Posterior predictive checks can and should be Bayesian: Comment on Gelman and Shalizi, ‘Philosophy and the practice of Bayesian statistics’

Bayesian inference is conditional on the space of models assumed by the analyst. The posterior distribution indicates only which of the available parameter values are less bad than the others, without indicating whether the best available parameter values really fit the data well. A posterior predictive check is important to assess whether the posterior predictions of the least bad parameters are discrepant from the actual data in systematic ways. Gelman and Shalizi (2012a) assert that the posterior predictive check, whether done qualitatively or quantitatively, is non‐Bayesian. I suggest that the qualitative posterior predictive check might be Bayesian, and the quantitative posterior predictive check should be Bayesian. In particular, I show that the ‘Bayesian p‐value’, from which an analyst attempts to reject a model without recourse to an alternative model, is ambiguous and inconclusive. Instead, the posterior predictive check, whether qualitative or quantitative, should be consummated with Bayesian estimation of an expanded model. The conclusion agrees with Gelman and Shalizi regarding the importance of the posterior predictive check for breaking out of an initially assumed space of models. Philosophically, the conclusion allows the liberation to be completely Bayesian instead of relying on a non‐Bayesian deus ex machina. Practically, the conclusion cautions against use of the Bayesian p‐value in favour of direct model expansion and Bayesian evaluation.     

Philosophy and the practice of Bayesian statistics

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico‐deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.     

Three case studies in the Bayesian analysis of cognitive models

Bayesian statistical inference offers a principled and comprehensive approach for relating psychological models to data. This article presents Bayesian analyses of three influential psychological models: multidimensional scaling models of stimulus representation, the generalized context model of category learning, and a signal detection theory model of decision making. In each case, the model is recast as a probabilistic graphical model and is evaluated in relation to a previously considered data set. In each case, it is shown that Bayesian inference is able to provide answers to important theoretical and empirical questions easily and coherently. The generality of the Bayesian approach and its potential for the understanding of models and data in psychology are discussed.     

An application of Carnapian inductive logic to an argument in the philosophy of statistics

I claim that an argument from the philosophy of statistics can be improved by using Carnapian inductive logic. Gelman and Shalizi [9] criticise a philosophical account of how statisticians ought to choose statistical models which they call ‘the received view of Bayesian inference’ and propose a different account inspired by falsificationist philosophy of science. I introduce another philosophical account inspired by Carnapian inductive logic and argue that it is even better than Gelman and Shalizi's falsificationist account.     

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer??s calculus of ??conditional density operators??. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.     

Bayesian nonparametric model selection and model testing

This article examines a Bayesian nonparametric approach to model selection and model testing, which is based on concepts from Bayesian decision theory and information theory. The approach can be used to evaluate the predictive-utility of any model that is either probabilistic or deterministic, with that model analyzed under either the Bayesian or classical-frequentist approach to statistical inference. Conditional on an observed set of data, generated from some unknown true sampling density, the approach identifies the “best” model as the one that predicts a sampling density that explains the most information about the true density. Furthermore, in the approach, the decision is to reject a model when it does not explain enough information about the true density (according to a straightforward calibration of the Kullback-Leibler divergence measure). The posterior estimate of the true density is based on a Bayesian nonparametric prior that can give positive support to the entire space of sampling densities (defined on some sample space). This article also discusses the theoretical and practical advantages of the Bayesian nonparametric approach over all other types of model selection procedures, and over any model testing procedure that depends on interpreting a p-value. Finally, the Bayesian nonparametric approach is illustrated on four real data sets, in the comparison and testing of order-constrained models, cognitive models, models of choice-behavior, and a test of a general psychometric model.     

A computational-level explanation of the speed of goal inference

The ability to understand the goals that drive another person’s actions is an important social and cognitive skill. This is no trivial task, because any given action may in principle be explained by different possible goals (e.g., one may wave ones arm to hail a cab or to swat a mosquito). To select which goal best explains an observed action is a form of abduction. To explain how people perform such abductive inferences, Baker, Tenenbaum, and Saxe (2007) proposed a computational-level theory that formalizes goal inference as Bayesian inverse planning (BIP). It is known that general Bayesian inference–be it exact or approximate–is computationally intractable (NP-hard). As the time required for computationally intractable computations grows excessively fast when scaled from toy domains to the real world, it seems that such models cannot explain how humans can perform Bayesian inferences quickly in real world situations. In this paper we investigate how the BIP model can nevertheless explain how people are able to make goal inferences quickly. The approach that we propose builds on taking situational constraints explicitly into account in the computational-level model. We present a methodology for identifying situational constraints that render the model tractable. We discuss the implications of our findings and reflect on how the methodology can be applied to alternative models of goal inference and Bayesian models in general.     

