How to become a Bayesian in eight easy steps: An annotated reading list
In this guide, we present a reading list to serve as a concise introduction to Bayesian data analysis. The introduction is geared toward reviewers, editors, and interested researchers who are new to Bayesian statistics. We provide commentary for eight recommended sources, which together cover the theoretical and practical cornerstones of Bayesian statistics in psychology and related sciences. The resources are presented in an incremental order, starting with theoretical foundations and moving on to applied issues. In addition, we outline an additional 32 articles and books that can be consulted to gain background knowledge about various theoretical specifics and Bayesian approaches to frequently used models. Our goal is to offer researchers a starting point for understanding the core tenets of Bayesian analysis, while requiring a low level of time commitment. After consulting our guide, the reader should understand how and why Bayesian methods work, and feel able to evaluate their use in the behavioral and social sciences.
KeywordsBayesian statistics Hypothesis testing
In recent decades, significant advances in computational software and hardware have allowed Bayesian statistics to rise to greater prominence in psychology (Van de Schoot, Winder, Ryan, Zondervan-Zwijnenburg, & Depaoli, in press). In the past few years, this rise has accelerated as a result of increasingly vocal criticism of p values in particular (Nickerson 2000; Wagenmakers 2007), and classical statistics in general (Trafimow and Marks 2015). When a formerly scarcely used statistical method rapidly becomes more common, editors and peer reviewers are expected to master it readily, and to adequately evaluate and judge manuscripts in which the method is applied. However, many researchers, reviewers, and editors in psychology are still unfamiliar with Bayesian methods.
We believe that this is at least partly due to the perception that a high level of difficulty is associated with proper use and interpretation of Bayesian statistics. Many seminal texts in Bayesian statistics are dense, mathematically demanding, and assume some background in mathematical statistics (e.g., Gelman et al., 2013). Even texts that are geared toward psychologists (e.g., Kruschke, 2015; Lee and Wagenmakers, 2014), while less mathematically difficult, require a radically different way of thinking than the classical statistical methods most researchers are familiar with. Furthermore, transitioning to a Bayesian framework requires a level of time commitment that is not feasible for many researchers. More approachable sources that survey the core tenets and reasons for using Bayesian methods exist, yet identifying these sources can prove difficult for researchers with little or no previous exposure to Bayesian statistics.
In this guide, we provide a small number of primary sources that editors, reviewers, and other interested researchers can study to gain a basic understanding of Bayesian statistics. Each of these sources was selected for their balance of accessibility with coverage of essential Bayesian topics. By focusing on interpretation, rather than implementation, the guide is able to provide an introduction to core concepts, from Bayes’ theorem through to Bayesian cognitive models, without getting mired in secondary details.
This guide is divided into two primary sections. The first, Theoretical sources, includes commentaries on three articles and one book chapter that explain the core tenets of Bayesian methods as well as their philosophical justification. The second, Applied sources, includes commentaries on four articles that cover the most commonly used methods in Bayesian data analysis at a primarily conceptual level. This section emphasizes issues of particular interest to reviewers, such as basic standards for conducting and reporting Bayesian analyses.
We suggest that for each source, readers first review our commentary, then consult the original source. The commentaries not only summarize the essential ideas discussed in each source but also give a sense of how those ideas fit into the bigger picture of Bayesian statistics. This guide is part of a larger special issue in Psychonomic Bulletin & Review on the topic of Bayesian inference that contains articles which elaborate on many of the same points we discuss here, so we will periodically point to these as potential next steps for the interested reader. For those who would like to delve further into the theory and practice of Bayesian methods, the Appendix provides a number of supplemental sources that would be of interest to researchers and reviewers. To facilitate readers’ selection of additional sources, each source is briefly described and has been given a rating by the authors that reflects its level of difficulty and general focus (i.e., theoretical versus applied; see Fig. 2). It is important to note that our reading list covers sources published up to the time of this writing (August, 2016).
Overall, the guide is designed such that a researcher might be able to read all eight of the highlighted articles1 and some supplemental readings within a week. After readers acquaint themselves with these sources, they should be well-equipped both to interpret existing research and to evaluate new research that relies on Bayesian methods.
In this section, we discuss the primary ideas underlying Bayesian inference in increasing levels of depth. Our first source introduces Bayes’ theorem and demonstrates how Bayesian statistics are based on a different conceptualization of probability than classical, or frequentist, statistics (Lindley 1993). These ideas are extended in our second source’s discussion of Bayesian inference as a reallocation of credibility between possible states of nature (Kruschke 2015). The third source demonstrates how the concepts established in the previous sources lead to many practical benefits for experimental psychology (Dienes 2011). The section concludes with an in-depth review of Bayesian hypothesis testing using Bayes factors with an emphasis on this technique’s theoretical benefits (Rouder, Speckman, Sun, Morey, & Iverson, 2009).
1. Conceptual introduction: What is Bayesian inference?
Source: Lindley (1993)—The analysis of experimental data: The appreciation of tea and wine
Lindley leads with a story in which renowned statistician Ronald A. Fisher is having his colleague, Dr. Muriel Bristol, over for tea. When Fisher prepared the tea—as the story goes—Dr. Bristol protested that Fisher had made the tea all wrong. She claims that tea tastes better when milk is added first and infusion second,2 rather than the other way around; she furthermore professes her ability to tell the difference. Fisher subsequently challenged Dr. Bristol to prove her ability to discern the two methods of preparation in a perceptual discrimination study. In Lindley’s telling of the story, which takes some liberties with the actual design of the experiment in order to emphasize a point, Dr. Bristol correctly identified five out of six cups where the tea was added either first or second. This result left Fisher faced with the question: Was his colleague merely guessing, or could she really tell the difference? Fisher then proceeded to develop his now classic approach in a sequence of steps, recognizing at various points that tests that seem intuitively appealing actually lead to absurdities, until he arrived at a method that consists of calculating the total probability of the observed result plus the probability of any more extreme results possible under the null hypothesis (i.e., the probability that she would correctly identify five or six cups by sheer guessing). This probability is the p value. If it is less than .05, then Fisher would declare the result significant and reject the null hypothesis of guessing.
