by Duanmu and Roy. Unlike the consideration of improper priors or other generalized notions of Bayes optimalitiy, the nonstandard extension is distinguished, in part, by having priors that can assign ‘infinitesimal’ mass in a sense that can be made rigorous using results from nonstandard analysis. With these additional priors, we find that, informally speaking, a decision procedure δ0 is admissible in the original statistical decision problem if and only if, in the nonstandard extension of the problem, the nonstandard extension of δ0 is Bayes optimal among the (extensions of) standard decision procedures with respect to a nonstandard prior that assigns at least infinitesimal mass to every standard parameter value. We use the above theorem to give further characterizations of admissibility, one related to Blyth’s method, one to a condition due to Stein which characterizes admissibility under some regularity assumptions; and finally, a characterization using finitely additive priors in decision problems meeting certain regularity requirements. Our results imply that Blyth’s method is a sound and complete method for establishing admissibility. Buoyed by this result, we revisit the univariate two-sample common-mean problem, and show that the Graybill–Deal estimator is admissible among a certain class of unbiased decision procedures. Joint work with Haosui Duanmu (HIT) and David Schrittesser (Toronto).

William Gregory: Improving Arctic Sea Ice Predictability with Gaussian Processes

Arctic sea ice is a major component of the Earth’s climate system, as well as an integral platform for travel, subsistence, and habitat. Since the late 1970s, significant advancements have been made in our ability to closely monitor the state of the ice cover at the polar regions through the launch of Earth-observation satellites. Subsequently, now over 4 decades of time-series data at our disposal, we have observed significant reductions in the spatial extent of Arctic sea ice, and more recently its thickness — directly in line with increasing anthropogenic CO2 emissions. The summer months, in particular, present the largest rate of decline in sea ice extent compared to other seasons, and also the largest pattern of inter-annual variability, making seasonal to inter-annual predictions difficult. Advanced predictions of the summer ice conditions are important as this is the time when the ice cover is at its minimum extent, and the Arctic becomes open to a whole host of traffic including coastal resupply vessels, eco-tourism, and the movement of local communities. This presentation explores Gaussian processes as a framework for both sea ice forecasting, and for optimally combining and interpolating multiple satellite observation sets. In the first instance, the spatio-temporal patterns of variability in past ice conditions are exploited using a framework of a complex network, which is then fed into a Gaussian process regression forecast model in the form of a random walk graph kernel, to predict regional and pan-Arctic (basin-wide) September sea ice extents with high skill. Following this, we will see how extensions to this work can be made in the form of spatial forecasts by adopting a multi-task learning approach. In the second application, the Gaussian process regression method is used to optimally combine (and interpolate) observations from 3 separate satellite altimeters in space and time, in order to produce the first-ever daily pan-Arctic observational data set of Arctic sea ice freeboard (the base product for deriving sea ice thickness). Following this, we will see how extensions to this work can be made through computational speed-ups by using relevant vector machines. In both the forecasting and interpolation applications, the hyperparameters of the models are learned through the empirical Bayes, or type-II maximum likelihood approach, which in the second application allows us to derive information relating to the spatio-temporal correlation length scales of Arctic sea ice thickness.