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A Statistical Primer for Ecologists

trained ecologists who object to treating states as random variables don’t mind using hypothesis tests that are grounded in the idea of a long-term frequency of observation in repeated observations, which don’t sensibly exist in many cases . . . .” Simple Bayesian Models • 81 5.1 Bayes’ Theorem The basic problem in ecological research is to understand processes that we cannot observe based on quantities that we can observe. We represent unobserved processes as models made up of parameters and latent states, which we notate here as θ . We make observations y to learn

6 Hierarch ica l Bayes ian Mode l s It is worthwhile to review the key points covered thus far. We started with the first principles rules of probability (sec. 3.2). We used those rules to develop Bayes’ theorem (sec. 5.1) and to show how we can factor joint distributions of observed and unobserved quantities into parts based on our knowledge of conditioning and independence (sec. 3.3). We learned about priors and their influence on the posterior (sec. 5.4). We now apply what we have learned to ecological examples of hierarchi- cal Bayesian models. These models

10 Wri t ing Bayes ian Mode l s Wespecify a model by writing it in mathematical and statistical notation. Recall that model specification occupies a central position in the modeling process, linking design to implementation (fig. 0.0.1). The crucial skill of specifying models is often neglected in statistical texts in general and texts on Bayesian modeling in particular. The central importance of model specification motivates this chapter. Building models is a learned craft, and no two modelers conduct their craft in exactly the same way. It can be daunting to

Volume 11, Issue 4 2012 Article 6 Statistical Applications in Genetics and Molecular Biology An Integrated Hierarchical Bayesian Model for Multivariate eQTL Mapping Marie Pier Scott-Boyer, Institut de recherches cliniques de Montréal (IRCM) and Université de Montréal Gregory C. Imholte, Fred Hutchinson Cancer Research Center Arafat Tayeb, Institut de recherches cliniques de Montréal (IRCM) and Université de Montréal Aurelie Labbe, University McGill Christian F. Deschepper, Institut de recherches cliniques de Montréal (IRCM) and Université de Montréal Raphael

-Dor, Zuk, and Domany (2006) and Zuk, Ein-Dor, and Domany (2007) suggested a Bayesian modelling of the observational noise whose assumptions include one about asymptotic independence of the sample correlations. Then, with regard to this modelling, the conclusion was that thousands of samples are needed to generate a robust gene-list for predicting outcome. The message of this work consists of two main points: (1) the assumptions of Ein-Dor, Zuk, and Domany (2006) and Zuk, Ein-Dor, and Domany (2007) are mathematically strong and hence they are less convincing from a

Volume 6, Issue 3 2010 Article 5 Journal of Quantitative Analysis in Sports Bayesian Modeling of Footrace Finishing Times Matthew S. Shotwell, Medical University of South Carolina Elizabeth H. Slate, Medical University of South Carolina Recommended Citation: Shotwell, Matthew S. and Slate, Elizabeth H. (2010) "Bayesian Modeling of Footrace Finishing Times," Journal of Quantitative Analysis in Sports: Vol. 6: Iss. 3, Article 5. DOI: 10.2202/1559-0410.1206 ©2010 American Statistical Association. All rights reserved. Bayesian Modeling of Footrace Finishing Times

Chapter 2 Turning a DSGE Model into a Bayesian Model Formally, a Bayesian model consists of a joint distribution of data Y and parameters θ. In the context of a DSGE model application Y might comprise time series for GDP growth, inflation, and interest rates, and θ stacks the structural pa- rameters that, for instance, appeared in the description of the small-scale DSGE model in Section 1.1 of Chapter 1. Throughout this book we will represent distributions by den- sities and denote the joint distribution by p(Y, θ). The joint distribution can be factored into a

pik X , Kiki5iki j = l k = l 1 = 1 That act (or an act if there is more than one) is performed for which Wj is a maximum. A Bayesian Model of Common-sense Deliberation 335 It is one thing to provide an abstract model of deliberation, but it is quite another thing, of course, to apply it to actual deliberations. The difficulty lies in devising a practical method of determining values for K and 5, i.e., for the kind and degree factors. Although I am not able to provide a complete solution to this problem, I shall try to indicate the general lines along which

Volume 6, Issue 1 2007 Article 11 Statistical Applications in Genetics and Molecular Biology A Bayesian Model of AFLP Marker Evolution and Phylogenetic Inference Ruiyan Luo, University of Wisconsin - Madison Andrew L. Hipp, The Morton Arboretum Bret Larget, University of Wisconsin - Madison Recommended Citation: Luo, Ruiyan; Hipp, Andrew L.; and Larget, Bret (2007) "A Bayesian Model of AFLP Marker Evolution and Phylogenetic Inference," Statistical Applications in Genetics and Molecular Biology: Vol. 6: Iss. 1, Article 11. DOI: 10.2202/1544-6115.1152 A Bayesian