Showing papers on "Bayes' theorem published in 2011"

Comparing isotopic niche widths among and within communities: SIBER - Stable Isotope Bayesian Ellipses in R.

Abstract: 1. The use of stable isotope data to infer characteristics of community structure and niche width of community members has become increasingly common. Although these developments have provided ecologists with new perspectives, their full impact has been hampered by an inability to statistically compare individual communities using descriptive metrics. 2. We solve these issues by reformulating the metrics in a Bayesian framework. This reformulation takes account of uncertainty in the sampled data and naturally incorporates error arising from the sampling process, propagating it through to the derived metrics. 3. Furthermore, we develop novel multivariate ellipse-based metrics as an alternative to the currently employed Convex Hull methods when applied to single community members. We show that unlike Convex Hulls, the ellipses are unbiased with respect to sample size, and their estimation via Bayesian inference allows robust comparison to be made among data sets comprising different sample sizes. 4. These new metrics, which we call SIBER (Stable Isotope Bayesian Ellipses in R), open up more avenues for direct comparison of isotopic niches across communities. The computational code to calculate the new metrics is implemented in the free-to-download package Stable Isotope Analysis for the R statistical environment.

Improving Marginal Likelihood Estimation for Bayesian Phylogenetic Model Selection

Abstract: The marginal likelihood is commonly used for comparing different evolutionary models in Bayesian phylogenet- ics and is the central quantity used in computing Bayes Factors for comparing model fit. A popular method for estimating marginal likelihoods, the harmonic mean (HM) method, can be easily computed from the output of a Markov chain Monte Carlo analysis but often greatly overestimates the marginal likelihood. The thermodynamic integration (TI) method is much more accurate than the HM method but requires more computation. In this paper, we introduce a new method, stepping- stone sampling (SS), which uses importance sampling to estimate each ratio in a series (the "stepping stones") bridging the posterior and prior distributions. We compare the performance of the SS approach to the TI and HM methods in simulation and using real data. We conclude that the greatly increased accuracy of the SS and TI methods argues for their use instead of the HM method, despite the extra computation needed. (Bayes factor; harmonic mean; phylogenetics, marginal likelihood; model selection; path sampling; thermodynamic integration; steppingstone sampling.)

Bayesian versus Orthodox statistics: which side are you on?

Abstract: Researchers are often confused about what can be inferred from significance tests. One problem occurs when people apply Bayesian intuitions to significance testing—two approaches that must be firmly separated. This article presents some common situations in which the approaches come to different conclusions; you can see where your intuitions initially lie. The situations include multiple testing, deciding when to stop running participants, and when a theory was thought of relative to finding out results. The interpretation of nonsignificant results has also been persistently problematic in a way that Bayesian inference can clarify. The Bayesian and orthodox approaches are placed in the context of different notions of rationality, and I accuse myself and others as having been irrational in the way we have been using statistics on a key notion of rationality. The reader is shown how to apply Bayesian inference in practice, using free online software, to allow more coherent inferences from data.

Survey of Branch Support Methods Demonstrates Accuracy, Power, and Robustness of Fast Likelihood-based Approximation Schemes

Abstract: Phylogenetic inference and evaluating support for inferred relationships is at the core of many studies testing evolutionary hypotheses. Despite the popularity of nonparametric bootstrap frequencies and Bayesian posterior probabil- ities, the interpretation of these measures of tree branch support remains a source of discussion. Furthermore, both meth- ods are computationally expensive and become prohibitive for large data sets. Recent fast approximate likelihood-based measures of branch supports (approximate likelihood ratio test (aLRT) and Shimodaira-Hasegawa (SH)-aLRT) provide a compelling alternative to these slower conventional methods, offering not only speed advantages but also excellent levels of accuracy and power. Here we propose an additional method: a Bayesian-like transformation of aLRT (aBayes). Consider- ing both probabilistic and frequentist frameworks, we compare the performance of the three fast likelihood-based methods with the standard bootstrap (SBS), the Bayesian approach, and the recently introduced rapid bootstrap. Our simulations and real data analyses show that with moderate model violations, all tests are sufficiently accurate, but aLRT and aBayes offer the highest statistical power and are very fast. With severe model violations aLRT, aBayes and Bayesian posteriors can produce elevated false-positive rates. With data sets for which such violation can be detected, we recommend using SH-aLRT, the nonparametric version of aLRT based on a procedure similar to the Shimodaira-Hasegawa tree selection. In general, the SBS seems to be excessively conservative and is much slower than our approximate likelihood-based methods. (Accuracy; aLRT; branch support methods; evolution; model violation; phylogenetic inference; power; SH-aLRT.)

