In recent years the amount of biological data has increased exponentially. Most of these data can be represented as matrices relating two different entity types, such as drug-target interactions (relating drugs to protein targets), gene expression profiles (relating drugs or cell lines to genes), and drug sensitivity values (relating drugs to cell lines). Not only the size of these datasets is increasing, but also the number of different entity types that they relate. Furthermore, not all values in these datasets are typically observed, and some are very sparse. Matrix factorisation is a popular group of methods that can be used to analyse these matrices. The idea is that each matrix can be decomposed into two or more smaller matrices, such that their product approximates the original one. This factorisation of the data reveals patterns in the matrix, and gives us a lower-dimensional representation. Not only can we use this technique to identify clusters and other biological signals, we can also predict the unobserved entries, allowing us to prune biological experiments. In this thesis we introduce and explore several Bayesian matrix factorisation models, focusing on how to best use them for predicting these missing values in biological datasets. Our main hypothesis is that matrix factorisation methods, and in particular Bayesian variants, are an extremely powerful paradigm for predicting values in biological datasets, as well as other applications, and especially for sparse and noisy data. We demonstrate the competitiveness of these approaches compared to other state-of-the-art methods, and explore the conditions under which they perform the best. We consider several aspects of the Bayesian approach to matrix factorisation. Firstly, the effect of inference approaches that are used to find the factorisation on predictive performance. Secondly, we identify different likelihood and Bayesian prior choices that we can use for these models, and explore when they are most appropriate. Finally, we introduce a Bayesian matrix factorisation model that can be used to integrate multiple biological datasets, and hence improve predictions. This model hybridly combines different matrix factorisation models and Bayesian priors. Through these models and experiments we support our hypothesis and provide novel insights into the best ways to use Bayesian matrix factorisation methods for predictive purposes.

Thanks to the affordability of DNA sequencing, hundreds of thousands of individuals with rare disorders are undergoing whole-genome sequencing in an effort to reveal novel disease aetiologies, increase our understanding of biological processes and improve patient care. However, the power to discover the genetic causes of many unexplained rare diseases is hindered by a paucity of cases with a shared molecular aetiology. This thesis presents research into statistical and computational methods for determining the genetic causes of rare diseases. Methods described herein treat important aspects of the nature of rare diseases, including genetic and phenotypic heterogeneity, phenotypes involving multiple organ systems, Mendelian modes of inheritance and the incorporation of complex prior information such as model organism phenotypes and evolutionary conservation. The complex nature of rare disease phenotypes and the need to aggregate patient data across many centres has led to the adoption of the Human Phenotype Ontology (HPO) as a means of coding patient phenotypes. The HPO provides a standardised vocabulary and captures relationships between disease features. I developed a suite of software packages dubbed 'ontologyX' in order to simplify analysis and visualisation of such ontologically encoded data, and enable them to be incorporated into complex analysis methods. An important aspect of the analysis of ontological data is quantifying the semantic similarity between ontologically annotated entities, which is implemented in the ontologyX software. We employed this functionality in a phenotypic similarity regression framework, 'SimReg', which models the relationship between ontologically encoded patient phenotypes of individuals and rare variation in a given genomic locus. It does so by evaluating support for a model under which the probability that a person carries rare alleles in a locus depends on the similarity between the person's ontologically encoded phenotype and a latent characteristic phenotype which can be inferred from data. A probability of association is computed by comparison of the two models, allowing prioritisation of candidate loci for involvement in disease with respect to a heterogeneous collection of disease phenotypes. SimReg includes a sophisticated treatment of HPO-coded phenotypic data but dichotomises the genetic data at a locus. Therefore, we developed an additional method, 'BeviMed', standing for Bayesian Evaluation of Variant Involvement in Mendelian Disease, which evaluates the evidence of association between allele configurations across rare variants within a genomic locus and a case/control label. It is capable of inferring the probability of association, and conditional on association, the probability of each mode of inheritance and probability of involvement of each variant. Inference is performed through a Bayesian comparison of multiple models: under a baseline model disease risk is independent of allele configuration at the given rare variant sites and under an alternate model disease risk depends on the configuration of alleles, a latent partition of variants into pathogenic and non-pathogenic groups and a mode of inheritance. The method can be used to analyse a dataset comprising thousands of individuals genotyped at hundreds of rare variant sites in a fraction of a second, making it much faster than competing methods and facilitating genome-wide application.

