Doing Bayesian Data Analysis (eBook)

A Tutorial Introduction with R

John Kruschke (Autor)

eBook Download: PDF | EPUB

2010 | 1. Auflage
672 Seiten
Elsevier Science (Verlag)
978-0-12-381486-9 (ISBN)

Infos

There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis obtainable to a wide audience. Doing Bayesian Data Analysis, A Tutorial Introduction with R and BUGS provides an accessible approach to Bayesian data analysis, as material is explained clearly with concrete examples. The book begins with the basics, including essential concepts of probability and random sampling, and gradually progresses to advanced hierarchical modeling methods for realistic data. The text delivers comprehensive coverage of all scenarios addressed by non-Bayesian textbooks--t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis).

This book is intended for first year graduate students or advanced undergraduates. It provides a bridge between undergraduate training and modern Bayesian methods for data analysis, which is becoming the accepted research standard. Prerequisite is knowledge of algebra and basic calculus.

Author website: http://www.indiana.edu/~kruschke/DoingBayesianDataAnalysis/

-Accessible, including the basics of essential concepts of probability and random sampling

-Examples with R programming language and BUGS software

-Comprehensive coverage of all scenarios addressed by non-bayesian textbooks- t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis).

-Coverage of experiment planning

-R and BUGS computer programming code on website

-Exercises have explicit purposes and guidelines for accomplishment

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There is an explosion of interest in Bayesian statistics, primarily because recently created computational methods have finally made Bayesian analysis tractable and accessible to a wide audience. Doing Bayesian Data Analysis, A Tutorial Introduction with R and BUGS, is for first year graduate students or advanced undergraduates and provides an accessible approach, as all mathematics is explained intuitively and with concrete examples. It assumes only algebra and 'rusty' calculus. Unlike other textbooks, this book begins with the basics, including essential concepts of probability and random sampling. The book gradually climbs all the way to advanced hierarchical modeling methods for realistic data. The text provides complete examples with the R programming language and BUGS software (both freeware), and begins with basic programming examples, working up gradually to complete programs for complex analyses and presentation graphics. These templates can be easily adapted for a large variety of students and their own research needs.The textbook bridges the students from their undergraduate training into modern Bayesian methods. Accessible, including the basics of essential concepts of probability and random sampling Examples with R programming language and BUGS software Comprehensive coverage of all scenarios addressed by non-bayesian textbooks- t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis). Coverage of experiment planning R and BUGS computer programming code on website Exercises have explicit purposes and guidelines for accomplishment

Textauszug

Chapter 2

Introduction
Models We Believe In

Contents

2.1 Models of Observations and Models of Beliefs

2.1.1 Prior and Posterior Beliefs

2.2 Three Goals for Inference from Data

2.2.1 Estimation of Parameter Values

2.2.2 Prediction of Data Values

2.2.3 Model Comparison

2.3 The R Programming Language

2.3.1 Getting and Installing R

2.3.2 Invoking R and Using the Command Line

2.3.3 A Simple Example of R in Action

2.3.4 Getting Help in R

2.3.5 Programming in R

2.4 Exercises

I just want someone who I can believe in,

Someone at home who will not leave me grievin’.

Show me a sign that you’ll always be true,

and I’ll be your model of faith and virtue.

Inferential statistical methods help us decide what to believe in. With inferential statistics, we don’t just introspect to find the truth. Instead, we rely on data from observations. Based on the data, what should we believe in? Should we believe that the tossed coin is fair if it comes up heads in 7 of 10 flips? Should we believe that we have cancer when the test comes back positive? Should we believe that she loves me when the daisy has 17 petals? Our beliefs can be modified when we have data, and this book is about techniques for making inferences from data to uncertain beliefs.

There might be some beliefs that cannot be decided by data, but such beliefs are dogmas that lie (double entendre intended) beyond the reach of evidence. If you are wondering about a belief that has no specific implications for concrete facts in the observable world, then inferential statistics won’t help.

Why do we need hefty tomes full of mathematics to help us make decisions based on data? After all, we make lots of decisions every day without math. If we’re driving, we look at the signal light and effortlessly decide whether it’s red or green. We don’t (consciously) go through a laborious process of mathematical statistics and finally conclude that it is probably the case that the light is red. Two attributes of this situation make the decision easy. First, the data about the light are numerous. An unobstructed view of the light results in a whole lot of photons striking our eyes. Second, there are only a few possible beliefs about the light that make distinct predictions about the photons: If the light is red, the photons are rather different than if the light is green. Consequently, the decision is easy because there is little variance in the data and little uncertainty across possible beliefs.

The math is most helpful when there is lots of variance in the data and lots of uncertainty in our beliefs. Data from scientific experiments, especially those involving humans or animals, are unmitigated heaps of variability. Theories in science tend to be rife with parameters of uncertain magnitude, and competing theories are numerous. In these situations, the mathematics of statistical inference provide precise numerical bounds on our uncertainty. The math allows us to determine accurately what the data imply for different possible beliefs. The math can tell us exactly how likely or unlikely each possibility is, even when there is an infinite spectrum of possibilities. It is this power of precisely defining our uncertainty that makes inferential statistics such a useful tool, worth the effort of learning.

