Department Seminars

PIMS 10th Anniversary Speaker Series 2007
Location: WMAX 110

Notes: Coffee and refreshments will be served half an hour before the talk.

Abstract: Statistical theory is often categorized as either "Bayesian" or "frequentist", and statisticians often self-identify in the same categories. During the development of theoretical statistics as a separate field in the twentieth century this categorisation led to a great deal of discussion, some of which was surprisingly bitter and antagonistic. With the development of several key results in the asymptotic theory of inference based on the likelihood function, it is becoming clear that the mathematical differences between Bayesian and frequentist methods are rather less important than the philosophical ones. Some of this work is based on efforts to construct priors which minimize the difference between the two approaches and some is based on an ongoing effort to develop so-called 'reference', or 'objective' or ;default' priors. Perhaps not surprisingly, even the correct terminology to be used in this setting has been the subject of debate!

I will give an overview of some of the asymptotic theory behind the development of approaches to constructing priors that minimize the differences between Bayesian and frequentist inference, with special attention to 'strong matching' priors that have been developed recently in joint work with Don Fraser and colleagues. The construction of these priors provides some insight into the exact points of departure between Bayesian and frequentist methods, at least from the mathematical point of view. The philosophical debate may well continue for some time.

PIMS 10th Anniversary Speaker Series 10th Anniversary Speaker Series
Location: WMAX 110

A computer/math model of a process often contains numerous unknown parameters. To utilize the model, these parameters either must be determined from expert knowledge or estimated from data. In the mathematical community, the most common methods of estimating such parameters are through regularized best fit of various forms, and these generally work fine in the presence of plentiful data. However, when the data is limited, as is typically the case, parameter estimates obtained in this way can be very uncertain and use of the model without recognizing this uncertainty is quite problematic. Bayesian methods of solving such inverse problems have been growing in popularity, because of their ability to account for this uncertainty. A number of issues and problems that arise in implementing Bayesian methods will be discussed.

Many aspects of mind and brain that interest psychologists involve continuous processes. For this reason, and due to the advancement of data collection technology, researchers often use measurements that are sampled over time and space (e.g., brain imaging, movement tracking, and continuous behavioral judgments). Such data present a challenge to traditional statistical techniques that make assumptions about the normal distribution and independence of collected data points. Correlations and regression analyses, for example, summarize the relations between entire data sets without the potential to reveal how those relations evolve over time. Functional Data Analysis (FDA, Ramsay & Silverman, 2002, 2005; Heckman, 2003) is ideal for analyzing data derived from continuous processes. These techniques model data as functions, and can be used to reveal the underlying dynamics that drive a set of measurements. Software and tutorials are available for free download to researchers who are interested in incorporating FDA tools into their analyses (www.functionaldata.org). I will describe some applications of FDA in music cognition research, including smoothing, registration, functional principal components analysis, functional regression analysis, and functional significance testing. I will demonstrate these techniques using data from a study investigating multi-modal perception of musical performances.

Time-course microarray data consist of mRNA expression from a common set of genes collected at different time points. Such data are thought to reflect underlying biological processes developing over time. In this paper we propose a method to examine gene network relationships using time course microarray data. We assume that a sample of gene expression profiles is a realization of a process where each profile is modeled as a random functional transformation of a common curve. We propose measures of functional similarity and time order based on estimated time transformation functions. This allows for novel inferences on gene networks which takes full account of the timing of the functional features of the gene expression profiles. We discuss the application of our model to simulated data as well as to microarray data on prostate cancer progression. This is joint work with D. Telesca, M. Neira, C. Nelson and M. Gleave.

With very noisy data, overfitting is a serious problem in pattern recognition. For nonlinear regression, having plentiful data eliminates overfitting, but for nonlinear principal component analysis (NLPCA), overfitting persists even with plentiful data. Thus simply minimizing the mean square error (MSE) is not a sufficient criterion for NLPCA to find good solutions in noisy data. A new information criterion is proposed which selects the NLPCA curve (computed using auto-associative neural networks) with the right amount of flexibility so it neither underfits nor overfits. This information criterion also automatically chooses between using an open or a closed curve fit for a dataset. Nonlinear canonical correlation analysis (NLCCA) can also be performed using neural network models. A more robust version using the biweight midcorrelation instead of the Pearson correlation has been developed to work on noisy data. These methods are applied to tropical Pacific and equatorial stratospheric climate data.