Spontaneous Cortical Activity Reveals Hallmarks of an Optimal Internal Model of the Environment

Abstract: The brain maintains internal models of its environment to interpret sensory inputs and to prepare actions. Although behavioral studies have demonstrated that these internal models are optimally adapted to the statistics of the environment, the neural underpinning of this adaptation is unknown. Using a Bayesian model of sensory cortical processing, we related stimulus-evoked and spontaneous neural activities to inferences and prior expectations in an internal model and predicted that they should match if the model is statistically optimal. To test this prediction, we analyzed visual cortical activity of awake ferrets during development. Similarity between spontaneous and evoked activities increased with age and was specific to responses evoked by natural scenes. This demonstrates the progressive adaptation of internal models to the statistics of natural stimuli at the neural level.

Review: Bayesian networks in environmental modelling

Abstract: Bayesian networks (BNs), also known as Bayesian belief networks or Bayes nets, are a kind of probabilistic graphical model that has become very popular to practitioners mainly due to the powerful probability theory involved, which makes them able to deal with a wide range of problems. The goal of this review is to show how BNs are being used in environmental modelling. We are interested in the application of BNs, from January 1990 to December 2010, in the areas of the ISI Web of Knowledge related to Environmental Sciences. It is noted that only the 4.2% of the papers have been published under this item. The different steps that configure modelling via BNs have been revised: aim of the model, data pre-processing, model learning, validation and software. Our literature review indicates that BNs have barely been used for Environmental Science and their potential is, as yet, largely unexploited.

A Bayesian Foundation for Individual Learning Under Uncertainty

Abstract: Computational learning models are critical for understanding mechanisms of adaptive behavior. However, the two major current frameworks, reinforcement learning (RL) and Bayesian learning, both have certain limitations. For example, many Bayesian models are agnostic of inter-individual variability and involve complicated integrals, making online learning difficult. Here, we introduce a generic hierarchical Bayesian framework for individual learning under multiple forms of uncertainty (e.g., environmental volatility and perceptual uncertainty). The model assumes Gaussian random walks of states at all but the first level, with the step size determined by the next higher level. The coupling between levels is controlled by parameters that shape the influence of uncertainty on learning in a subject-specific fashion. Using variational Bayes under a mean field approximation and a novel approximation to the posterior energy function, we derive trial-by-trial update equations which (i) are analytical and extremely efficient, enabling real-time learning, (ii) have a natural interpretation in terms of RL, and (iii) contain parameters representing processes which play a key role in current theories of learning, e.g., precision-weighting of prediction error. These parameters allow for the expression of individual differences in learning and may relate to specific neuromodulatory mechanisms in the brain. Our model is very general: it can deal with both discrete and continuous states and equally accounts for deterministic and probabilistic relations between environmental events and perceptual states (i.e., situations with and without perceptual uncertainty). These properties are illustrated by simulations and analyses of empirical time series. Overall, our framework provides a novel foundation for understanding normal and pathological learning that contextualizes RL within a generic Bayesian scheme and thus connects it to principles of optimality from probability theory.

Bayesian Fundamentalism or Enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition.