2.1 Models of Observations and Models of Beliefs

Suppose we flip a coin to decide which team kicks off. The teams agree to this procedure for deciding the kickoff because they believe that the coin is fair. But how do we determine whether the coin really is fair? Even if we could study the exact minting process of the coin and x-ray every nuance of the coin’s interior, we would still need to test whether the coin really is fair when it’s actually flipped. Ultimately, all we can do is flip the coin a few times and watch its behavior. From these observations, we can modify our beliefs about the fairness of the coin.

Suppose we have a coin from our friend the numismatist.1 We notice that on the obverse is embossed the head of Tanit (of ancient Carthage) and on the reverse side is embossed a horse. The coin is gold and shows the date 350 BCE. Do you believe that the coin is fair? Maybe you do, but maybe you’re not very certain.2 Let’s flip it a few times. Suppose we flip it 10 times and we obtain two heads and eight tails. Now what do you think? Do you have a suspicion that maybe the coin is biased to come up tails more often than heads?

We’ve seen that the coin comes up horse tails a lot. Whoa! Let’s dismount and have a heart-to-heart ‘round the campfire. In that simple coin-flipping scenario we have made two sets of assumptions. First, we have assumed that the coin has some inherent fairness or bias that we can’t directly observe. All we can actually observe is an inherently probabilistic effect of that bias, namely, whether the coin comes up heads or tails on any given flip. We’ve made lots of assumptions about exactly how the observable head or tail relates to the unobservable bias of the coin. For instance, we’ve assumed that the bias stays the same, flip after flip. We’ve assumed that the coin can’t remember what way it came up last flip, so that its flip this time is uncorrupted by its previous landings. All these assumptions are about the process that converts the unobservable bias into a probabilistic observable event. This collection of assumptions about the coin-flipping process is our model of the head-tail observations.

The second set of assumptions is about our beliefs regarding the bias of the coin. We assume that we believe most strongly in the coin being fair, but we also allow for the possibility that the coin could be biased. Thus, we have a set of assumptions about how likely it is for the coin to be fair or to be biased to different amounts. This collection of assumptions is our model of our beliefs.

When we want to get specific about our model assumptions, then we have to use mathematical descriptions. A “formal” model uses mathematical formulas to precisely describe something. In this book, we’ll almost always be using formal models, so whenever the term “model” comes up, you can assume it means a mathematical description. In the context of statistical models, the models are typically models of probabilities. Some models describe the probabilities of observable events (e.g., we can have a formula that describes the probability that a coin will come up heads). Other models describe the extent to which we believe in various underlying possibilities (e.g., we can have a formula that describes how much we believe in each possible bias of the coin).

Mathematical models are formulas with variables. Some variables have values supplied as data from the world. For example, the data variable of a coin flip can have the values head or tail. But other variables refer to underlying characteristics, such as the bias of the coin. Variables that refer to underlying characteristics are called parameters. A parameter can take on many possible values. For example, the bias parameter of a coin can take on a value anywhere on the continuum between zero and one.

2.1.1 Prior and Posterior Beliefs

We could believe that the coin is fair—that is, that the probability of coming up heads is 50%. We could instead have other beliefs about the coin, especially if it’s dated 350 BCE, which no coin would be labeled if it were really minted BCE (because the people alive in 350 BCE didn’t yet know they were BCE). Perhaps, therefore, we also think it’s possible for the coin to be biased to come up heads 20% of the time, or 80% of the time. Before observing the coin flips, we might believe that each of these three dispositions is equally likely—that is, we believe that there is a one-in-three chance that the bias is 20%, a one-in-three chance that the bias is 50%, and a one-in-three chance that the bias is 80%.

After flipping the coin and observing 2 heads in 10 flips, we will want to modify our beliefs. It makes sense that we should now believe more strongly that the bias is 20%, because we observed 20% heads in the sample. This book is about determining exactly how much more strongly we should believe that the bias is 20%.

Before observing the flips of the coin, we had certain beliefs about the possible biases of the coin. These are called prior beliefs because they are our beliefs before taking into account some particular set of observations. After observing the flips of the coin, we modified our beliefs. These are called the posterior beliefs because they are computed after taking into account a particular set of observations. Bayesian inference gets us from prior to posterior beliefs.

There is an infelicity in the terms “prior” and “posterior,” however. The terms connote the passage of time, as if the prior beliefs were held temporally before the posterior beliefs. But that is a misconception. There is no temporal ordering in the prior and posterior beliefs! Rather,...

Produkt-Details

• Erscheint lt. Verlag: 25.11.2010
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 0-12-381486-3 / 0123814863
• ISBN-13: 978-0-12-381486-9 / 9780123814869