Statistics seminars are informal events where new ideas can be presented and discussed. The aim is to provide a venue for discussion and exchange of ideas between different statistics research groups and others interested in statistics. All seminars are in English. The program is updated continuously so look out for new titles.
Note! The Spring term seminars are cancelled for the time being.
Time: Fortnightly on Tuesdays at 12:15 - 13:00.
Place: Exactum C124.
All interested are warmly welcome!
Abstract: Consider the consequences of an alternative history. What if Leonhard Euler had happened to read the posthumous publication of the paper by Thomas Bayes on “An Essay towards solving a Problem in the Doctrine of Chances”? This paper was published in 1763 in the Philosophical Transactions of the Royal Society, so if Euler had read this article, we can wonder whether the section in his three volume book Institutionum calculi integralis, published in 1768, on numerical solution of differential equations might have been quite different.
Would the awareness by Euler of the “Bayesian” proposition of characterising uncertainty due to unknown quantities using the probability calculus have changed the development of numerical methods and their analysis to one that is more inherently statistical?
Fast forward the clock two centuries to the late 1960s in America, when the mathematician F.M. Larkin published a series of papers on the definition of Gaussian Measures in infinite dimensional Hilbert spaces, culminating in the 1972 work on “Gaussian Measure on Hilbert Space and Applications in Numerical Analysis”. In that work the formal definition of the mathematical tools required to consider average case errors in Hilbert spaces for numerical analysis were laid down and methods such as Bayesian Quadrature or Bayesian Monte Carlo were developed in full, long before their independent reinvention in the 1990s and 2000s brought them to a wider audience.
Now in 2017 the question of viewing numerical analysis as a problem of Statistical Inference in many ways seems natural and is being demanded by applied mathematicians, engineers and physicists who need to carefully and fully account for all sources of uncertainty in mathematical modelling and numerical simulation.
Now we have a research frontier that has emerged in scientific computation founded on the principle that error in numerical methods, which for example solves differential equations, entails uncertainty that ought to be subjected to statistical analysis. This viewpoint raises exciting challenges for contemporary statistical and numerical analysis, including the design of statistical methods that enable the coherent propagation of probability measures through a computational and inferential pipeline.
Abstract
Researchers often find that nonlinear regression models are more applicable for modelling various biological, physical and chemical processes than are linear ones since they tend to fit the data well and since these models (and model parameters) are more scientifically meaningful. These researchers are thus often in a position of requiring optimal or near-optimal designs for a given nonlinear model. A common shortcoming of most optimal designs for nonlinear models used in practical settings, however, is that these designs typically focus only on (first-order) parameter variance or predicted variance, and thus ignore the inherent nonlinear of the assumed model function. Another shortcoming of optimal designs is that they often have only support points, where is the number of model parameters.
Furthermore, measures of marginal curvature, first introduced in Clarke (1987) and extended in Haines et al (2004), provide a useful means of assessing this nonlinearity. Other relevant developments are the second-order volume design criterion introduced in Hamilton and Watts (1985) and extended in O’Brien (2010), and the second-order MSE criterion developed and illustrated in Clarke and Haines (1995).
In the context of applied statistical modelling, this talk examines various robust design criteria and those based on second-order (curvature) considerations. These techniques, coded in popular software packages, are illustrated with several examples including one from a preclinical dose-response setting encountered in a recent
consulting session.