Search Results

Bayesian Statistical Inference

Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray

in Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data

This chapter introduces the most important aspects of Bayesian statistical inference and techniques for performing such calculations in practice. It first reviews the basic steps in Bayesian ... More

This chapter introduces the most important aspects of Bayesian statistical inference and techniques for performing such calculations in practice. It first reviews the basic steps in Bayesian inference in early sections of the chapter, and then illustrates them with several examples in sections that follow. Numerical techniques for solving complex problems are next discussed, and the final section provides a summary of pros and cons for classical and Bayesian method. It argues that most users of Bayesian estimation methods are likely to use a mix of Bayesian and frequentist tools. The reverse is also true—frequentist data analysts, even if they stay formally within the frequentist framework, are often influenced by “Bayesian thinking,” referring to “priors” and “posteriors.” The most advisable position is to know both paradigms well, in order to make informed judgments about which tools to apply in which situations.Less

Developing predictive models

E. J. Milner-Gulland and Marcus Rowcliffe

in Conservation and Sustainable Use: A Handbook of Techniques

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may ... More

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may simply be in the form of a conceptual model, but is much more powerful when formalized as a mathematical model. This chapter introduces methods for building a model of the system that can be used to predict future sustainability with or without management interventions. The emphasis is on the simulation of biological and bioeconomic dynamics, for which step-by-step worked examples are given. These examples start with conceptual models, then show how to formalize these as mathematical equations, build these into computer code; test model sensitivity, validity, and alternative structures; and finally, explore future scenarios. Methods for modelling stochasticity and human behaviour are also introduced, as well as the use of Bayesian methods for understanding dynamic systems and exploring management interventions.Less

Hierarchical Bayesian Models

N. Thompson Hobbs and Mevin B. Hooten

in Bayesian Models: A Statistical Primer for Ecologists

This chapter seeks to explain hierarchical models and how they differ from simple Bayesian models and to illustrate building hierarchical models using mathematically correct expressions. It begins ... More

This chapter seeks to explain hierarchical models and how they differ from simple Bayesian models and to illustrate building hierarchical models using mathematically correct expressions. It begins with the definition of hierarchical models. Next, the chapter introduces four general classes of hierarchical models that have broad application in ecology. These classes can be used individually or in combination to attack virtually any research problem. Examples are used to show how to draw Bayesian networks that portray stochastic relationships between observed and unobserved quantities. The chapter furthermore shows how to use network drawings as a guide for writing posterior and joint distributions.Less

Thinking about Evidence1

DAVID LAGNADO

in Evidence, Inference and Enquiry

This chapter argues that people reason about legal evidence using small-scale qualitative networks. These cognitive networks are typically qualitative and incomplete, and based on people's causal ... More

This chapter argues that people reason about legal evidence using small-scale qualitative networks. These cognitive networks are typically qualitative and incomplete, and based on people's causal beliefs about the specifics of the case as well as the workings of the physical and social world in general. A key feature of these networks is their ability to represent qualitative relations between hypotheses and evidence, allowing reasoners to capture the concepts of dependency and relevance critical in legal contexts. In support of this claim, the chapter introduces some novel empirical and formal work on alibi evidence, showing that people's reasoning conforms to the dictates of a qualitative Bayesian model. However, people's inferences do not always conform to Bayesian prescripts. Empirical studies are also discussed in which people over-extend the discredit of one item of evidence to other unrelated items. This bias is explained in terms of the propensity to group positive and negative evidence separately and the use of coherence-based inference mechanisms. It is argued that these cognitive processes are a natural response to deal with the complexity of legal evidence.Less

Simple Bayesian Models

N. Thompson Hobbs and Mevin B. Hooten

in Bayesian Models: A Statistical Primer for Ecologists

This chapter lays out the basic principles of Bayesian inference, building on the concepts of probability developed in Chapter 3. It seeks to use the rules of probability to show how Bayes' theorem ... More