Lindley’s paper essentially continues Fisher’s work, showing that Fisher’s classic procedure is inadequate and itself leads to absurdities because it hinges upon the nonexistent ability to define what other unobserved results would count as “more extreme” than the actual observations. That is, if Fisher had set out to serve Dr. Bristol six cups (and only six cups) and she is correct five times, then we get a p value of .109, which is not statistically significant. According to Fisher, in this case we should not reject the null hypothesis that Dr. Bristol is guessing. But had he set out to keep giving her additional cups until she was correct five times, which incidentally required six cups, we get a p value of .031, which is statistically significant. According to Fisher, we should now reject the null hypothesis. Even though the data observed in both cases are exactly the same, we reach different conclusions because our definition of “more extreme” results (that did not occur) changes depending on which sampling plan we use. Absurdly, the p value, and with it our conclusion about Dr. Bristol’s ability, depends on how we think about results that might have occurred but never actually did, and that in turn depends on how we planned the experiment (rather than only on how it turned out).
A key takeaway from this paper is that Lindley’s Bayesian approach depends only on the observed data, so the results are interpretable regardless of whether the sampling plan was rigid or flexible or even known at all. Another key point is that the Bayesian approach is inherently comparative: Hypotheses are tested against one another and never in isolation. Lindley further concludes that, since the posterior probability that the null is true will often be higher than the p value, the latter metric will discount null hypotheses more easily in general.
2. Bayesian credibility assessments
In this book chapter, Kruschke explains the fundamental Bayesian principle of reallocation of probability, or “credibility,” across possible states of nature. Kruschke uses an example featuring Sherlock Holmes to demonstrate that the famous detective essentially used Bayesian reasoning to solve his cases. Suppose that Holmes has determined that there exist only four different possible causes (A, B, C, and D) of a committed crime which, for simplicity in the example, he holds to be equally credible at the outset. This translates to equal prior probabilities for each of the four possible causes (i.e., a prior probability of 1/4 for each). Now suppose that Holmes gathers evidence that allows him to rule out cause A with certainty. This development causes the probability assigned to A to drop to zero, and the probability that used to be assigned to cause A to be then redistributed across the other possible causes. Since the probabilities for the four alternatives need to sum to one, the probability for each of the other causes is now equal to 1/3 (Figure 2.1, p. 17). What Holmes has done is reallocate credibility across the different possible causes based on the evidence he has gathered. His new state of knowledge is that only one of the three remaining alternatives can be the cause of the crime and that they are all equally plausible. Holmes, being a man of great intellect, is eventually able to completely rule out two of the remaining three causes, leaving him with only one possible explanation—which has to be the cause of the crime (as it now must have probability equal to 1), no matter how improbable it might have seemed at the beginning of his investigation.
“How often have Isaid to you that when all other 𝜃 yield P(x|𝜃) of 0, whatever remains, however low its P(𝜃), must have P(𝜃|x) = 1?” – Sherlock Holmes, paraphrased
The reader might object that it is rather unrealistic to assume that data can be gathered that allow a researcher to completely rule out contending hypotheses. In real applications, psychological data are noisy, and outcomes are only probabilistically linked to the underlying causes. In terms of reallocation of credibility, this means that possible hypotheses can rarely be ruled out completely (i.e., reduced to zero probability), however, their credibility can be greatly diminished, leading to a substantial increase in the credibility of other possible hypotheses. Although a hypothesis has not been eliminated, something has been learned: Namely, that one or more of the candidate hypotheses has had their probabilities reduced and are now less likely than the others.
In a statistical context, the possible hypotheses are parameter values in mathematical models that serve to describe the observed data in a useful way. For example, a scientist could assume that their observations are normally distributed and be interested in which range of values for the mean is most credible. Sherlock Holmes only considered a set of discrete possibilities, but in many cases it would be very restrictive to only allow a few alternatives (e.g., when estimating the mean of a normal distribution). In the Bayesian framework one can easily consider an infinite continuum of possibilities, across which credibility may still be reallocated. It is easy to extend this framework of reallocation of credibility to hypothesis testing situations where one parameter value is seen as “special” and receives a high amount of prior probability compared to the alternatives (as in Lindley’s tea example above).
Kruschke (2015) serves as a good first introduction to Bayesian thinking, as it requires only basic statistical knowledge (a natural follow-up is Kruschke & Liddell, this issue). In this chapter, Kruschke also provides a concise introduction to mathematical models and parameters, two core concepts which our other sources will build on. One final key takeaway from this chapter is the idea of sequential updating from prior to posterior (Figure 2.1, p. 17) as data are collected. As Dennis Lindley famously said: “Today’s posterior is tomorrow’s prior” (Lindley 1972, p. 2).
3. Implications of Bayesian statistics for experimental psychology
Source: Dienes (2011) — Bayesian versus orthodox statistics: Which side are you on?
Dienes explains several differences between the frequentist (which Dienes calls orthodox and we have called classical; we use these terms interchangeably) and Bayesian paradigm which have practical implications for how experimental psychologists conduct experiments, analyze data, and interpret results (a natural follow-up to the discussion in this section is available in Dienes & McLatchie, this issue). Throughout the paper, Dienes also discusses subjective (or context-dependent) Bayesian methods which allow for inclusion of relevant problem-specific knowledge in to the formation of one’s statistical model.
The probabilities of data given theory and of theory given data
When testing a theory, both the frequentist and Bayesian approaches use probability theory as the basis for inference, yet in each framework, the interpretation of probability is different. It is important to be aware of the implications of this difference in order to correctly interpret frequentist and Bayesian analyses. One major contrast is a result of the fact that frequentist statistics only allow for statements to be made about P(data∣theory):4 Assuming the theory is correct, the probability of observing the obtained (or more extreme) data is evaluated. Dienes argues that often the probability of the data assuming a theory is correct is not the probability the researcher is interested in. What researchers typically want to know is P(theory∣data): Given that the data were those obtained, what is the probability that the theory is correct? At first glance, these two probabilities might appear similar, but Dienes illustrates their fundamental difference with the following example: The probability that a person is dead (i.e., data) given that a shark has bitten the person’s head off (i.e., theory) is 1. However, given that a person is dead, the probability that a shark has bitten this person’s head off is very close to zero (see Senn, 2013, for an intuitive explanation of this distinction). It is important to keep in mind that a p value does not correspond to P(theory∣data); in fact, statements about this probability are only possible if one is willing to attach prior probabilities (degrees of plausibility or credibility) to theories—which can only be done in the Bayesian paradigm.
In the following sections, Dienes explains how the Bayesian approach is more liberating than the frequentist approach with regard to the following concepts: stopping rules, planned versus post hoc comparisons, and multiple testing. For those new to the Bayesian paradigm, these proposals may seem counterintuitive at first, but Dienes provides clear and accessible explanations for each.
