Perceptual decision making can be described as a process of accumulating evidence to a bound which has been formalized within drift-diffusion models (DDMs). Recently, an equivalent Bayesian model has been proposed. In contrast to standard DDMs, this Bayesian model directly links information in the stimulus to the decision process. Here, we extend this Bayesian model further and allow inter-trial variability of two parameters following the extended version of the DDM. We derive parameter distributions for the Bayesian model and show that they lead to predictions that are qualitatively equivalent to those made by the extended drift-diffusion model (eDDM). Further, we demonstrate the usefulness of the extended Bayesian model (eBM) for the analysis of concrete behavioral data. Specifically, using Bayesian model selection, we find evidence that including additional inter-trial parameter variability provides for a better model, when the model is constrained by trial-wise stimulus features. This result is remarkable because it was derived using just 200 trials per condition, which is typically thought to be insufficient for identifying variability parameters in DDMs. In sum, we present a Bayesian analysis, which provides for a novel and promising analysis of perceptual decision making experiments.

Perceptual decision making can be described as a process of accumulating evidence to a bound which has been formalized within drift-diffusion models (DDMs). Recently, an equivalent Bayesian model has been proposed. In contrast to standard DDMs, this Bayesian model directly links information in the stimulus to the decision process. Here, we extend this Bayesian model further and allow inter-trial variability of two parameters following the extended version of the DDM. We derive parameter distributions for the Bayesian model and show that they lead to predictions that are qualitatively equivalent to those made by the extended drift-diffusion model (eDDM). Further, we demonstrate the usefulness of the extended Bayesian model (eBM) for the analysis of concrete behavioral data. Specifically, using Bayesian model selection, we find evidence that including additional inter-trial parameter variability provides for a better model, when the model is constrained by trial-wise stimulus features. This result is remarkable because it was derived using just 200 trials per condition, which is typically thought to be insufficient for identifying variability parameters in DDMs. In sum, we present a Bayesian analysis, which provides for a novel and promising analysis of perceptual decision making experiments.

The reliability analysis of plates and shells with respect to plastic collapse or to inadaptation is formulated on the basis of limit and shakedown theorems. The loading, the material strength and the shell thickness are considered as random variables. Based on a direct definition of the limit state function, the nonlinear problems may be efficiently solved by using the First and Second Order Reliability Methods (FORM/SORM). The sensitivity analyses in FORM/SORM can be based on the sensitivities of the deterministic shakedown problem. The problem of reliability of structural systems is also handled by the application of a special barrier technique which permits to find all the design points corresponding to all the failure modes. The direct plasticity approach reduces considerably the necessary knowledge of uncertain input data, computing costs and the numerical error. / Die Zuverlässigkeitsanalyse von Platten und Schalen in Bezug auf plastischen Kollaps oder Nicht-Anpassung wird mit den Traglast- und Einspielsätzen formuliert. Die Lasten, die Werkstofffestigkeit und die Schalendicke werden als Zufallsvariablen betrachtet. Auf der Grundlage einer direkten Definition der Grenzzustandsfunktion kann die Berechnung der Versagenswahrscheinlichkeit effektiv mit den Zuverlässigkeitsmethoden erster und zweiter Ordnung (FROM/SORM) gelöst werden. Die Sensitivitätsanalysen in FORM/SORM lassen sich auf der Basis der Sensitivitäten des deterministischen Einspielproblems berechnen. Die Schwierigkeiten bei der Ermittlung der Zuverlässigkeit von strukturellen Systemen werden durch Anwendung einer speziellen Barrieremethode behoben, die es erlaubt, alle Auslegungspunkte zu allen Versagensmoden zu finden. Die Anwendung direkter Plastizitätsmethoden führt zu einer beträchtlichen Verringerung der notwendigen Kenntnis der unsicheren Eingangsdaten, des Berechnungsaufwandes und der numerischen Fehler.

My thesis addresses a systematic approach to stochastic models in population genetics; in particular, the Wright-Fisher models affected only by the random genetic drift. I used various mathematical methods such as Probability, PDE, and Geometry to answer an important question: \"How do genetic change factors (random genetic drift, selection, mutation, migration, random environment, etc.) affect the behavior of gene frequencies or genotype frequencies in generations?”. In a Hardy-Weinberg model, the Mendelian population model of a very large number of individuals without genetic change factors, the answer is simple by the Hardy-Weinberg principle: gene frequencies remain unchanged from generation to generation, and genotype frequencies from the second generation onward remain also unchanged from generation to generation. With directional genetic change factors (selection, mutation, migration), we will have a deterministic dynamics of gene frequencies, which has been studied rather in detail. With non-directional genetic change factors (random genetic drift, random environment), we will have a stochastic dynamics of gene frequencies, which has been studied with much more interests. A combination of these factors has also been considered. We consider a monoecious diploid population of fixed size N with n + 1 possible alleles at a given locus A, and assume that the evolution of population was only affected by the random genetic drift. The question is that what the behavior of the distribution of relative frequencies of alleles in time and its stochastic quantities are. When N is large enough, we can approximate this discrete Markov chain to a continuous Markov with the same characteristics. In 1931, Kolmogorov first introduced a nice relation between a continuous Markov process and diffusion equations. These equations called the (backward/forward) Kolmogorov equations which have been first applied in population genetics in 1945 by Wright. Note that these equations are singular parabolic equations (diffusion coefficients vanish on boundary). To solve them, we use generalized hypergeometric functions. To know more about what will happen after the first exit time, or more general, the behavior of whole process, in joint work with J. Hofrichter, we define the global solution by moment conditions; calculate the component solutions by boundary flux method and combinatorics method. One interesting property is that some statistical quantities of interest are solutions of a singular elliptic second order linear equation with discontinuous (or incomplete) boundary values. A lot of papers, textbooks have used this property to find those quantities. However, the uniqueness of these problems has not been proved. Littler, in his PhD thesis in 1975, took up the uniqueness problem but his proof, in my view, is not rigorous. In joint work with J. Hofrichter, we showed two different ways to prove the uniqueness rigorously. The first way is the approximation method. The second way is the blow-up method which is conducted by J. Hofrichter. By applying the Information Geometry, which was first introduced by Amari in 1985, we see that the local state space is an Einstein space, and also a dually flat manifold with the Fisher metric; the differential operator of the Kolmogorov equation is the affine Laplacian which can be represented in various coordinates and on various spaces. Dynamics on the whole state space explains some biological phenomena.