Abstract: The prominence of Bayesian modeling of cognition has increased recently largely because of mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations. We note commonalities between this rational approach and other movements in psychology - namely, Behaviorism and evolutionary psychology - that set aside mechanistic explanations or make use of optimality assumptions. Through these comparisons, we identify a number of challenges that limit the rational program's potential contribution to psychological theory. Specifically, rational Bayesian models are significantly unconstrained, both because they are uninformed by a wide range of process-level data and because their assumptions about the environment are generally not grounded in empirical measurement. The psychological implications of most Bayesian models are also unclear. Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the hypothesis sets and the approximation algorithms used to derive model predictions, without a clear delineation between psychological commitments and implementational details. Comparing multiple Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we argue they must be developed in conjunction with mechanistic considerations to offer substantive explanations of cognition. We lay out several means for such an integration, which take into account the representations on which Bayesian inference operates, as well as the algorithms and heuristics that carry it out. We argue this unification will better facilitate lasting contributions to psychological theory, avoiding the pitfalls that have plagued previous theoretical movements.

Bayesian Assessment of Null Values Via Parameter Estimation and Model Comparison

Abstract: Psychologists have been trained to do data analysis by asking whether null values can be rejected. Is the difference between groups nonzero? Is choice accuracy not at chance level? These questions have been traditionally addressed by null hypothesis significance testing (NHST). NHST has deep problems that are solved by Bayesian data analysis. As psychologists transition to Bayesian data analysis, it is natural to ask how Bayesian analysis assesses null values. The article explains and evaluates two different Bayesian approaches. One method involves Bayesian model comparison (and uses Bayes factors). The second method involves Bayesian parameter estimation and assesses whether the null value falls among the most credible values. Which method to use depends on the specific question that the analyst wants to answer, but typically the estimation approach (not using Bayes factors) provides richer information than the model comparison approach.

Bayesian Modeling of Uncertainty in Low-Level Vision

Abstract: Over the last decade, many low-level vision algorithms have been devised for extracting depth from one or more intensity images. The output of such algorithms usually contains no indication of the uncertainty associated with the scene reconstruction. In other areas of computer vision and robotics, the need for such error modeling is becoming recognized, both because of the uncertainty inherent in sensing and because of the desire to integrate information from different sensors or viewpoints. In this thesis, we develop a new Bayesian model for the dense fields that are commonly used in low-level vision. The Bayesian model consists of three components: a prior model, a sensor model, and a posterior model. The prior model captures any a priori information about the structure of the dense field. We construct this model by using the smoothness constraints for regularization to define a Markov Random Field. The sensor model describes the behaviour and noise characteristics of our measurement system. We develop a number of sensor models for both sparse depth measurements and dense flow or intensity measurements. The posterior model combines the information from the prior and sensor models using Bayes' Rule, and can be used as the input to later stages of processing. We show how to compute optimal estimates from the posterior model, and also how to compute the uncertainty (variance) in these estimates. This thesis applies Bayesian modeling to a number of low-level vision problems. The main application is the on-line extraction of depth from motion. For this application, we use a two-dimensional generalization of the Kalman filter to convert the current posterior model into a prior model for the next estimate. The resulting incremental algorithm provides a dense on-line estimate of depth whose uncertainty and error are reduced over time. Other applications of Bayesian modeling, include the choice of optimal smoothing parameter for interpolation; the determination of observer motion from sparse depth measurements without correspondence; and the construction of multiresolution relative surface representations. The approach to uncertainty modeling which we develop, and the utility of this approach in various applications, support our thesis that Bayesian modeling is a useful and practical framework for low-level vision.

Lack of confidence in approximate Bayesian computation model choice

Abstract: Approximate Bayesian computation (ABC) have become an essential tool for the analysis of complex stochastic models. Grelaud et al. [(2009) Bayesian Anal 3:427–442] advocated the use of ABC for model choice in the specific case of Gibbs random fields, relying on an intermodel sufficiency property to show that the approximation was legitimate. We implemented ABC model choice in a wide range of phylogenetic models in the Do It Yourself-ABC (DIY-ABC) software [Cornuet et al. (2008) Bioinformatics 24:2713–2719]. We now present arguments as to why the theoretical arguments for ABC model choice are missing, because the algorithm involves an unknown loss of information induced by the use of insufficient summary statistics. The approximation error of the posterior probabilities of the models under comparison may thus be unrelated with the computational effort spent in running an ABC algorithm. We then conclude that additional empirical verifications of the performances of the ABC procedure as those available in DIY-ABC are necessary to conduct model choice.