In the classical statistical paradigm, it is necessary to specify in advance how the data will be collected. In practice, one usually has to specify how many participants will be collected; stopping data collection early or continuing after the pre-specified number of participants has been reached is not permitted. One reason why collecting additional participants is not permitted in the typical frequentist paradigm is that, given the null hypothesis is true, the p value is not driven in a particular direction as more observations are gathered. In fact, in many cases the distribution of the p value is uniform when the null hypothesis is true, meaning that every p value is equally likely under the null. This implies that even if there is no effect, a researcher is guaranteed to obtain a statistically significant result if they simply continue to collect participants and stop when the p value is sufficiently low. In contrast, the Bayes factor, the most common Bayesian method of hypothesis testing, will approach infinite support in favor of the null hypothesis as more observations are collected if the null hypothesis is true. Furthermore, since Bayesian inference obeys the likelihood principle, one is allowed to continue or stop collecting participants at any time while maintaining the validity of one’s results (p. 276; see also Cornfield, 1966; Rouder, 2014, and Royall, 2004 in the appended Further Reading section).
Planned versus post hoc comparisons
In the classical hypothesis-testing approach, a distinction is made between planned and post hoc comparisons: It matters whether the hypothesis was formulated before or after data collection. In contrast, Dienes argues that adherence to the likelihood principle entails that a theory does not necessarily need to precede the data when a Bayesian approach is adopted; since this temporal information does not enter into the likelihood function for the data, the evidence for or against the theory will be the same no matter its temporal relation to the data.
When conducting multiple tests in the classical approach, it is important to correct for the number of tests performed (see Gelman & Loken, 2014). Dienes points out that within the Bayesian approach, the number of hypotheses tested does not matter—it is not the number of tests that is important, but the evaluation of how accurately each hypothesis predicts the observed data. Nevertheless, it is crucial to consider all relevant evidence, including so-called “outliers,” because “cherry picking is wrong on all statistical approaches” (Dienes 2011, p. 280).
Context-dependent Bayes factors
The last part of the article addresses how problem-specific knowledge may be incorporated in the calculation of the Bayes factor. As is also explained in our next highlighted source (Rouder et al. 2009), there are two main schools of Bayesian thought: default (or objective) Bayes and context-dependent (or subjective) Bayes. In contrast to the default Bayes factors for general application that are designed to have certain desirable mathematical properties (e.g., Jeffreys, 1961; Ly, Verhagen, & Wagenmakers, 2016; Rouder & Morey, 2012; Rouder, Morey, Speckman, & Province, 2012; Rouder et al., 2009), Dienes provides an online calculator5 that enables one to obtain context-dependent Bayes factors that incorporate domain knowledge for several commonly used statistical tests. In contrast to the default Bayes factors, which are typically designed to use standardized effect sizes, the context-dependent Bayes factors specify prior distributions in terms of the raw effect size. Readers who are especially interested in prior elicitation should see the appendix of Dienes’ article for a short review of how to appropriately specify prior distributions that incorporate relevant theoretical information (and Dienes, 2014, for more details and worked examples).
4. Structure and motivation of Bayes factors
Source: Rouder et al. (2009) — Bayesian t tests for accepting and rejecting the null hypothesis
In many cases, a scientist’s primary interest is in showing evidence for an invariance, rather than a difference. For example, researchers may want to conclude that experimental and control groups do not differ in performance on a task (e.g., van Ravenzwaaij, Boekel, Forstmann, Ratcliff, & Wagenmakers, 2014), that participants were performing at chance (Dienes & Overgaard, 2015), or that two variables are unrelated (Rouder & Morey, 2012). In classical statistics this is generally not possible as significance tests are asymmetric; they can only serve to reject the null hypothesis and never to affirm it. One benefit of Bayesian analysis is that inference is perfectly symmetric, meaning evidence can be obtained that favors the null hypothesis as well as the alternative hypothesis (see Gallistel, 2009, as listed in our Further Reading Appendix). This is made possible by the use of Bayes factors.6 The section covering the shortcomings of classical statistics (“Critiques of Inference by Significance Tests”) can safely be skipped, but readers particularly interested in the motivation of Bayesian inference are advised to read it.
What is a Bayes factor?
The Bayes factor is a representation of the relative predictive success of two or more models, and it is a fundamental measure of relative evidence. The way Bayesians quantify predictive success of a model is to calculate the probability of the data given that model—also called the marginal likelihood or sometimes simply the evidence. The ratio of two such probabilities is the Bayes factor. Rouder and colleagues (2009) denote the probability of the data given some model, represented by H i , as f(data∣H i ).7 The Bayes factor for H 0 versus H 1 is simply the ratio of f(data∣H 0) and f(data∣H 1) written B 01 (or B F 01), where the B (or BF) indicates a Bayes factor, and the subscript indicates which two models are being compared (see p. 228). If the result of a study is B 01 = 10 then the data are ten times more probable under H 0 than under H 1. Researchers should report the exact value of the Bayes factor since it is a continuous measure of evidence, but various benchmarks have been suggested to help researchers interpret Bayes factors, with values between 1 and 3, between 3 and 10, and greater than 10 generally taken to indicate inconclusive, weak, and strong evidence, respectively (see Jeffreys, 1961; Wagenmakers, 2007; Etz & Vandekerckhove, 2016), although different researchers may set different benchmarks. Care is need when interpreting Bayes factors against these benchmarks, as they are not meant to be bright lines against which we judge a study’s success (as opposed to how a statistical significance criterion is sometimes treated); the difference between a Bayes factor of, say, 8 and 12 is more a difference of degree than of category. Furthermore, Bayes factors near 1 indicate the data are uninformative, and should not be interpreted as even mild evidence for either of the hypotheses under consideration.
Readers who are less comfortable with reading mathematical notation may skip over most of the equations without too much loss of clarity. The takeaway is that to evaluate which model is better supported by the data, we need to find out which model has done the best job predicting the data we observe. To a Bayesian, the probability a model assigns to the observed data constitutes its predictive success (see Morey, Romeijn, & Rouder, 2016); a model that assigns a high probability to the data relative to another model is best supported by the data. The goal is then to find the probability a given model assigns the data, f(data∣H i ). Usually, the null hypothesis specifies that the true parameter is a particular value of interest (e.g., 0), so we can easily find f(data∣H 0). However, we generally do not know the value of the parameter if the null model is false, so we do not know what probability it assigns the data. To represent our uncertainty with regard to the true value of the parameter if the null hypothesis is false, Bayesians specify a range of plausible values that the parameter might take under the alternative hypothesis. All of these parameter values are subsequently used in computing an average probability of the data given the alternative hypothesis, f(data∣H 1) (for an intuitive illustration, see Gallistel, 2009 as listed in our Further Reading Appendix). If the prior distribution gives substantial weight to parameter values that assign high probability to the data, then the average probability the alternative hypothesis assigns to the data will be relatively high—the model is effectively rewarded for its accurate predictions with a high value for f(data∣H 1).
