Towards end-to-end likelihood-free inference with convolutional neural networks

Complex simulator-based models with non-standard sampling distributions require sophisticated design choices for reliable approximate parameter inference. We introduce a fast, end-to-end approach for approximate Bayesian computation (ABC) based on fully convolutional neural networks. The method enables users of ABC to derive simultaneously the posterior mean and variance of multidimensional posterior distributions directly from raw simulated data. Once trained on simulated data, the convolutional neural network is able to map real data samples of variable size to the first two posterior moments of the relevant parameter's distributions. Thus, in contrast to other machine learning approaches to ABC, our approach allows us to generate reusable models that can be applied by different researchers employing the same model. We verify the utility of our method on two common statistical models (i.e., a multivariate normal distribution and a multiple regression scenario), for which the posterior parameter distributions can be derived analytically. We then apply our method to recover the parameters of the leaky competing accumulator (LCA) model and we reference our results to the current state-of-the-art technique, which is the probability density estimation (PDA). Results show that our method exhibits a lower approximation error compared with other machine learning approaches to ABC. It also performs similarly to PDA in recovering the parameters of the LCA model.     

Hierarchical Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper, we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function.     

A generalized,likelihood-free method for posterior estimation

Recent advancements in Bayesian modeling have allowed for likelihood-free posterior estimation. Such estimation techniques are crucial to the understanding of simulation-based models, whose likelihood functions may be difficult or even impossible to derive. However, current approaches are limited by their dependence on sufficient statistics and/or tolerance thresholds. In this article, we provide a new approach that requires no summary statistics, error terms, or thresholds and is generalizable to all models in psychology that can be simulated. We use our algorithm to fit a variety of cognitive models with known likelihood functions to ensure the accuracy of our approach. We then apply our method to two real-world examples to illustrate the types of complex problems our method solves. In the first example, we fit an error-correcting criterion model of signal detection, whose criterion dynamically adjusts after every trial. We then fit two models of choice response time to experimental data: the linear ballistic accumulator model, which has a known likelihood, and the leaky competing accumulator model, whose likelihood is intractable. The estimated posterior distributions of the two models allow for direct parameter interpretation and model comparison by means of conventional Bayesian statistics—a feat that was not previously possible.     

A simple introduction to Markov Chain Monte–Carlo sampling

Markov Chain Monte–Carlo (MCMC) is an increasingly popular method for obtaining information about distributions, especially for estimating posterior distributions in Bayesian inference. This article provides a very basic introduction to MCMC sampling. It describes what MCMC is, and what it can be used for, with simple illustrative examples. Highlighted are some of the benefits and limitations of MCMC sampling, as well as different approaches to circumventing the limitations most likely to trouble cognitive scientists.     

One and Done? Optimal Decisions From Very Few Samples

In many learning or inference tasks human behavior approximates that of a Bayesian ideal observer, suggesting that, at some level, cognition can be described as Bayesian inference. However, a number of findings have highlighted an intriguing mismatch between human behavior and standard assumptions about optimality: People often appear to make decisions based on just one or a few samples from the appropriate posterior probability distribution, rather than using the full distribution. Although sampling‐based approximations are a common way to implement Bayesian inference, the very limited numbers of samples often used by humans seem insufficient to approximate the required probability distributions very accurately. Here, we consider this discrepancy in the broader framework of statistical decision theory, and ask: If people are making decisions based on samples—but as samples are costly—how many samples should people use to optimize their total expected or worst‐case reward over a large number of decisions? We find that under reasonable assumptions about the time costs of sampling, making many quick but locally suboptimal decisions based on very few samples may be the globally optimal strategy over long periods. These results help to reconcile a large body of work showing sampling‐based or probability matching behavior with the hypothesis that human cognition can be understood in Bayesian terms, and they suggest promising future directions for studies of resource‐constrained cognition.     

Exemplar models as a mechanism for performing Bayesian inference

Probabilistic models have recently received much attention as accounts of human cognition. However, most research in which probabilistic models have been used has been focused on formulating the abstract problems behind cognitive tasks and their optimal solutions, rather than on mechanisms that could implement these solutions. Exemplar models are a successful class of psychological process models in which an inventory of stored examples is used to solve problems such as identification, categorization, and function learning. We show that exemplar models can be used to perform a sophisticated form of Monte Carlo approximation known as importance sampling and thus provide a way to perform approximate Bayesian inference. Simulations of Bayesian inference in speech perception, generalization along a single dimension, making predictions about everyday events, concept learning, and reconstruction from memory show that exemplar models can often account for human performance with only a few exemplars, for both simple and relatively complex prior distributions. These results suggest that exemplar models provide a possible mechanism for implementing at least some forms of Bayesian inference.     