A Random-Finite-Set Approach to Bayesian SLAM

Abstract: This paper proposes an integrated Bayesian frame work for feature-based simultaneous localization and map building (SLAM) in the general case of uncertain feature number and data association. By modeling the measurements and feature map as random finite sets (RFSs), a formulation of the feature-based SLAM problem is presented that jointly estimates the number and location of the features, as well as the vehicle trajectory. More concisely, the joint posterior distribution of the set-valued map and vehicle trajectory is propagated forward in time as measurements arrive, thereby incorporating both data association and feature management into a single recursion. Furthermore, the Bayes optimality of the proposed approach is established. A first-order solution, which is coined as the probability hypothesis density (PHD) SLAM filter, is derived, which jointly propagates the posterior PHD of the map and the posterior distribution of the vehicle trajectory. A Rao-Blackwellized (RB) implementation of the PHD-SLAM filter is proposed based on the Gaussian-mixture PHD filter (for the map) and a particle filter (for the vehicle trajectory). Simulated and experimental results demonstrate the merits of the proposed approach, particularly in situations of high clutter and data association ambiguity.

Hierarchical Commensurate and Power Prior Models for Adaptive Incorporation of Historical Information in Clinical Trials

Abstract: Bayesian clinical trial designs offer the possibility of a substantially reduced sample size, increased statistical power, and reductions in cost and ethical hazard. However when prior and current information conflict, Bayesian methods can lead to higher than expected type I error, as well as the possibility of a costlier and lengthier trial. This motivates an investigation of the feasibility of hierarchical Bayesian methods for incorporating historical data that are adaptively robust to prior information that reveals itself to be inconsistent with the accumulating experimental data. In this article, we present several models that allow for the commensurability of the information in the historical and current data to determine how much historical information is used. A primary tool is elaborating the traditional power prior approach based upon a measure of commensurability for Gaussian data. We compare the frequentist performance of several methods using simulations, and close with an example of a colon cancer trial that illustrates a linear models extension of our adaptive borrowing approach. Our proposed methods produce more precise estimates of the model parameters, in particular, conferring statistical significance to the observed reduction in tumor size for the experimental regimen as compared to the control regimen.

Tweedie's Formula and Selection Bias.

Abstract: We suppose that the statistician observes some large number of estimates zi, each with its own unobserved expectation parameter μi. The largest few of the zi’s are likely to substantially overestimate their corresponding μi’s, this being an example of selection bias, or regression to the mean. Tweedie’s formula, first reported by Robbins in 1956, offers a simple empirical Bayes approach for correcting selection bias. This article investigates its merits and limitations. In addition to the methodology, Tweedie’s formula raises more general questions concerning empirical Bayes theory, discussed here as “relevance” and “empirical Bayes information.” There is a close connection between applications of the formula and James–Stein estimation.

Approximate Likelihood Calculation on a Phylogeny for Bayesian Estimation of Divergence Times

Abstract: The molecularclockprovidesapowerfulwaytoestimatespeciesdivergencetimes.Ifinformationonsomespeciesdivergence times is available from the fossil or geological record, it can be used to calibrate a phylogeny and estimate divergence times for all nodes in the tree. The Bayesian method provides a natural framework to incorporate different sources of information concerningdivergencetimes,such asinformationinthefossiland molecular data.Currentmodels ofsequenceevolutionare intractable in a Bayesian setting, and Markov chain Monte Carlo (MCMC) is used to generate the posterior distribution of divergence timesandevolutionaryrates.This methodiscomputationallyexpensive,asitinvolvesthe repeatedcalculationof the likelihoodfunction.Here, we explorethe use ofTaylor expansionto approximatethe likelihoodduring MCMC iteration. The approximation is much faster than conventionallikelihood calculation. However, the approximationis expected to be poor when the proposed parameters are far from the likelihood peak. We explore the use of parameter transforms (square root, logarithm,andarcsine)toimprove theapproximationto the likelihoodcurve. We foundthat thenew methods, particularly thearcsine-basedtransform,providedvery good approximationsunder relaxedclockmodelsandalsounderthe global clock model when the global clock is not seriously violated. The approximationis poorer for analysis under the global clock when the globalclockisseriously wrong andshouldthus notbe used. The resultssuggest that theapproximatemethodmay be useful for Bayesiandating analysisusing large data sets.