The role of priors
The form of the prior can have important consequences on the resulting Bayes factor. As discussed in our third source (Dienes 2011), there are two primary schools of Bayesian thought: default (objective) Bayes (Berger 2006) and context-dependent (subjective) Bayes (Goldstein et al., 2006; Rouder, Morey, & Wagenmakers, 2016). The default Bayesian tries to specify prior distributions that convey little information while maintaining certain desirable properties. For example, one desirable property is that changing the scale of measurement should not change the way the information is represented in the prior, which is accomplished by using standardized effect sizes. Context-dependent prior distributions are often used because they more accurately encode our prior information about the effects under study, and can be represented with raw or standardized effect sizes, but they do not necessarily have the same desirable mathematical properties (although sometimes they can).
Choosing a prior distribution for the standardized effect size is relatively straightforward for the default Bayesian. One possibility is to use a normal distribution centered at 0 and with some standard deviation (i.e., spread) σ. If σ is too large, the Bayes factor will always favor the null model, so such a choice would be unwise (see also DeGroot, 1982; Robert, 2014). This happens because such a prior distribution assigns weight to very extreme values of the effect size, when in reality, the effect is most often reasonably small (e.g., almost all psychological effects are smaller than Cohen’s d = 2). The model is penalized for low predictive success. Setting σ to 1 is reasonable and common—this is called the unit information prior. However, using a Cauchy distribution (which resembles a normal distribution but with less central mass and fatter tails) has some better properties than the unit information prior, and is now a common default prior on the alternative hypothesis, giving rise to what is now called the default Bayes factor (see Rouder & Morey, 2012 for more details; see also Wagenmakers, Love, et al., this issue and Wagenmakers, Marsman, et al., this issue). To use the Cauchy distribution, like the normal distribution, again one must specify a scaling factor. If it is too large, the same problem as before occurs where the null model will always be favored. Rouder and colleagues suggest a scale of 1, which implies that the effect size has a prior probability of 50% to be between d = −1 and d = 1. For some areas, such as social psychology, this is not reasonable, and the scale should be reduced. However, slight changes to the scale often do not make much difference in the qualitative conclusions one draws.
Readers are advised to pay close attention to the sections “Subjectivity in priors” and “Bayes factors with small effects.” The former explains how one can tune the scale of the default prior distribution to reflect more contextually relevant information while maintaining the desirable properties attached to prior distributions of this form, a practice that is a reasonable compromise between the default and context-dependent schools. The latter shows why the Bayes factor will often show evidence in favor of the null hypothesis if the observed effect is small and the prior distribution is relatively diffuse.
At this point, the essential concepts of Bayesian probability, Bayes’ theorem, and the Bayes factor have been discussed in depth. In the following four sources, these concepts are applied to real data analysis situations. Our first source provides a broad overview of the most common methods of model comparison, including the Bayes factor, with a heavy emphasis on its proper interpretation (Vandekerckhove, Matzke, & Wagenmakers, 2015). The next source begins by demonstrating Bayesian estimation techniques in the context of developmental research, then provides some guidelines for reporting Bayesian analyses (van de Schoot et al. 2014). Our final two sources discuss issues in Bayesian cognitive modeling, such as the selection of appropriate priors (Lee & Vanpaemel, this issue), and the use of cognitive models for theory testing (Lee 2008).
Before moving on to our final four highlighted sources, it will be useful if readers consider some differences in perspective among practitioners of Bayesian statistics. The application of Bayesian methods is very much an active field of study, and as such, the literature contains a multitude of deep, important, and diverse viewpoints on how data analysis should be done, similar to the philosophical divides between Neyman–Pearson and Fisher concerning proper application of classical statistics (see Lehmann, 1993). The divide between subjective Bayesians, who elect to use priors informed by theory, and objective Bayesians, who instead prefer “uninformative” or default priors, has already been mentioned throughout the Theoretical sources section above.
A second division of note exists between Bayesians who see a place for hypothesis testing in science, and those who see statistical inference primarily as a problem of estimation. The former believe statistical models can stand as useful surrogates for theoretical positions, whose relative merits are subsequently compared using Bayes factors and other such “scoring” metrics (as reviewed in Vandekerckhove et al., 2015, discussed below; for additional examples, see Jeffreys, 1961and Rouder, Morey, Verhagen, Province, & Wagenmakers, 2016). The latter would rather delve deeply into a single model or analysis and use point estimates and credible intervals of parameters as the basis for their theoretical conclusions (as demonstrated in Lee, 2008, discussed below; for additional examples, see Gelman & Shalizi, 2013 and McElreath, 2016).8
Novice Bayesians may feel surprised that such wide divisions exist, as statistics (of any persuasion) is often thought of as a set of prescriptive, immutable procedures that can be only right or wrong. We contend that debates such as these should be expected due to the wide variety of research questions—and diversity of contexts—to which Bayesian methods are applied. As such, we believe that the existence of these divisions speaks to the intellectual vibrancy of the field and its practitioners. We point out these differences here so that readers might use this context to guide their continued reading.
5. Bayesian model comparison methods
Source: Vandekerckhove et al. (2015) — Model comparison and the principle of parsimony
John von Neumann famously said: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” (as quoted in Mayer, Khairy, & Howard, 2010, p. 698), pointing to the natural tension between model parsimony and goodness of fit. The tension occurs because it is always possible to decrease the amount of error between a model’s predictions and the observed data by simply adding more parameters to the model. In the extreme case, any data set of N observations can be reproduced perfectly by a model with N parameters. Such practices, however, termed overfitting, result in poor generalization and greatly reduce the accuracy of out-of-sample predictions. Vandekerckhove and colleagues (2015) take this issue as a starting point to discuss various criteria for model selection. How do we select a model that both fits the data well and generalizes adequately to new data?
Putting the problem in perspective, the authors discuss research on recognition memory that relies on multinomial processing trees, which are simple, but powerful, cognitive models. Comparing these different models using only the likelihood term is ill-advised, because the model with the highest number of parameters will—all other things being equal—yield the best fit. As a first step to addressing this problem, Vandekerckhove et al. (2015) discuss the popular Akaike information criterion (AIC) and Bayesian information criterion (BIC).
