Data Analysis: A Bayesian Tutorial provides such a text, putting emphasis as This difference in approach makes the text ideal as a tutorial guide forsenior. This book attempts to remedy the situation by expounding a logical and unified approach to the whole subject of data analysis. This text is intended as a tutorial. Statistics lectures have been a source of much bewilderment and frustration for generations of students. This book attempts to remedy the.
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Download Citation on ResearchGate | Data Analysis: A Bayesian Tutorial the parameter α can be estimated as follows : If we assign a uniform pdf for the. Editorial Reviews. Review. "Review from previous edition Providing a clear rationale for some Data Analysis: A Bayesian Tutorial 2nd Edition, site Edition. A modern Bayesian physicist, Steve Gull from Cambridge, described data analysis The training in data analysis that most of us are given as undergraduates consists of being . us to relate this probability distribution function (pdf) to others that are .. D. S. Sivia, Data analysis – a Bayesian tutorial, Oxford University Press.
At the top of both panels, the discrete model-index parameter is denoted M. In other words, the prior distributions on the model parameters are an integral part of the meanings of the models. If the prior distribution of a model gives high credibility to parameter values that happen to fit the data well, then the posterior probability of that model index will tend to be high.
But if the prior distribution of a model dilutes its prior distribution over a wide range of parameter values that do not fit the data well, then the posterior probability of that model index will tend to be low. Bayesian model comparison automatically takes into account model complexity.
This is important because a complex model will always be able to fit data better than a restricted model nested in the complex model , even when the simpler restricted model actually generated the data. A quartic polynomial will always be able to fit the data better than a linear trend because the quartic will over-fit random noise in the data. We would like the model comparison method to be able to declare the simpler model as better than the complex model in this sort of situation.
Bayesian methods inherently do this. The reason is that complex models involve more parameters than restricted models, and higher-dimensional parameter spaces require the prior distribution to be diluted over a larger space.
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The diluted prior distribution means that the prior probability is relatively small for any particular parameter value that happens to fit the data. Therefore the posterior probability of a complex model tends to be downweighted by its diluted prior probability distribution. A complex model can win, however, if the data are much better fit by parameter values in the complex model that are not accessible by the restricted model. In summary, Bayesian model comparison is an excellent method for assessing models because it is both intuitively informative just read off the posterior probabilities and automatically takes into account model complexity.
There is no need to compute p values by generating sampling distributions of imaginary data from restricted models. Bayesian model comparison does, however, require the analyst to think carefully about the prior distributions on the parameters within the models, and to think carefully about the prior distribution on the model index. One way to keep the prior distributions within the two models on an equal playing field is by informing both models with the same representative previous data.
That is, both models are started with a diffuse proto-prior, and both models are updated with the same previous data. The resulting posterior distributions from the previous data act as the prior distributions for the model comparison Kruschke , Section For further reading about Bayesian model comparison and setting useful priors within models, see examples in Kary et al.
The setting of prior probabilities on the model index is also important but less often considered. Bayesian methods also allow expressing uncertainty in the prior probabilities of the model indices see, e.
Two approaches to assessing null values In many fields of science, research focuses on magnitudes of phenomena. For example, psychometricians might be interested in where people or questionnaire items fall on scales of abilities, attitudes, or traits. But in other domains, questions might focus on presence versus absence of an effect, without much concern for magnitudes. Is the effect of treatment different from the control group or not? Researchers would like to know whether the estimated underlying effect is credibly different from the null value of zero or chance.
In this section, we will consider two Bayesian approaches to assessing null values. We will see that the two approaches correspond to different levels in the model-comparison diagram in Fig. Science should work the other way around, and posit theories that are challenged more severely by more precise data. A theory should instead be framed such that increased precision of data yields a greater challenge to the theory.
A solution was described by Serlin and Lapsley , : Theories should predict a magnitude of effect, with a region of practical equivalence ROPE around the predicted magnitude. When that range falls outside the ROPE the theory is disconfirmed, and when the range of uncertainty falls entirely within the ROPE then the theory is confirmed for practical purposes.