It has been long hypothesized that due to the inherent ambiguities of visual input and the limitations of the visual system, vision is a form of “unconscious inference” whereby the brain relies on assumptions (aka expectations) to interpret the external world. This hypothesis has been recently formalized into Bayesian models of perception (the “Bayesian brain”) that represent these expectations as prior probabilities. In this thesis, I focus on a particular kind of expectation that humans are thought to possess – that objects in the world tend to be still or move slowly – known as the “slow speed prior”. Through a combination of experimental and theoretical work, I investigate how the speed prior is acquired and how it impacts motion perception. The first part of my work consists of an experiment where subjects are exposed to simple "training" stimuli moving more often at high speeds than at low speeds. By subsequently testing the subjects with slow-moving stimuli of high uncertainty (low contrast), I find that their perception gradually changes in a manner consistent with the progressive acquisition of an expectation that favours progressively higher speeds. Thus subjects appear to gradually internalize the speed statistics of the stimulus ensemble over the duration of the experiment. I model these results using an existing Bayesian model of motion perception that incorporates a speed prior with a peak at zero, extending the model so that the mean gradually shifts away from zero. Although the first experiment presents evidence for the plasticity of the speed prior, the experimental paradigm and the constraints of the model limit the accuracy and precision in the reconstruction of observers’ priors. To address these limitations, I perform a different experiment where subjects compare the speed of moving gratings of different contrasts. The new paradigm allows more precise measurements of the contrast-dependent biases in perceived speed. Using a less constrained Bayesian model, I extract the priors of subjects and find considerable interindividual variability. Furthermore, noting that the Bayesian model cannot account for certain subtleties in the data, I combine the model with a non-Bayesian, physiologically motivated model of speed tuning of cortical neurons and show that the combination offers an improved description of the data. Using the paradigm of the second experiment, I then explore the role of visual experience on the form of the speed prior. By recruiting avid video gamers (who are routinely exposed to high speeds) and nongamers of both sexes, I study the differences in the prior among groups and find, surprisingly, that subjects’ speed priors depend more on gender than on gaming experience. In a final series of experiments similar to the first, I also test subjects on variations of the trained stimulus configuration – namely different orientations and motion directions. Subjects’ responses suggest that they are able to apply the changed prior to different orientations and, furthermore, that the changed prior persists for at least a week after the end of the experiment. These results provide further support for the plasticity of the speed prior but also suggest that the learned prior may be used only across similar stimulus configurations, whereas in sufficiently different configurations or contexts a “default” prior may be used instead.

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This dissertation explores the modeling and estimation of learning in strategic and individual choice settings. While learning has been extensively used in economics, I introduce the concept into standard models in unorthodox ways. In each case, changing the perspective of what learning is drastically changes standard models. Estimation proceeds using advanced Bayesian techniques which perform very well in simulated data. The first chapter proposes a framework called Experienced-Based Ability (EBA) in which players increase the payoffs of a particular strategy in the future through using the strategy today. This framework is then introduced into a model of differentiated duopoly in which firms can utilize price or quantity contracts, and I explore how the resulting equilibrium is affected by changes in model parameters. The second chapter extends the EBA model into an evolutionary setting. This new model offers a simple and intuitive way to theoretically explain complicated dynamics. Moreover, this chapter demonstrates how to estimate posterior distributions of the model's parameters using a particle filter and Metropolis-Hastings algorithm, a technique that can also be used in estimating standard evolutionary models. This allows researchers to recover estimates of unobserved fitness and skill across time while only observing population share data. The third chapter investigates individual learning in a dynamic discrete choice setting. This chapter relaxes the assumption that individuals base decisions off an optimal policy and investigates the importance of policy learning. Q-learning is proposed as a model of individual choice when optimal policies are unknown, and I demonstrate how it can be used in the estimation of dynamic discrete choice (DDC) models. Using Bayesian Markov chain Monte Carlo techniques on simulated data, I show that the Q-learning model performs well at recovering true parameter values and thus functions as an alternative structural DDC model for researchers who want to move away from the rationality assumption. In addition, the simulated data are used to illustrate possible issues with standard structural estimation if the rationality assumption is incorrect. Lastly, using marginal likelihood analysis, I demonstrate that the Q-learning model can be used to test for the significance of learning effects if this is a concern.

For the empirical macroeconomist, accounting for nonlinearities in data series by using regime switching techniques has a long history. Over the past 25 years, there have been tremendous advances in both the estimation of regime switching and the incorporation of regime switching into macroeconomic models. In this dissertation, I apply techniques from this literature to study two topics that are of particular relevance to the conduct of monetary policy: asset bubbles and the Federal Reserve’s policy reaction function. My first chapter utilizes a recently developed Markov-Switching model in order to test for asset bubbles in simulated data. I find that this flexible model is able to detect asset bubbles in about 75% of simulations. In my second and third chapters, I focus on the Federal Reserve’s policy reaction function. My second chapter advances the literature in two important directions. First, it uses meeting- based timing to more properly account for the target Federal Funds rate; second, it allows for the inclusion of up to 14 economic variables. I find that the long-run inflation response coefficient is larger than had been found in previous studies, and that increasing the number of economic variables that can enter the model improves both in-sample fit and out-of-sample forecasting ability. In my third chapter, I introduce a new econometric model that allows for Markov-Switching, but can also remove variables from the model, or enforce a restriction that there is no regime switching. My findings indicate that the majority of coefficients in the Federal Reserve’s policy reaction function have not changed over time.