Though derived from different philosophies (for an overview, see Aho, Derryberry, & Peterson, 2014), both AIC and BIC try to solve the trade-off between goodness-of-fit and parsimony by combining the likelihood with a penalty for model complexity. However, this penalty is solely a function of the number of parameters and thus neglects the functional form of the model, which can be informative in its own right. As an example, the authors mention Fechner’s law and Steven’s law. The former is described by a simple logarithmic function, which can only ever fit negatively accelerated data. Steven’s law, however, is described by an exponential function, which can account for both positively and negatively accelerated data. Additionally, both models feature just a single parameter, nullifying the benefit of the complexity penalty in each of the two aforementioned information criteria.
The Bayes factor yields a way out. It extends the simple likelihood ratio test by integrating the likelihood with respect to the prior distribution, thus taking the predictive success of the prior distribution into account (see also Gallistel, 2009, in the Further Reading Appendix). Essentially, the Bayes factor is a likelihood ratio test averaged over all possible parameter values for the model, using the prior distributions as weights: It is the natural extension of the likelihood ratio test to a Bayesian framework. The net effect of this is to penalize complex models. While a complex model can predict a wider range of possible data points than a simple model can, each individual data point is less likely to be observed under the complex model. This is reflected in the prior distribution being more spread out in the complex model. By weighting the likelihood by the corresponding tiny prior probabilities, the Bayes factor in favor of the complex model decreases. In this way, the Bayes factor instantiates an automatic Ockham’s Razor (see also, Myung & Pitt, 1997 in the appended Further Reading section).
However, the Bayes factor can be difficult to compute because it often involves integration over very many dimensions at once. Vandekerckhove and colleagues (2015) advocate two methods to ease the computational burden: importance sampling and the Savage-Dickey density ratio (see also Wagenmakers, Lodewyckx, Kuriyal, & Grasman, 2010) in our in our Further reading Appendix); additional common computational methods include the Laplace approximation (Kass & Raftery, 1995), bridge sampling (Gronau et al. 2017; Meng & Wong, 1996), and the encompassing prior approach (Hoijtink, Klugkist, & Boelen, 2008). They also provide code to estimate parameters in multinomial processing tree models and to compute the Bayes factor to select among them. Overall, the chapter provides a good overview of different methods used to tackle the tension between goodness-of-fit and parsimony in a Bayesian framework. While it is more technical then the sources reviewed above, this article can greatly influence how one thinks about models and methods for selecting among them.
6. Bayesian estimation
Source: van de Schoot et al. (2014) — A gentle introduction to Bayesian analysis: Applications to developmental research
This source approaches practical issues related to parameter estimation in the context of developmental research. This setting offers a good basis for discussing the choice of priors and how those choices influence the posterior estimates for parameters of interest. This is a topic that matters to reviewers and editors alike: How does the choice of prior distributions for focal parameters influence the statistical results and theoretical conclusions that are obtained? The article discusses this issue on a basic and illustrative level.
At this point we feel it is important to note that the difference between hypothesis testing and estimation in the Bayesian framework is much greater than it is in the frequentist framework. In the frequentist framework there is often a one-to-one relationship between the null hypothesis falling outside the sample estimate’s 95% confidence interval and rejection of the null hypothesis with a significance test (e.g., when doing a t-test). This is not so in the Bayesian framework; one cannot test a null hypothesis by simply checking if the null value is inside or outside a credible interval. A detailed explanation of the reason for this deserves more space than we can afford to give it here, but in short: When testing hypotheses in the Bayesian framework one should calculate a model comparison metric. See Rouder and Vandekerckhove (this issue) for an intuitive introduction to (and synthesis of) the distinction between Bayesian estimation and testing.
van de Schoot and colleagues (2014) begin by reviewing the main differences between frequentist and Bayesian approaches. Most of this part can be skipped by readers who are comfortable with basic terminology at that point. The only newly introduced term is Markov chain Monte Carlo (MCMC) methods, which refers to the practice of drawing samples from the posterior distribution instead of deriving the distribution analytically (which may not be feasible for many models; see also van Ravenzwaaij, Cassey, & Brown, this issue and Matzke, Boehm, & Vandekerckhove, this issue). After explaining this alternative approach (p. 848), Bayesian estimation of focal parameters and the specification of prior distributions is discussed with the aid of two case examples.
The first example concerns estimation of an ordinary mean value and the variance of reading scores and serves to illustrate how different sources of information can be used to inform the specification of prior distributions. The authors discuss how expert domain knowledge (e.g., reading scores usually fall within a certain range), statistical considerations (reading scores are normally distributed), and evidence from previous studies (results obtained from samples from similar populations) may be jointly used to define adequate priors for the mean and variance model parameters. The authors perform a prior sensitivity analysis to show how using priors based on different considerations influence the obtained results. Thus, the authors examine and discuss how the posterior distributions of the mean and variance parameters are dependent on the prior distributions used.
The second example focuses on a data set from research on the longitudinal reciprocal associations between personality and relationships. The authors summarize a series of previous studies and discuss how results from these studies may or may not inform prior specifications for the latest obtained data set. Ultimately, strong theoretical considerations are needed to decide whether data sets that were gathered using slightly different age groups can be used to inform inferences about one another.
The authors fit a model with data across two time points and use it to discuss how convergence of the MCMC estimator can be supported and checked. They then evaluate overall model fit via a posterior predictive check. In this type of model check, data simulated from the specified model are compared to the observed data. If the model is making appropriate predictions, the simulated data and the observed data should appear similar. The article concludes with a brief outline of guidelines for reporting Bayesian analyses and results in a manuscript. Here, the authors emphasize the importance of the specification of prior distributions and of convergence checks (if MCMC sampling is used) and briefly outline how both might be reported. Finally, the authors discuss the use of default priors and various options for conducting Bayesian analyses with common software packages (such as Mplus and WinBUGS).
The examples in the article illustrate different considerations that should be taken into account for choosing prior specifications, the consequences they can have on the obtained results, and how to check whether and how the choice of priors influenced the resulting inferences.
7. Prior elicitation
Source: Lee and Vanpaemel (this issue) — Determining priors for cognitive models
Statistics does not operate in a vacuum, and often prior knowledge is available that can inform one’s inferences. In contrast to classical statistics, Bayesian statistics allows one to formalize and use this prior knowledge for analysis. The paper by Lee and Vanpaemel (this issue) fills an important gap in the literature: What possibilities are there to formalize and uncover prior knowledge?