On the half-Cauchy prior for a global scale parameter

Abstract: This paper argues that the half-Cauchy distribution should replace the inverse-Gamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the original case made by Gelman (2006) in support of the folded-t family of priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these priors are used for a top-level normal variance in a Bayesian hierarchical model. We go on to prove a proposition that, together with the results for moments and marginals, allows us to characterize the frequentist risk of the Bayes estimators under all global-shrinkage priors in the class. These theoretical results, in turn, allow us to study the frequentist properties of the half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies a sensible 'middle ground' within this class: it performs very well near the origin, but does not lead to drastic compromises in other parts of the parameter space. This provides an alternative, classical justification for the repeated, routine use of this prior. We also consider situations where the underlying mean vector is sparse, where we argue that the usual conjugate choice of an inverse-gamma prior is particularly inappropriate, and can lead to highly distorted posterior inferences. Finally, we briefly summarize some open issues in the specification of default priors for scale terms in hierarchical models.

The Bayesian lasso for genome-wide association studies

Abstract: Motivation: Despite their success in identifying genes that affect complex disease or traits, current genome-wide association studies (GWASs) based on a single SNP analysis are too simple to elucidate a comprehensive picture of the genetic architecture of phenotypes. A simultaneous analysis of a large number of SNPs, although statistically challenging, especially with a small number of samples, is crucial for genetic modeling. Method: We propose a two-stage procedure for multi-SNP modeling and analysis in GWASs, by first producing a ‘preconditioned’ response variable using a supervised principle component analysis and then formulating Bayesian lasso to select a subset of significant SNPs. The Bayesian lasso is implemented with a hierarchical model, in which scale mixtures of normal are used as prior distributions for the genetic effects and exponential priors are considered for their variances, and then solved by using the Markov chain Monte Carlo (MCMC) algorithm. Our approach obviates the choice of the lasso parameter by imposing a diffuse hyperprior on it and estimating it along with other parameters and is particularly powerful for selecting the most relevant SNPs for GWASs, where the number of predictors exceeds the number of observations. Results: The new approach was examined through a simulation study. By using the approach to analyze a real dataset from the Framingham Heart Study, we detected several significant genes that are associated with body mass index (BMI). Our findings support the previous results about BMI-related SNPs and, meanwhile, gain new insights into the genetic control of this trait. Availability: The computer code for the approach developed is available at Penn State Center for Statistical Genetics web site, http://statgen.psu.edu. Contact: rwu@hes.hmc.psu.edu Supplementary information:Supplementary data are available at Bioinformatics online.

Risk, Unexpected Uncertainty, and Estimation Uncertainty: Bayesian Learning in Unstable Settings

Abstract: Recently, evidence has emerged that humans approach learning using Bayesian updating rather than (model-free) reinforcement algorithms in a six-arm restless bandit problem. Here, we investigate what this implies for human appreciation of uncertainty. In our task, a Bayesian learner distinguishes three equally salient levels of uncertainty. First, the Bayesian perceives irreducible uncertainty or risk: even knowing the payoff probabilities of a given arm, the outcome remains uncertain. Second, there is (parameter) estimation uncertainty or ambiguity: payoff probabilities are unknown and need to be estimated. Third, the outcome probabilities of the arms change: the sudden jumps are referred to as unexpected uncertainty. We document how the three levels of uncertainty evolved during the course of our experiment and how it affected the learning rate. We then zoom in on estimation uncertainty, which has been suggested to be a driving force in exploration, in spite of evidence of widespread aversion to ambiguity. Our data corroborate the latter. We discuss neural evidence that foreshadowed the ability of humans to distinguish between the three levels of uncertainty. Finally, we investigate the boundaries of human capacity to implement Bayesian learning. We repeat the experiment with different instructions, reflecting varying levels of structural uncertainty. Under this fourth notion of uncertainty, choices were no better explained by Bayesian updating than by (model-free) reinforcement learning. Exit questionnaires revealed that participants remained unaware of the presence of unexpected uncertainty and failed to acquire the right model with which to implement Bayesian updating.