In particular, a null value surrounded by a ROPE can be accepted, not only be rejected. Among frequentists, this approach is used in the method called equivalence testing e.
A related framework is also used in clinical non-inferiority testing e. The approach forms an intuitive basis for our first way to assess null values with a Bayesian posterior distribution. Thus, this choice of ROPE says that any value of effect size that is less than half of small is practically equivalent to zero. The ROPE is merely a decision threshold, and its limits are chosen always in the context of current theory and measurement precision. Serlin and Lapsley , p. It depends on the state of the art of the theory We know the general background IQ has a mean of and a standard deviation of The resulting posterior distribution of the effect size is shown in panel A of Fig.
The null value of 0 is marked by a vertical dotted line, annotated with the percentage of the posterior distribution that falls below it and above it. The ROPE limits are also marked with vertical dotted lines and the percentage of the posterior distribution that falls below, within, and above the ROPE.
This distribution is just a different perspective on the same posterior distribution shown in Fig. Recall that the posterior distribution is a joint distribution on the multidimensional parameter space. Every point in parameter space also has a corresponding posterior credibility. The posterior distribution of effect sizes is shown in Fig.
Unlike a p value, the statement is about the probability of parameter values, not about the probability of fictitious samples of data from a null hypothesis. The resulting posterior distribution on effect size is shown in panel B of Fig.
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Notice that this conclusion is based on having sufficient precision in the estimate of the effect size so we can safely say that more extreme values for the effect size are not very credible. Comparing spike prior to alternative prior A different way of assessing a null value is by expressing the null hypothesis as a particular prior distribution over the parameters and comparing it to an alternative prior distribution e.
Bayesian inference assesses the relative credibility of the two prior distributions as accounts of the data. This framing for Bayesian hypothesis testing is really a special case of Bayesian model comparison that was illustrated in Fig. The two models in this case are the spike-prior null hypothesis and the broad-prior alternative hypothesis.
Both models involve the same parameters but different prior distributions.
Bayesian inference re-allocates credibility across all the parameters simultaneously, including the model-index parameter and the parameters within the models. The posterior probabilities on the model indices indicate the relative credibilities of the null and alternative hypotheses.
This framework is illustrated in Fig.
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This is a special case of model comparison that was shown in Fig. Both models involve the same parameters but differ in their assumptions about the prior distributions on those parameters. In panel B, the posterior distribution shows that credibility has been re-allocated across the possible parameter values.
Within both models the other parameters not shown have also had their distributions re-allocated, differently in each model In principle, the decision rule about the models would focus on the posterior probabilities of the model indices. For example, we might decide to accept a model if its posterior probability is at least ten times greater than the next most probable model.
In practice, the decision rule is often instead based on how much the two model probabilities have shifted, not on where the model probabilities ended up. The degree of shift is called the Bayes factor, which is technically defined as the ratio of probabilities of the data as predicted by each model.
Put another way, the Bayes factor is a multiplier that gets us from the prior odds ratio to the posterior odds ratio. A Bayes factor of When the Bayes factor exceeds a critical threshold, say 10, we decide to accept the winning model and reject the losing model. The decision threshold for the Bayes factor is set by practical considerations. Dienes used a Bayes factor of 3 for making decisions.
As concrete examples, consider the data corresponding to panel A of Fig. For the data corresponding to panel B of Fig. There has recently been a flurry of articles promoting Bayes-factor tests of null hypotheses e. Despite the many appealing qualities described in those articles, we urge caution when using model comparison for assessing null hypotheses and Bayes factors in particular , for the following main reasons: 1. The magnitude and direction of a Bayes factor can change, sometimes dramatically, depending on the choice of alternative prior.
Examples of the sensitivity of the Bayes factor to the alternative-hypothesis prior distribution are provided by Kruschke , Ch.
Proponents of the Bayes factor approach to hypothesis testing are well aware of this issue, of course. One way to address this issue is by establishing default families of alternative priors e. Default alternative priors often are not representative of theoretically meaningful alternative priors.