The authors start by noting a fundamental point: Cognitive modeling (as introduced in our final source, Lee, 2008) is an extension of general purpose statistical modeling (e.g., linear regression). Cognitive models are designed to instantiate theory, and thus may need to use richer information and assumptions than general purpose models (see also Franke, 2016). A consequence of this is that the prior distribution, just like the likelihood, should be seen as an integral part of the model. As Jaynes (2003) put it: “If one fails to specify the prior information, a problem of inference is just as ill-posed as if one had failed to specify the data” (p. 373).
What information can we use to specify a prior distribution? Because the parameters in such a cognitive model usually have a direct psychological interpretation, theory may be used to constrain parameter values. For example, a parameter interpreted as a probability of correctly recalling a word must be between 0 and 1. To make this point clear, the authors discuss three cognitive models and show how the parameters instantiate relevant information about psychological processes. Lee and Vanpaemel also discuss cases in which all of the theoretical content is carried by the prior, while the likelihood does not make any strong assumptions. They also discuss the principle of transformation invariance, that is, prior distributions for parameters should be invariant to the scale they are measured on (e.g., measuring reaction time using seconds versus milliseconds).
Lee and Vanpaemel also discuss specific methods of prior specification. These include the maximum entropy principle, the prior predictive distribution, and hierarchical modeling. The prior predictive distribution is the model-implied distribution of the data, weighted with respect to the prior. Recently, iterated learning methods have been employed to uncover an implicit prior held by a group of participants. These methods can also be used to elicit information that is subsequently formalized as a prior distribution. (For a more in-depth discussion of hierarchical cognitive modeling, see Lee, 2008, discussed below.)
In sum, the paper gives an excellent overview of why and how one can specify prior distributions for cognitive models. Importantly, priors allow us to integrate domain-specific knowledge, and thus build stronger theories (Platt 1964; Vanpaemel 2010). For more information on specifying prior distributions for data-analytic statistical models rather than cognitive models see Rouder, Morey, Verhagen, Swagman, and Wagenmakers (in press) and Rouder, Engelhardt, McCabe, and Morey (2016).
8. Bayesian cognitive modeling
Source: Lee (2008) — Three case studies in the Bayesian analysis of cognitive models
Our final source (Lee 2008) further discusses cognitive modeling, a more tailored approach within Bayesian methods. Often in psychology, a researcher will not only expect to observe a particular effect, but will also propose a verbal theory of the cognitive process underlying the expected effect. Cognitive models are used to formalize and test such verbal theories in a precise, quantitative way. For instance, in a cognitive model, psychological constructs, such as attention and bias, are expressed as model parameters. The proposed psychological process is expressed as dependencies among parameters and observed data (the “structure” of the model).
In peer-reviewed work, Bayesian cognitive models are often presented in visual form as a graphical model. Model parameters are designated by nodes, where the shape, shading, and style of border of each node reflect various parameter characteristics. Dependencies among parameters are depicted as arrows connecting the nodes. Lee gives an exceptionally clear and concise description of how to read graphical models in his discussion of multidimensional scaling (Lee 2008, p. 2).
After a model is constructed, the observed data are used to update the priors and generate a set of posterior distributions. Because cognitive models are typically complex, posterior distributions are almost always obtained through sampling methods (i.e., MCMC; see van Ravenzwaaij et al., this issue), rather than through direct, often intractable, analytic calculations.
Lee demonstrates the construction and use of cognitive models through three case studies. Specifically, he shows how three popular process models may be implemented in a Bayesian framework. In each case, he begins by explaining the theoretical basis of each model, then demonstrates how the verbal theory may be translated into a full set of prior distributions and likelihoods. Finally, Lee discusses how results from each model may be interpreted and used for inference.
Each case example showcases a unique advantage of implementing cognitive models in a Bayesian framework (see also Bartlema, Voorspoels, Rutten, Tuerlinckx, & Vanpaemel, this issue). For example, in his discussion of signal detection theory, Lee highlights how Bayesian methods are able to account for individual differences easily (see also Rouder & Lu, 2005, in the Further reading Appendix). Throughout, Lee emphasizes that Bayesian cognitive models are useful because they allow the researcher to reach new theoretical conclusions that would be difficult to obtain with non-Bayesian methods. Overall, this source not only provides an approachable introduction to Bayesian cognitive models, but also provides an excellent example of good reporting practices for research that employs Bayesian cognitive models.
By focusing on interpretation, rather than implementation, we have sought to provide a more accessible introduction to the core concepts and principles of Bayesian analysis than may be found in introductions with a more applied focus. Ideally, readers who have read through all eight of our highlighted sources, and perhaps some of the supplementary reading, may now feel comfortable with the fundamental ideas in Bayesian data analysis, from basic principles (Kruschke 2015; Lindley 1993) to prior distribution selection (Lee & Vanpaemel, this issue), and with the interpretation of a variety of analyses, including Bayesian analogs of classical statistical tests (e.g., t-tests; Rouder et al., 2009), estimation in a Bayesian framework (van de Schoot et al. 2014), Bayes factors and other methods for hypothesis testing (Dienes 2011; Vandekerckhove et al. 2015), and Bayesian cognitive models (Lee 2008).
Reviewers and editors unfamiliar with Bayesian methods may initially feel hesitant to evaluate empirical articles in which such methods are applied (Wagenmakers, Love, et al., this issue). Ideally, the present article should help ameliorate this apprehension by offering an accessible introduction to Bayesian methods that is focused on interpretation rather than application. Thus, we hope to help minimize the amount of reviewer reticence caused by authors’ choice of statistical framework.
Our overview was not aimed at comparing the advantages and disadvantages of Bayesian and classical methods. However, some conceptual conveniences and analytic strategies that are only possible or valid in the Bayesian framework will have become evident. For example, Bayesian methods allow for the easy implementation of hierarchical models for complex data structures (Lee 2008), they allow multiple comparisons and flexible sampling rules during data collection without correction of inferential statistics (Dienes, 2011; see also Schönbrodt, Wagenmakers, Zehetleitner, & Perugini, 2015), as listed in our Further reading Appendix, and also Schönbrodt & Wagenmakers, this issue), and they allow inferences that many researchers in psychology are interested in but are not able to answer with classical statistics such as providing support for a null hypothesis (for a discussion, see Wagenmakers, 2007). Thus, the inclusion of more research that uses Bayesian methods in the psychological literature should be to the benefit of the entire field (Etz & Vandekerckhove, 2016). In this article, we have provided an overview of sources that should allow a novice to understand how Bayesian statistics allow for these benefits, even without prior knowledge of Bayesian methods.
Links to freely available versions of each article are provided in the References section.