An overview of Bayesian reasoning in the analysis of delay‐discounting data

Statistical inference (including interval estimation and model selection) is increasingly used in the analysis of behavioral data. As with many other fields, statistical approaches for these analyses traditionally use classical (i.e., frequentist) methods. Interpreting classical intervals and p‐values correctly can be burdensome and counterintuitive. By contrast, Bayesian methods treat data, parameters, and hypotheses as random quantities and use rules of conditional probability to produce direct probabilistic statements about models and parameters given observed study data. In this work, we reanalyze two data sets using Bayesian procedures. We precede the analyses with an overview of the Bayesian paradigm. The first study reanalyzes data from a recent study of controls, heavy smokers, and individuals with alcohol and/or cocaine substance use disorder, and focuses on Bayesian hypothesis testing for covariates and interval estimation for discounting rates among various substance use disorder profiles. The second example analyzes hypothetical environmental delay‐discounting data. This example focuses on using historical data to establish prior distributions for parameters while allowing subjective expert opinion to govern the prior distribution on model preference. We review the subjective nature of specifying Bayesian prior distributions but also review established methods to standardize the generation of priors and remove subjective influence while still taking advantage of the interpretive advantages of Bayesian analyses. We present the Bayesian approach as an alternative paradigm for statistical inference and discuss its strengths and weaknesses.     

Sensitivity to sampling in Bayesian word learning

We report a new study testing our proposal that word learning may be best explained as an approximate form of Bayesian inference (Xu & Tenenbaum, in press). Children are capable of learning word meanings across a wide range of communicative contexts. In different contexts, learners may encounter different sampling processes generating the examples of word-object pairings they observe. An ideal Bayesian word learner could take into account these differences in the sampling process and adjust his/her inferences about word meaning accordingly. We tested how children and adults learned words for novel object kinds in two sampling contexts, in which the objects to be labeled were sampled either by a knowledgeable teacher or by the learners themselves. Both adults and children generalized more conservatively in the former context; that is, they restricted the label to just those objects most similar to the labeled examples when the exemplars were chosen by a knowledgeable teacher, but not when chosen by the learners themselves. We discuss how this result follows naturally from a Bayesian analysis, but not from other statistical approaches such as associative word-learning models.     

A Bayesian Way to Make Stopping Rules Matter

Disputes between advocates of Bayesians and more orthodox approaches to statistical inference presuppose that Bayesians must regard must regard stopping rules, which play an important role in orthodox statistical methods, as evidentially irrelevant.In this essay, I show that this is not the case and that the stopping rule is evidentially relevant given some Bayesian confirmation measures that have been seriously proposed. However, I show that accepting a confirmation measure of this sort comes at the cost of rejecting two useful ancillaryBayesian principles.     

When can non‐commutative statistical inference be Bayesian?

Based on recalling two characteristic features of Bayesian statistical inference in commutative probability theory, a stability property of the inference is pointed out, and it is argued that that stability of the Bayesian statistical inference is an essential property which must be preserved under generalization of Bayesian inference to the non‐commutative case. Mathematical no‐go theorems are recalled then which show that, in general, the stability can not be preserved in non‐commutative context. Two possible interpretations of the impossibility of generalization of Bayesian statistical inference to the non‐commutative case are offered, none of which seems to be completely satisfying.     

Theory-based Bayesian models of inductive learning and reasoning

Inductive inference allows humans to make powerful generalizations from sparse data when learning about word meanings, unobserved properties, causal relationships, and many other aspects of the world. Traditional accounts of induction emphasize either the power of statistical learning, or the importance of strong constraints from structured domain knowledge, intuitive theories or schemas. We argue that both components are necessary to explain the nature, use and acquisition of human knowledge, and we introduce a theory-based Bayesian framework for modeling inductive learning and reasoning as statistical inferences over structured knowledge representations.     

Coherent psychometric modelling with Bayesian nonparametrics

In this paper we argue that model selection, as commonly practised in psychometrics, violates certain principles of coherence. On the other hand, we show that Bayesian nonparametrics provides a coherent basis for model selection, through the use of a ‘nonparametric’ prior distribution that has a large support on the space of sampling distributions. We illustrate model selection under the Bayesian nonparametric approach, through the analysis of real questionnaire data. Also, we present ways to use the Bayesian nonparametric framework to define very flexible psychometric models, through the specification of a nonparametric prior distribution that supports all distribution functions for the inverse link, including the standard logistic distribution functions. The Bayesian nonparametric approach provides a coherent method for model selection that can be applied to any statistical model, including psychometric models. Moreover, under a ‘non‐informative’ choice of nonparametric prior, the Bayesian nonparametric approach is easy to apply, and selects the model that maximizes the log likelihood. Thus, under this choice of prior, the approach can be extended to non‐Bayesian settings where the parameters of the competing models are estimated by likelihood maximization, and it can be used with any psychometric software package that routinely reports the model log likelihood.     