Directed Convergence in Stable Percept Acquisition

We view a perceptual capacity as a nondeductive inference, represented as a function from a set of premises to a set of conclusions. The application of the function to a single premise to produce a single conclusion is called a "percept" or "instantaneous percept." We define a stable percept as a convergent sequence of instantaneous percepts. Assuming that the sets of premises and conclusions are metric spaces, we introduce a strategy for acquiring stable percepts, called directed convergence. We consider probabilistic inferences, where the premise and conclusion sets are spaces of probability measures, and in this context we study Bayesian probabilistic/recursive inference. In this type of Bayesian inference the premises are probability measures, and the prior as well as the posterior is updated nontrivially at each iteration. This type of Bayesian inference is distinguished from classical Bayesian statistical inference where the prior remains fixed, and the posterior evolves by conditioning on successively more punctual premises. We indicate how the directed convergence procedure may be implemented in the context of Bayesian probabilistic/recursive inference. We discuss how the L(infinity) metric can be used to give numerical control of this type of Bayesian directed convergence. Copyright 2001 Academic Press.     

Where do hypotheses come from?

Why are human inferences sometimes remarkably close to the Bayesian ideal and other times systematically biased? In particular, why do humans make near-rational inferences in some natural domains where the candidate hypotheses are explicitly available, whereas tasks in similar domains requiring the self-generation of hypotheses produce systematic deviations from rational inference. We propose that these deviations arise from algorithmic processes approximating Bayes’ rule. Specifically in our account, hypotheses are generated stochastically from a sampling process, such that the sampled hypotheses form a Monte Carlo approximation of the posterior. While this approximation will converge to the true posterior in the limit of infinite samples, we take a small number of samples as we expect that the number of samples humans take is limited. We show that this model recreates several well-documented experimental findings such as anchoring and adjustment, subadditivity, superadditivity, the crowd within as well as the self-generation effect, the weak evidence, and the dud alternative effects. We confirm the model’s prediction that superadditivity and subadditivity can be induced within the same paradigm by manipulating the unpacking and typicality of hypotheses. We also partially confirm our model’s prediction about the effect of time pressure and cognitive load on these effects.     

Bayesian Fundamentalism or Enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition

The prominence of Bayesian modeling of cognition has increased recently largely because of mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations. We note commonalities between this rational approach and other movements in psychology - namely, Behaviorism and evolutionary psychology - that set aside mechanistic explanations or make use of optimality assumptions. Through these comparisons, we identify a number of challenges that limit the rational program's potential contribution to psychological theory. Specifically, rational Bayesian models are significantly unconstrained, both because they are uninformed by a wide range of process-level data and because their assumptions about the environment are generally not grounded in empirical measurement. The psychological implications of most Bayesian models are also unclear. Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the hypothesis sets and the approximation algorithms used to derive model predictions, without a clear delineation between psychological commitments and implementational details. Comparing multiple Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we argue they must be developed in conjunction with mechanistic considerations to offer substantive explanations of cognition. We lay out several means for such an integration, which take into account the representations on which Bayesian inference operates, as well as the algorithms and heuristics that carry it out. We argue this unification will better facilitate lasting contributions to psychological theory, avoiding the pitfalls that have plagued previous theoretical movements.     

Spectral Decompositions of Multiple Time Series: A Bayesian Non-parametric Approach

We consider spectral decompositions of multiple time series that arise in studies where the interest lies in assessing the influence of two or more factors. We write the spectral density of each time series as a sum of the spectral densities associated to the different levels of the factors. We then use Whittle’s approximation to the likelihood function and follow a Bayesian non-parametric approach to obtain posterior inference on the spectral densities based on Bernstein–Dirichlet prior distributions. The prior is strategically important as it carries identifiability conditions for the models and allows us to quantify our degree of confidence in such conditions. A Markov chain Monte Carlo (MCMC) algorithm for posterior inference within this class of frequency-domain models is presented. We illustrate the approach by analyzing simulated and real data via spectral one-way and two-way models. In particular, we present an analysis of functional magnetic resonance imaging (fMRI) brain responses measured in individuals who participated in a designed experiment to study pain perception in humans.     

Bayesian inference and the classical test theory model: Reliability and true scores

A general one-way analysis of variance components with unequal replication numbers is used to provide unbiased estimates of the true and error score variance of classical test theory. The inadequacy of the ANOVA theory is noted and the foundations for a Bayesian approach are detailed. The choice of prior distribution is discussed and a justification for the Tiao-Tan prior is found in the particular context of the “n-split” technique. The posterior distributions of reliability, error score variance, observed score variance and true score variance are presented with some extensions of the original work of Tiao and Tan. Special attention is given to simple approximations that are available in important cases and also to the problems that arise when the ANOVA estimate of true score variance is negative. Bayesian methods derived by Box and Tiao and by Lindley are studied numerically in relation to the problem of estimating true score. Each is found to be useful and the advantages and disadvantages of each are discussed and related to the classical test-theoretic methods. Finally, some general relationships between Bayesian inference and classical test theory are discussed. Supported in part by the National Institute of Child Health and Human Development under Research Grant 1 PO1 HDO1762. Reproduction, translation, use or disposal by or for the United States Government is permitted.     