This chapter lays out the basic principles of Bayesian inference, building on the concepts of probability developed in Chapter 3. It seeks to use the rules of probability to show how Bayes' theorem works, by making use of the conditional rule of probability and the law of total probability. The chapter begins with the central, underpinning tenet of the Bayesian view: the world can be divided into quantities that are observed and quantities that are unobserved. Unobserved quantities include parameters in models, latent states predicted by models, missing data, effect sizes, future states, and data before they are observed. We wish to learn about these quantities using observations. The Bayesian framework for achieving that understanding is applied in exactly the same way regardless of the specifics of the research problem at hand or the nature of the unobserved quantities.Less

Writing Bayesian Models

N. Thompson Hobbs and Mevin B. Hooten

in Bayesian Models: A Statistical Primer for Ecologists

This chapter offers a general set of steps for writing models to assist the researcher in formulating their own approach to the Bayesian model. The crucial skill of specifying models is often ... More

This chapter offers a general set of steps for writing models to assist the researcher in formulating their own approach to the Bayesian model. 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 also motivates this chapter. The overarching challenge in building models is to specify the components of the posterior distribution and the joint distribution and to factor the joint distribution into sensible parts. This chapter first lays out a framework for doing just that, albeit in somewhat abstract terms, before moving on to a more concrete example—the effects of grazing by livestock and wild ungulates on structure and function of a sagebrush steppe ecosystem.Less

Humans' Multisensory Perception, from Integration to Segregation, Follows Bayesian Inference

Ladan Shams and Ulrik Beierholm

in Sensory Cue Integration

This chapter first discusses experimental findings showing that multisensory perception encompasses a spectrum of phenomena ranging from full integration (or fusion), to partial integration, to ... More

This chapter first discusses experimental findings showing that multisensory perception encompasses a spectrum of phenomena ranging from full integration (or fusion), to partial integration, to complete segregation. Next, it describes two Bayesian causal-inference models that can account for the entire range of combinations of two or more sensory cues. It shows that one of these models, which is a hierarchical Bayesian model, is a special form of the other one (which is a nonhierarchical model). It then compares the predictions of these models with human data in multiple experiments and shows that Bayesian causal-inference models can account for the human data remarkably well. Finally, a study is presented that investigates the stability of priors in the face of drastic change in sensory conditions.Less

Spatial variation in risk

Dirk U. Pfeiffer, Timothy P. Robinson, Mark Stevenson, Kim B. Stevens, David J. Rogers, and Archie C. A. Clements

in Spatial Analysis in Epidemiology

This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues ... More

This chapter discusses spatial variation in risk. Epidemiological disease investigations should include an assessment of the spatial variation of disease risk, as this may provide important clues leading to causal explanations. The objective is to produce a map representation of the important spatial effects present in the data while simultaneously removing any distracting noise or extreme values. The resulting smoothed map should have increased precision without introducing significant bias. The method used to analyse the data depends on how they have been recorded. Smoothing based on kernel functions, smoothing based and on Bayesian models, and spatial interpolation are discussed.Less

Characterizing Uncertainty of Future Climate Change Projections using Hierarchical Bayesian Models *

Claudia Tebaldi, Bruno Sansó, and Richard L. Smith

in Bayesian Statistics 9

The use of projections from ensembles of climate models to characterize fu ture climate change at regional scales has become the most widely adopted framework, as opposed to what was standard ... More

The use of projections from ensembles of climate models to characterize fu ture climate change at regional scales has become the most widely adopted framework, as opposed to what was standard practice until just a few years ago when a single model's projections constituted the basis for arguing about future changes and their impacts. It is believed that comparing and synthe sizing simulations of multiple models is key to quantifying a best estimate of the future changes and its uncertainty. In the last few years there has been an explosion of literature in climate change science where mostly heuristic meth ods of synthesizing the output of multiple models have been proposed, and the statistical literature is showing more involvement by our community as well, of late. In this paper we give a brief overview of the mainstreams of research in this area and then focus on our recent work, through which we have proposed the framework of hierarchical Bayesian models to combine information from model simulations and observations, in order to derive posterior probabilities of temperature and precipitation change at regional scales.Less