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For example, Kruschke showed a case of testing extrasensory perception in which the Bayes factor changed direction, from favoring the null to favoring the alternative, when the alternative prior was changed from a default to a distribution based on idealized previous results. Therefore, for a hypothesis test to be meaningful, the alternative prior distribution must be meaningful and the prior on the other parameters in the null hypothesis also must be meaningful.
Perhaps the most meaningful alternative prior distribution is one that is obtainable as a Bayesian posterior distribution from previous data after starting with a diffuse proto-prior Kruschke , pp. The same procedure can be used to make a meaningful null-hypothesis distribution.
A similar approach was taken by Verhagen and Wagenmakers when applying Bayesian hypothesis testing to replication analysis. By contrast, alternative prior distributions that are centered at zero, or arbitrarily shaped such as half normals or restricted uniform intervals Dienes , typically would not be obtained as a Bayesian posterior from previous data and a diffuse proto-prior. The Bayes factor by itself does not use the prior probabilities of the hypotheses, hence does not indicate the relative posterior probabilities of the hypotheses, and therefore can be misleading.
To be clear, here we are referring to the prior probabilities of the model indices, shown as the bars on M in panel A of Fig. Consider, for example, the standard introductory example of disease diagnosis.
Lastly, suppose we test a person at random and the result is positive. The Bayes factor of the result is 0. By considering the Bayes factor alone, we would decide that the patient has the disease. But the Bayes factor ignores the prior probability of having the disease. If the disease were rare, with only 0. While the posterior probability is 19 times higher than the prior probability of 0.
Thus, using the Bayes factor to make decisions is dangerous because it ignores the prior probabilities of the models. When applied to null hypothesis testing, if either the null hypothesis or the alternative hypothesis has minuscule prior probability, then an enormous Bayes factor would be required to reverse the prior probabilities of the hypotheses.
The Bayes factor indicates nothing about the magnitude of the effect or the precision of the estimate of the magnitude.
In this way, using a Bayes factor alone is analogous to using a p value alone without a point estimate or confidence interval. Interval estimates should be given for any effect sizes Wilkinson , p. Bayesian Data Analysis, Third Edition Massachusetts Institute of Technology, Today's Web-enabled deluge of electronic data calls for automated methods of data analysis.
Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive Academic Press, For a brief discussion of several benefits of Bayesian data analysis, along with a worked example, and an emphasis that Bayesian data analysis is not Bayesian modeling of mind, see Kruschke c.
For a lengthier exposition that explains one of the primary pitfalls of null hypothesis significance testing and has a discussion of Bayesian null hypothesis testing, along with different examples, see Kruschke a. Before arriving, install necessary software Well be doing the analyses, so bring your notebook computer. There will not be internet access from the tutorial room, so you must prepare your computer before arriving at the tutorial. Please visit the tutorial web site, framed at the top of this page, before arriving at the tutorial.
Follow the instructions on the web site to install the free software on your computer. Figure 2: Hierarchical model for Bayesian linear regression. The data are denoted yi at the bottom of the diagram. The model assumes that the data are generated probabilistically from a normal distribution, governed by parameters 0 the intercept , 1 the slope , and the noise.
Bayesian estimation keeps track of combinations of 0 , 1 , and that credibly account for the data. The instructor John Kruschke has taught introductory Bayesian statistics to graduate students for several years and traditional statistics and mathematical modeling for over 20 years.
He has written an introductory textbook on Bayesian data analysis Kruschke, b ; see also the articles linked above. His research interests include models of attention in learning, which he has developed in both connectionist and Bayesian formalisms.Parameter estimation I.
In this section, we will consider two Bayesian approaches to assessing null values. The article clarifies misconceptions about Bayesian methods that newcomers might have acquired elsewhere. The final sections focus on disabusing possible misconceptions that newcomers might have. Please see the companion article by Kruschke and Liddell for further discussion.
Example 5: Gaussian noise revisited. Along with a complete reorganization of the material, this edition concentrates more on hierarchical Bayesian modeling as implemented via If the disease were rare, with only 0. CRC Press,