As a historical note: Distinguishing milk-first from infusion-first tea preparation was not a particular affectation of Dr. Bristol’s, but a cultural debate that has persisted for over three centuries (e.g., Orwell, 1946).
If the null hypothesis is not initially considered tenable, then we can proceed without assigning separate probability to it and instead focus on estimating the parameters of interest (e.g., the taster’s accuracy in distinguishing wines, as in Lindley’s second example; see Lindley’s Figure 1, and notice that the amount of probability assigned to p = .5 is gone). Additionally, if a range of values of the parameter is considered impossible—such as rates that are below chance—then this range may be given zero prior probability.
The conditional probability (P) of the data, given (∣) theory.
Readers for whom Rouder and colleagues’ (2009) treatment is too technical could focus on Dienes’ conceptual ideas and motivations underlying the Bayes factor.
The probability (f) of the observed data, given (∣) hypothesis i (H i ), where i indicates one of the candidate hypotheses (e.g., 0, 1, A, etc.). The null hypothesis is usually denoted H 0 and the alternative hypothesis is usually denoted either H 1 or H A .
This divide in Bayesian statistics may be seen as a parallel to the recent discussions about use of classical statistics in psychology (e.g., Cumming, 2014), where a greater push has been made to adopt an estimation approach over null hypothesis significance testing (NHST). Discussions on the merits of hypothesis testing have been running through all of statistics for over a century, with no end in sight.
The authors would like to thank Jeff Rouder, E.-J. Wagenmakers, and Joachim Vandekerckhove for their helpful comments. AE and BB were supported by grant #1534472 from NSF’s Methods, Measurements, and Statistics panel. AE was further supported by the National Science Foundation Graduate Research Fellowship Program (#DGE1321846).
- Bartlema, A., Voorspoels, W., Rutten, F., Tuerlinckx, F., & Vanpaemel, W. (this issue). Sensitivity to the prototype in children with high-functioning autism spectrum disorder: An example of Bayesian cognitive psychometrics. Psychonomic Bulletin and Review.Google Scholar
- Berger, J. O., & Delampady, M. (1987). Testing precise hypotheses. Statistical Science, 317–335. Retrieved from https://projecteuclid.org/euclid.ss/1177013238
- DeGroot, M. H. (1982). Lindley’s paradox: Comment. Journal of the American Statistical Association, 336–339. Retrieved from http://www.jstor.org/stable/2287246
- Dienes, Z (2008). Understanding psychology as a science: An introduction to scientific and statistical inference. Palgrave Macmillan.Google Scholar
- Dienes, Z. (2014). Using Bayes to get the most out of nonsignificant results. Frontiers in Psychology, 5. Retrieved from http://journal.frontiersin.org/article/10.3389/fpsyg.2014.00781/full
- Dienes, Z., & McLatchie, N. (this issue). Four reasons to prefer Bayesian over orthodox statistical analyses. Psychonomic Bulletin and Review.Google Scholar
- Dienes, Z., & Overgaard, M. (2015). How Bayesian statistics are needed to determine whether mental states are unconscious. Behavioural Methods in Consciousness Research, 199–220. Retrieved from http://tinyurl.com/dienes2015
- Etz, A., & Vandekerckhove, J. (this issue). Introduction to Bayesian inference for psychology. Psychonomic Bulletin and Review.Google Scholar
- Etz, A., & Wagenmakers, E.-J. (in press). J. B. S. Haldane’s contribution to the Bayes factor hypothesis test. Statistical Science.Google Scholar
- Franke, M. (2016). Task types, link functions & probabilistic modeling in experimental pragmatics. In F. Salfner & U. Sauerland (Eds.), Preproceedings of ‘trends in experimental pragmatics’ (pp. 56–63).Google Scholar
- Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis (Vol. 3). Chapman & Hall/CRC.Google Scholar
- Gronau, Q. F., Sarafoglou, A., Matzke, D., Ly, A., Boehm, U., Marsman, M., ..., & Steingroever, H. (2017). A tutorial on bridge sampling. arXiv:1703.05984
- Hoijtink, H., Klugkist, I., & Boelen, P. (2008). Bayesian evaluation of informative hypotheses. Springer Science & Business Media.Google Scholar
- Jaynes, E. T. (1986). Bayesian methods: General background In In J.H. Justice, & E.T. Jaynes (Eds.), Maximum entropy and Bayesian methods in applied statistics, (pp. 1–25). Cambridge: Cambridge University Press. Retrieved from http://tinyurl.com/jaynes1986
- Jeffreys, H. (1936). Xxviii. on some criticisms of the theory of probability. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 22(146), 337–359. doi: 10.1080/14786443608561691. Retrieved from http://www.tandfonline.com/doi/pdf/10.1080/14786443608561691.CrossRefGoogle Scholar
- Jeffreys, H. (1961). Theory of probability, 3rd edn. Oxford, UK: Oxford University Press.Google Scholar
- Kaplan, D., & Depaoli, S. (2012). Bayesian structural equation modeling In In R. Hoyle, D. Kaplan, & S. Depaoli (Eds.), Handbook of structural equation modeling, (pp. 650–673). New York, NY: Guilford. Retrieved from http://tinyurl.com/kaplan2012
- Kruschke, J. K. (2015). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press. Retrieved from http://tinyurl.com/kruschke2015
- Kruschke, J. K., & Liddell, T. (this issue). Bayesian data analysis for newcomers. Psychonomic Bulletin and Review.Google Scholar
- Lee, M. D., & Vanpaemel, W. (this issue). Determining priors for cognitive models. Psychonomic Bulletin & Review. Retrieved from https://webfiles.uci.edu/mdlee/LeeVanpaemel2016.pdf
- Lee, M. D., & Wagenmakers, E.J. (2014). Bayesian cognitive modeling: A practical course. Cambridge: Cambridge University Press.Google Scholar
- Love, J., Selker, R., Marsman, M., Jamil, T., Dropmann, D., Verhagen, J., ..., & Wagenmakers, E.-J. (2015). JASP (version 0.7.1.12). Computer Software.Google Scholar