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abstract: Statistical mediation analysis has been widely used in the social sciences in order to examine the indirect effects of an independent variable on a dependent variable. The statistical properties of the single mediator model with manifest and latent variables have been studied using simulation studies. However, the single mediator model with latent variables in the Bayesian framework with various accurate and inaccurate priors for structural and measurement model parameters has yet to be evaluated in a statistical simulation. This dissertation outlines the steps in the estimation of a single mediator model with latent variables as a Bayesian structural equation model (SEM). A Monte Carlo study is carried out in order to examine the statistical properties of point and interval summaries for the mediated effect in the Bayesian latent variable single mediator model with prior distributions with varying degrees of accuracy and informativeness. Bayesian methods with diffuse priors have equally good statistical properties as Maximum Likelihood (ML) and the distribution of the product. With accurate informative priors Bayesian methods can increase power up to 25% and decrease interval width up to 24%. With inaccurate informative priors the point summaries of the mediated effect are more biased than ML estimates, and the bias is higher if the inaccuracy occurs in priors for structural parameters than in priors for measurement model parameters. Findings from the Monte Carlo study are generalizable to Bayesian analyses with priors of the same distributional forms that have comparable amounts of (in)accuracy and informativeness to priors evaluated in the Monte Carlo study. / Dissertation/Thesis / Doctoral Dissertation Psychology 2017

abstract: Research methods based on the frequentist philosophy use prior information in a priori power calculations and when determining the necessary sample size for the detection of an effect, but not in statistical analyses. Bayesian methods incorporate prior knowledge into the statistical analysis in the form of a prior distribution. When prior information about a relationship is available, the estimates obtained could differ drastically depending on the choice of Bayesian or frequentist method. Study 1 in this project compared the performance of five methods for obtaining interval estimates of the mediated effect in terms of coverage, Type I error rate, empirical power, interval imbalance, and interval width at N = 20, 40, 60, 100 and 500. In Study 1, Bayesian methods with informative prior distributions performed almost identically to Bayesian methods with diffuse prior distributions, and had more power than normal theory confidence limits, lower Type I error rates than the percentile bootstrap, and coverage, interval width, and imbalance comparable to normal theory, percentile bootstrap, and the bias-corrected bootstrap confidence limits. Study 2 evaluated if a Bayesian method with true parameter values as prior information outperforms the other methods. The findings indicate that with true values of parameters as the prior information, Bayesian credibility intervals with informative prior distributions have more power, less imbalance, and narrower intervals than Bayesian credibility intervals with diffuse prior distributions, normal theory, percentile bootstrap, and bias-corrected bootstrap confidence limits. Study 3 examined how much power increases when increasing the precision of the prior distribution by a factor of ten for either the action or the conceptual path in mediation analysis. Power generally increases with increases in precision but there are many sample size and parameter value combinations where precision increases by a factor of 10 do not lead to substantial increases in power. / Dissertation/Thesis / Masters Thesis Psychology 2014

Forestry in the UK is dominated by coniferous plantations. Sitka spruce (Picea sitchensis) and Scots pine (Pinus sylvestris) are the most prevalent species and are mostly grown in single age mono-culture stands. Forest strategy for Scotland, England, and Wales all include efforts to achieve further afforestation. The aim of this afforestation is to provide a multi-functional forest with a broad range of benefits. Due to the time scale involved in forestry, accurate forecasts of stand productivity (along with clearly defined uncertainties) are essential to forest managers. These can be provided by a range of approaches to modelling forest growth. In this project model comparison, Bayesian calibration, and data assimilation methods were all used to attempt to improve forecasts and understanding of uncertainty therein of the two most important conifers in UK forestry. Three different forest growth models were compared in simulating growth of Scots pine. A yield table approach, the process-based 3PGN model, and a Stand Level Dynamic Growth (SLeDG) model were used. Predictions were compared graphically over the typical productivity range for Scots pine in the UK. Strengths and weaknesses of each model were considered. All three produced similar growth trajectories. The greatest difference between models was in volume and biomass in unthinned stands where the yield table predicted a much larger range compared to the other two models. Future advances in data availability and computing power should allow for greater use of process-based models, but in the interim more flexible dynamic growth models may be more useful than static yield tables for providing predictions which extend to non-standard management prescriptions and estimates of early growth and yield. A Bayesian calibration of the SLeDG model was carried out for both Sitka spruce and Scots pine in the UK for the first time. Bayesian calibrations allow both model structure and parameters to be assessed simultaneously in a probabilistic framework, providing a model with which forecasts and their uncertainty can be better understood and quantified using posterior probability distributions. Two different structures for including local productivity in the model were compared with a Bayesian model comparison. A complete calibration of the more probable model structure was then completed. Example forecasts from the calibration were compatible with existing yield tables for both species. This method could be applied to other species or other model structures in the future. Finally, data assimilation was investigated as a way of reducing forecast uncertainty. Data assimilation assumes that neither observations nor models provide a perfect description of a system, but combining them may provide the best estimate. SLeDG model predictions and LiDAR measurements for sub-compartments within Queen Elizabeth Forest Park were combined with an Ensemble Kalman Filter. Uncertainty was reduced following the second data assimilation in all of the state variables. However, errors in stand delineation and estimated stand yield class may have caused observational uncertainty to be greater thus reducing the efficacy of the method for reducing overall uncertainty.