Choosing among partition models in Bayesian phylogenetics.

Abstract: Bayesian phylogenetic analyses often depend on Bayes factors (BFs) to determine the optimal way to partition the data. The marginal likelihoods used to compute BFs, in turn, are most commonly estimated using the harmonic mean (HM) method, which has been shown to be inaccurate. We describe a new more accurate method for estimating the marginal likelihood of a model and compare it with the HM method on both simulated and empirical data. The new method generalizes our previously described stepping-stone (SS) approach by making use of a reference distribution parameterized using samples from the posterior distribution. This avoids one challenging aspect of the original SS method, namely the need to sample from distributions that are close (in the Kullback–Leibler sense) to the prior. We specifically address the choice of partition models and find that using the HM method can lead to a strong preference for an overpartitioned model. In contrast to the HM method and the original SS method, we show using simulated data that the generalized SS method is strikingly more precise (repeatable BF values of the same data and partition model) and yields BF values that are much more reasonable than those produced by the HM method. Comparisons of HM and generalized SS methods on an empirical data set demonstrate that the generalized SS method tends to choose simpler partition schemes that are more in line with expectation based on inferred patterns of molecular evolution. The generalized SS method shares with thermodynamic integration the need to sample from a series of distributions in addition to the posterior. Such dedicated path-based Markov chain Monte Carlo analyses appear to be a cost of estimating marginal likelihoods accurately.

Textual and Visual Content-Based Anti-Phishing: A Bayesian Approach

Abstract: A novel framework using a Bayesian approach for content-based phishing web page detection is presented. Our model takes into account textual and visual contents to measure the similarity between the protected web page and suspicious web pages. A text classifier, an image classifier, and an algorithm fusing the results from classifiers are introduced. An outstanding feature of this paper is the exploration of a Bayesian model to estimate the matching threshold. This is required in the classifier for determining the class of the web page and identifying whether the web page is phishing or not. In the text classifier, the naive Bayes rule is used to calculate the probability that a web page is phishing. In the image classifier, the earth mover's distance is employed to measure the visual similarity, and our Bayesian model is designed to determine the threshold. In the data fusion algorithm, the Bayes theory is used to synthesize the classification results from textual and visual content. The effectiveness of our proposed approach was examined in a large-scale dataset collected from real phishing cases. Experimental results demonstrated that the text classifier and the image classifier we designed deliver promising results, the fusion algorithm outperforms either of the individual classifiers, and our model can be adapted to different phishing cases.

Bayesian models: the structure of the world, uncertainty, behavior, and the brain

Abstract: Experiments on humans and other animals have shown that uncertainty due to unreliable or incomplete information affects behavior. Recent studies have formalized uncertainty and asked which behaviors would minimize its effect. This formalization results in a wide range of Bayesian models that derive from assumptions about the world, and it often seems unclear how these models relate to one another. In this review, we use the concept of graphical models to analyze differences and commonalities across Bayesian approaches to the modeling of behavioral and neural data. We review behavioral and neural data associated with each type of Bayesian model and explain how these models can be related. We finish with an overview of different theories that propose possible ways in which the brain can represent uncertainty.