- Ly, A., Verhagen, A. J., & Wagenmakers, E.-J. (2016). Harold Jeffreys’s default Bayes factor hypothesis tests: Explanation, extension, and application in psychology. Journal of Mathematical Psychology, 72, 19–32. Retrieved from http://tinyurl.com/zyvgp9y http://tinyurl.com/zyvgp9y CrossRefGoogle Scholar
- Matzke, D., Boehm, U., & Vandekerckhove, J. (this issue). Bayesian inference for psychology, Part III: Parameter estimation in nonstandard models. Psychonomic Bulletin and Review.Google Scholar
- McElreath, R. (2016). Statistical rethinking: A Bayesian course with examples in R and Stan (Vol. 122). Boca Raton: CRC Press.Google Scholar
- Meng, X.-L., & Wong, W. H. (1996). Simulating ratios of normalizing constants via a simple identity: A theoretical exploration. Statistica Sinica, 831–860.Google Scholar
- Morey, R. D., Romeijn, J.-W., & Rouder, J. N. (2016). The philosophy of Bayes factors and the quantification of statistical evidence. Journal of Mathematical Psychology. Retrieved from http://tinyurl.com/BFphilo
- Orwell, G. (1946). A nice cup of tea. Evening Standard, January.Google Scholar
- Rouder, J. N., Morey, R. D., Verhagen, J., Swagman, A. R., & Wagenmakers, E.-J (in press). Bayesian analysis of factorial designs. Psychological Methods. Retrieved from http://tinyurl.com/zh4bkt8
- Rouder, J. N., Morey, R. D., & Wagenmakers, E.-J. (2016). The interplay between subjectivity, statistical practice, and psychological science. Collabra, 2(1). Retrieved from http://www.collabra.org/article/10.1525/collabra.28/
- Rouder, J. N., & Vandekerckhove, J. (this issue). Bayesian inference for psychology, Part IV: Parameter estimation and Bayes factors. Psychonomic Bulletin and Review.Google Scholar
- Royall, R. (1997). Statistical evidence: A likelihood paradigm (Vol. 77). Boca Raton: CRC Press.Google Scholar
- Royall, R. (2004). The likelihood paradigm for statistical inference In In M.L. Taper, & S.R. Lele (Eds.), The nature of scientific evidence: Statistical, philosophical and empirical considerations, (pp. 119–152). Chicago: The University of Chicago Press. Retrieved from http://tinyurl.com/royall2004
- Schönbrodt, F. D., & Wagenmakers, E.-J. (this issue). Bayes factor design analysis: Planning for compelling evidence. Psychonomic Bulletin and Review.Google Scholar
- Schönbrodt, F. D., Wagenmakers, E.-J., Zehetleitner, M., & Perugini, M. (2015). Sequential hypothesis testing with Bayes factors: Efficiently testing mean differences. Psychological Methods. Retrieved from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2604513. doi: 10.1037/met0000061
- Senn, S. (2013). Invalid inversion. Significance, 10(2), 40–42. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/j.1740-9713.2013.00652.x/full CrossRefGoogle Scholar
- Sorensen, T., Hohenstein, S., & Vasishth, S. (2016). Bayesian linear mixed models using Stan: A tutorial for psychologists, linguists, and cognitive scientists. The Quantitative Methods for Psychology (3). Retrieved from http://www.tqmp.org/RegularArticles/vol12-3/p175/p175.pdf. doi: 10.20982/tqmp.12.3.p175
- Stone, J. V. (2013). Bayes’ rule: A tutorial introduction to Bayesian analysis. Sebtel Press.Google Scholar
- Vandekerckhove, J., Matzke, D., & Wagenmakers, E.-J. (2015). Model comparison and the principle of parsimony In In J. Busemeyer, J. Townsend, Z. J. Wang, A. Eidels, J. Vandekerckhove, D. Matzke, & E.-J. Wagenmakers (Eds.), Oxford handbook of computational and mathematical psychology (pp. 300–317). Oxford: Oxford University Press. Retrieved from http://tinyurl.com/vandekerckhove2015
- van de Schoot, R., Kaplan, D., Denissen, J., Asendorpf, J. B., Neyer, F. J., & Aken, M. A (2014). A gentle introduction to Bayesian analysis: Applications to developmental research . Child Development, 85 (3), 842–860. Retrieved from http://tinyurl.com/vandeschoot
- Van de Schoot, R., Winder, S., Ryan, O., Zondervan-Zwijnenburg, M., & Depaoli, S. (in press). A systematic review of Bayesian papers in psychology: The last 25 years. Psychological Methods.Google Scholar
- van Ravenzwaaij, D., Cassey, P., & Brown, S. (this issue). A simple introduction to Markov chain Monte-Carlo sampling. Psychonomic Bulletin and Review.Google Scholar
- van Ravenzwaaij, D., Boekel, W., Forstmann, B. U., Ratcliff, R., & Wagenmakers, E.- J. (2014). Action video games do not improve the speed of information processing in simple perceptual tasks. Journal of Experimental Psychology: General, 143(5), 1794–1805. doi: 10.1037/a0036923. Retrieved from http://tinyurl.com/vanRavenzwaaij.
- Wagenmakers, E.-J., Lodewyckx, T., Kuriyal, H., & Grasman, R. (2010). Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method. Cognitive Psychology, 60(3), 158–189. doi: 10.1016/j.cogpsych.2009.12.001. Retrieved from http://tinyurl.com/wagenmakers2010.CrossRefPubMedGoogle Scholar
- Wagenmakers, E.-J., Love, J., Marsman, M., Jamil, T., Ly, A., Verhagen, J., ..., & Morey, R. D. (this issue). Bayesian inference for psychology, Part II: Example applications with JASP. Psychonomic Bulletin and Review.Google Scholar
- Wagenmakers, E.-J., Marsman, M., Jamil, T., Ly, A., Verhagen, J., Love, J., ..., & Morey, R. (this issue). Bayesian inference for psychology, Part I: Theoretical advantages and practical ramifications. Psychonomic Bulletin and Review.Google Scholar
- Wagenmakers, E.-J., Morey, R. D., & Lee, M. (2016). Bayesian benefits for the pragmatic researcher. Current Directions in Psychological Science, 25(3). Retrieved from https://osf.io/3tdh9/
- Wagenmakers, E.-J., Verhagen, J., & Ly, A. (2015). How to quantify the evidence for the absence of a correlation. Behavior Research Methods, 1–14.Google Scholar
- Wetzels, R., Matzke, D., Lee, M. D., Rouder, J. N., Iverson, G. J., & Wagenmakers, E.- J. (2011). Statistical evidence in experimental psychology: An empirical comparison using 855 t-tests. Perspectives on Psychological Science, 6 (3), 291–298. doi: 10.1177/1745691611406923. Retrieved from http://tinyurl.com/wetzels2011.CrossRefPubMedGoogle Scholar
- Winkler, R. L. (2003). An introduction to Bayesian inference and decision, 2nd edn. Holt, Rinehart and Winston: New York.Google Scholar