Preview

N. Thompson Hobbs and Mevin B. Hooten

in Bayesian Models: A Statistical Primer for Ecologists

This chapter sketches an approach to inference applicable to an enormous range of research problems—one that can be understood from first principles and that can be unambiguously communicated to ... More

This chapter sketches an approach to inference applicable to an enormous range of research problems—one that can be understood from first principles and that can be unambiguously communicated to other scientists, managers, and policy makers. In doing research, it is important that one is able to ask important questions and provide compelling answers to them. Doing so depends on establishing a line of inference that extends from current thinking, theory, and questions to new insight qualified by uncertainty. This chapter introduces a highly general, flexible approach to establishing this line of inference. It offers a somewhat abstract overview of this framework followed by a concrete example to properly illustrate this framework.Less

Optimal Random Exploration for Trade‐related Nonindigenous Species Risk

Michael Springborn, Christopher Costello, and Peyton Ferrier

in Bioinvasions and Globalization: Ecology, Economics, Management, and Policy

This chapter identifies variables from the port inspection setting that influence the gains to exploration via random inspections. It begins by describing a Bayesian learning model of trade-related ... More

This chapter identifies variables from the port inspection setting that influence the gains to exploration via random inspections. It begins by describing a Bayesian learning model of trade-related non-indigenous species (NIS) risk in Section 10.2, which captures uncertainty over the true probability that trade from a given source is infested, and provides a framework for updating these beliefs as observations accrue. The formal inspection allocation decision problem is expressed in Section 10.3, where the computational demands of the central task are made clear. The analysis in Section 10.4 begins with the simplest possible nontrivial version of the problem. Elements of real-world complexity are subsequently added to build intuition for the ultimate task of exploring random inspection policy in an empirically-based setting. The workhorse method applies Monte Carlo simulation under various policies to characterize performance in terms of interceptions and to identify optimal choices for design and intensity of exploration.Less

Surviving fully Bayesian nonparametric regression models

Timothy E. Hanson and Alejandro Jara

in Bayesian Theory and Applications

This chapter compares two Bayesian nonparametric models that generalize the accelerated failure time model, based on recent work on probability models for predictor-dependent probability ... More

This chapter compares two Bayesian nonparametric models that generalize the accelerated failure time model, based on recent work on probability models for predictor-dependent probability distributions. It begins by reviewing commonly used semiparametric survival models. It then discusses the Bayesian nonparametric priors used in the generalizations of the accelerated failure time (AFT) model. Next, the two generalizations of the accelerated failure time model are introduced and compared by means of real-life data analyses. The models correspond to generalizations of AFT models based on dependent extensions of the Dirichlet process (DP) and Polya tree (PT) priors. Advantages of the induced survival regression models include ease of interpretability and computational tractability.Less

Investigating the genetic association between diabetes and malaria: an application of Bayesian ecological regression models with errors in covariates

L. Bernardinelli, C. Pascutto, C. Montomoli, and W. Gilks

in Spatial Epidemiology: Methods and Applications

This chapter describes a Bayesian hierarchical model and applies it to a new dataset on insulin-dependent diabetes mellitus (IDDM) prevalence among 18-year-old males born in Sardinia between 1936 and ... More

This chapter describes a Bayesian hierarchical model and applies it to a new dataset on insulin-dependent diabetes mellitus (IDDM) prevalence among 18-year-old males born in Sardinia between 1936 and 1973, using malaria prevalence in 1938–40 as the ecological covariate. It shows how to deal with the potential bias associated with using such a proxy by extending the Bayesian model to allow for covariate measurement error. It provides a method for choosing the hyperprior distributions for the spatial variation parameters of the model, and discusses the sensitivity of the results to different choices.Less

Bayesianism

Brian D. Haig

in The Philosophy of Quantitative Methods: Understanding Statistics

Chapter 4 focuses on Bayesian confirmation theory, a formal theory of reasoning based on probability theory. It deals with important, and related, general ideas, such as rationality, confirmation, ... More