Link prediction in complex networks: a local na\"{\i}ve Bayes model

Abstract: Common-neighbor-based method is simple yet effective to predict missing links, which assume that two nodes are more likely to be connected if they have more common neighbors. In such method, each common neighbor of two nodes contributes equally to the connection likelihood. In this Letter, we argue that different common neighbors may play different roles and thus lead to different contributions, and propose a local na\"{\i}ve Bayes model accordingly. Extensive experiments were carried out on eight real networks. Compared with the common-neighbor-based methods, the present method can provide more accurate predictions. Finally, we gave a detailed case study on the US air transportation network.

Different models of genetic variation and their effect on genomic evaluation

Abstract: The theory of genomic selection is based on the prediction of the effects of quantitative trait loci (QTL) in linkage disequilibrium (LD) with markers. However, there is increasing evidence that genomic selection also relies on "relationships" between individuals to accurately predict genetic values. Therefore, a better understanding of what genomic selection actually predicts is relevant so that appropriate methods of analysis are used in genomic evaluations. Simulation was used to compare the performance of estimates of breeding values based on pedigree relationships (Best Linear Unbiased Prediction, BLUP), genomic relationships (gBLUP), and based on a Bayesian variable selection model (Bayes B) to estimate breeding values under a range of different underlying models of genetic variation. The effects of different marker densities and varying animal relationships were also examined. This study shows that genomic selection methods can predict a proportion of the additive genetic value when genetic variation is controlled by common quantitative trait loci (QTL model), rare loci (rare variant model), all loci (infinitesimal model) and a random association (a polygenic model). The Bayes B method was able to estimate breeding values more accurately than gBLUP under the QTL and rare variant models, for the alternative marker densities and reference populations. The Bayes B and gBLUP methods had similar accuracies under the infinitesimal model. Our results suggest that Bayes B is superior to gBLUP to estimate breeding values from genomic data. The underlying model of genetic variation greatly affects the predictive ability of genomic selection methods, and the superiority of Bayes B over gBLUP is highly dependent on the presence of large QTL effects. The use of SNP sequence data will outperform the less dense marker panels. However, the size and distribution of QTL effects and the size of reference populations still greatly influence the effectiveness of using sequence data for genomic prediction.

Bayesian sampling in visual perception

Abstract: It is well-established that some aspects of perception and action can be understood as probabilistic inferences over underlying probability distributions In some situations, it would be advantageous for the nervous system to sample interpretations from a probability distribution rather than commit to a particular interpretation In this study, we asked whether visual percepts correspond to samples from the probability distribution over image interpretations, a form of sampling that we refer to as Bayesian sampling To test this idea, we manipulated pairs of sensory cues in a bistable display consisting of two superimposed moving drifting gratings, and we asked subjects to report their perceived changes in depth ordering We report that the fractions of dominance of each percept follow the multiplicative rule predicted by Bayesian sampling Furthermore, we show that attractor neural networks can sample probability distributions if input currents add linearly and encode probability distributions with probabilistic population codes

Robust Predictions in Games with Incomplete Information

Abstract: We analyze games of incomplete information and offer equilibrium predictions which are valid for all possible private information structures that the agents may have. Our characterization of these robust predictions relies on an epistemic result which establishes a relationship between the set of Bayes Nash equilibria and the set of Bayes correlated equilibria.We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior information of the analyst refines the set of equilibrium distribution. As an application, we obtain new results regarding the optimal information sharing policy of firms under demand uncertainty.Finally, we reverse the perspective and investigate the identification problem under concerns for robustness to private information. We show how the presence of private information leads to partial rather than complete identification of the structural parameters of the game. As a prominent example we analyze the canonical problem of demand and supply identification.

How to Analyze Paired Comparison Data

Abstract: : Thurstone's Law of Comparative Judgment provides a method to convert subjective paired comparisons into one-dimensional quality scores. Applications include judging quality of different image reconstructions, or different products, or different web search results, etc. This tutorial covers the popular Thurstone-Mosteller Case V model and the Bradley-Terry logistic variant. We describe three approaches to model- fitting: standard least-squares, maximum likelihood, and Bayesian approaches. This tutorial assumes basic knowledge of random variables and probability distributions.