A tutorial introduction to Bayesian models of cognitive development

We present an introduction to Bayesian inference as it is used in probabilistic models of cognitive development. Our goal is to provide an intuitive and accessible guide to the what, the how, and the why of the Bayesian approach: what sorts of problems and data the framework is most relevant for, and how and why it may be useful for developmentalists. We emphasize a qualitative understanding of Bayesian inference, but also include information about additional resources for those interested in the cognitive science applications, mathematical foundations, or machine learning details in more depth. In addition, we discuss some important interpretation issues that often arise when evaluating Bayesian models in cognitive science.     

How to use probabilities in reasoning

Conclusions Probabilities are important in belief updating, but probabilistic reasoning does not subsume everything else (as the Bayesian would have it). On the contrary, Bayesian reasoning presupposes knowledge that cannot itself be obtained by Bayesian reasoning, making generic Bayesianism an incoherent theory of belief updating. Instead, it is indefinite probabilities that are of principal importance in belief updating. Knowledge of such indefinite probabilities is obtained by some form of statistical induction, and inferences to non-probabilistic conclusions are carried out in accordance with the statistical syllogism. Such inferences have been the focus of much attention in the nonmonotonic reasoning literature, but the logical complexity of such inference has not been adequately appreciated.     

Word learning as Bayesian inference

The authors present a Bayesian framework for understanding how adults and children learn the meanings of words. The theory explains how learners can generalize meaningfully from just one or a few positive examples of a novel word's referents, by making rational inductive inferences that integrate prior knowledge about plausible word meanings with the statistical structure of the observed examples. The theory addresses shortcomings of the two best known approaches to modeling word learning, based on deductive hypothesis elimination and associative learning. Three experiments with adults and children test the Bayesian account's predictions in the context of learning words for object categories at multiple levels of a taxonomic hierarchy. Results provide strong support for the Bayesian account over competing accounts, in terms of both quantitative model fits and the ability to explain important qualitative phenomena. Several extensions of the basic theory are discussed, illustrating the broader potential for Bayesian models of word learning.     

A Bayesian random effects model for testlets

Standard item response theory (IRT) models fit to dichotomous examination responses ignore the fact that sets of items (testlets) often come from a single common stimuli (e.g. a reading comprehension passage). In this setting, all items given to an examinee are unlikely to be conditionally independent (given examinee proficiency). Models that assume conditional independence will overestimate the precision with which examinee proficiency is measured. Overstatement of precision may lead to inaccurate inferences such as prematurely ending an examination in which the stopping rule is based on the estimated standard error of examinee proficiency (e.g., an adaptive test). To model examinations that may be a mixture of independent items and testlets, we modified one standard IRT model to include an additional random effect for items nested within the same testlet. We use a Bayesian framework to facilitate posterior inference via a Data Augmented Gibbs Sampler (DAGS; Tanner & Wong, 1987). The modified and standard IRT models are both applied to a data set from a disclosed form of the SAT. We also provide simulation results that indicates that the degree of precision bias is a function of the variability of the testlet effects, as well as the testlet design.The authors wish to thank Robert Mislevy, Andrew Gelman and Donald B. Rubin for their helpful suggestions and comments, Ida Lawrence and Miriam Feigenbaum for providing us with the SAT data analyzed in section 5, and to the two anonymous referees for their careful reading and thoughtful suggestions on an earlier draft. We are also grateful to the Educational Testing service for providing the resources to do this research.     

Robust Bayesian growth curve modelling using conditional medians

Growth curve models have been widely used to analyse longitudinal data in social and behavioural sciences. Although growth curve models with normality assumptions are relatively easy to estimate, practical data are rarely normal. Failing to account for non-normal data may lead to unreliable model estimation and misleading statistical inference. In this work, we propose a robust approach for growth curve modelling using conditional medians that are less sensitive to outlying observations. Bayesian methods are applied for model estimation and inference. Based on the existing work on Bayesian quantile regression using asymmetric Laplace distributions, we use asymmetric Laplace distributions to convert the problem of estimating a median growth curve model into a problem of obtaining the maximum likelihood estimator for a transformed model. Monte Carlo simulation studies have been conducted to evaluate the numerical performance of the proposed approach with data containing outliers or leverage observations. The results show that the proposed approach yields more accurate and efficient parameter estimates than traditional growth curve modelling. We illustrate the application of our robust approach using conditional medians based on a real data set from the Virginia Cognitive Aging Project.