Chapter 4 focuses on Bayesian confirmation theory, a formal theory of reasoning based on probability theory. It deals with important, and related, general ideas, such as rationality, confirmation, and inductive inference, including statistical inference. The chapter also provides a selective discussion of Bayesian statistics. The chapter traces some of the broad contours of Bayesian confirmation theory and then presents an evaluation of a philosophy of Bayesian statistical practice. Psychology’s attitudes to Bayesianism are briefly discussed. Considered is the question of whether Bayesianism provides an illuminating account of the approach to theory evaluation known as inference to the best explanation. The chapter offers some broad recommendations for research practice.Less

Simplicity and Model Selection

Jan Sprenger and Stephan Hartmann

in Bayesian Philosophy of Science

Is simplicity a virtue of a good scientific theory, and are simpler theories more likely to be true or predictively successful? If so, how much should simplicity count vis-à-vis predictive accuracy? ... More

Is simplicity a virtue of a good scientific theory, and are simpler theories more likely to be true or predictively successful? If so, how much should simplicity count vis-à-vis predictive accuracy? We address this question using Bayesian inference, focusing on the context of statistical model selection and an interpretation of simplicity via the degree of freedoms of a model. We rebut claims to prove the epistemic value of simplicity by means of showing its particular role in Bayesian model selection strategies (e.g., the BIC or the MML). Instead, we show that Bayesian inference in the context of model selection is usually done in a philosophically eclectic, instrumental fashion that is more tuned to practical applications than to philosophical foundations. Thus, these techniques cannot justify a particular “appropriate weight of simplicity in model selection”.Less

Domestic Coalition Shifts in War Termination

in Paths to Peace: Domestic Coalition Shifts, War Termination and the Korean War

This chapter outlines the existing literature about war termination. It demonstrates that there is a causal connection between the difficulty of ending wars and the fact that they are started and ... More

This chapter outlines the existing literature about war termination. It demonstrates that there is a causal connection between the difficulty of ending wars and the fact that they are started and ended by politicians. Realpolitik, domestic politics, and bargaining models can be used to illustrate categories about war termination. Bayesian models of war termination theorized more explicitly about lags in the updating process that can result in protracted stalemates. These models showed that a lag in updating can delay war termination, they do not elaborate on how states get beyond this lag. Ending war needed settling with the enemy as well as at home. It is noted that policy stability can result governing coalitions being ineffective for altering the status quo, even when such changes are necessary or desirable. Furthermore, a coalition shift can overpower the obstacles to peace and thus allows a war to end.Less

Analysis of Short-term Selection Experiments: 2. Mixed-model and Bayesian Approaches

Bruce Walsh and Michael Lynch

in Evolution and Selection of Quantitative Traits

When the full pedigree of individuals whose values (records) were used in the selection decisions during an experiment (or breeding program) is known, LS analysis can be replaced by mixed models and ... More

When the full pedigree of individuals whose values (records) were used in the selection decisions during an experiment (or breeding program) is known, LS analysis can be replaced by mixed models and their Bayesian extensions. In this setting, REML can be used to estimate genetic variances and BLUP can be used to estimate the mean breeding value in any given generation. The latter allows for genetic trends to be separated from environmental trends without the need for a control population. Under the infinitesimal model setting (wherein selection-induced allele-frequency changes are small during the course of the experiment), the use of the relationship matrix in a BLUP analysis accounts for drift, nonrandom mating, and linkage disequilibrium.Less

Bayesian Hierarchical Modeling of Public Health Surveillance Data: A Case Study of Air Pollution and Mortality

Scott L. Zeger, Francesca Dominici, Aidan Mcdermott, and Jonathan M. Samet

in Monitoring the Health of Populations: Statistical Principles and Methods for Public Health Surveillance

This chapter illustrates the use of log-linear regression and hierarchical models to estimate the association of daily mortality with acute exposure to particulate air pollution. It focuses on ... More

This chapter illustrates the use of log-linear regression and hierarchical models to estimate the association of daily mortality with acute exposure to particulate air pollution. It focuses on multistage models of daily mortality data in the eighty-eight largest cities in the United States to illustrate the main ideas. These models have been used to quantify the risks of shorter-term exposure to particulate pollution and to address key causal questions.Less