Vision as Bayesian inference: analysis by synthesis?

We argue that the study of human vision should be aimed at determining how humans perform natural tasks with natural images. Attempts to understand the phenomenology of vision from artificial stimuli, although worthwhile as a starting point, can lead to faulty generalizations about visual systems, because of the enormous complexity of natural images. Dealing with this complexity is daunting, but Bayesian inference on structured probability distributions offers the ability to design theories of vision that can deal with the complexity of natural images, and that use 'analysis by synthesis' strategies with intriguing similarities to the brain. We examine these strategies using recent examples from computer vision, and outline some important implications for cognitive science.     

A Bayesian Power Analysis Procedure Considering Uncertainty in Effect Size Estimates from a Meta-analysis

In conventional frequentist power analysis, one often uses an effect size estimate, treats it as if it were the true value, and ignores uncertainty in the effect size estimate for the analysis. The resulting sample sizes can vary dramatically depending on the chosen effect size value. To resolve the problem, we propose a hybrid Bayesian power analysis procedure that models uncertainty in the effect size estimates from a meta-analysis. We use observed effect sizes and prior distributions to obtain the posterior distribution of the effect size and model parameters. Then, we simulate effect sizes from the obtained posterior distribution. For each simulated effect size, we obtain a power value. With an estimated power distribution for a given sample size, we can estimate the probability of reaching a power level or higher and the expected power. With a range of planned sample sizes, we can generate a power assurance curve. Both the conventional frequentist and our Bayesian procedures were applied to conduct prospective power analyses for two meta-analysis examples (testing standardized mean differences in example 1 and Pearson's correlations in example 2). The advantages of our proposed procedure are demonstrated and discussed.     

A hierarchical model for estimating response time distributions

We present a statistical model for inference with response time (RT) distributions. The model has the following features. First, it provides a means of estimating the shape, scale, and location (shift) of RT distributions. Second, it is hierarchical and models between-subjects and within-subjects variability simultaneously. Third, inference with the model is Bayesian and provides a principled and efficient means of pooling information across disparate data from different individuals. Because the model efficiently pools information across individuals, it is particularly well suited for those common cases in which the researcher collects a limited number of observations from several participants. Monte Carlo simulations reveal that the hierarchical Bayesian model provides more accurate estimates than several popular competitors do. We illustrate the model by providing an analysis of the symbolic distance effect in which participants can more quickly ascertain the relationship between nonadjacent digits than that between adjacent digits.     

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 Semiparametric Structural Equation Models with Latent Variables

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.     

Language evolution by iterated learning with bayesian agents

Languages are transmitted from person to person and generation to generation via a process of iterated learning: people learn a language from other people who once learned that language themselves. We analyze the consequences of iterated learning for learning algorithms based on the principles of Bayesian inference, assuming that learners compute a posterior distribution over languages by combining a prior (representing their inductive biases) with the evidence provided by linguistic data. We show that when learners sample languages from this posterior distribution, iterated learning converges to a distribution over languages that is determined entirely by the prior. Under these conditions, iterated learning is a form of Gibbs sampling, a widely-used Markov chain Monte Carlo algorithm. The consequences of iterated learning are more complicated when learners choose the language with maximum posterior probability, being affected by both the prior of the learners and the amount of information transmitted between generations. We show that in this case, iterated learning corresponds to another statistical inference algorithm, a variant of the expectation-maximization (EM) algorithm. These results clarify the role of iterated learning in explanations of linguistic universals and provide a formal connection between constraints on language acquisition and the languages that come to be spoken, suggesting that information transmitted via iterated learning will ultimately come to mirror the minds of the learners.     

Bayesian estimation of semiparametric nonlinear dynamic factor analysis models using the Dirichlet process prior

Parameters in time series and other dynamic models often show complex range restrictions and their distributions may deviate substantially from multivariate normal or other standard parametric distributions. We use the truncated Dirichlet process (DP) as a non‐parametric prior for such dynamic parameters in a novel nonlinear Bayesian dynamic factor analysis model. This is equivalent to specifying the prior distribution to be a mixture distribution composed of an unknown number of discrete point masses (or clusters). The stick‐breaking prior and the blocked Gibbs sampler are used to enable efficient simulation of posterior samples. Using a series of empirical and simulation examples, we illustrate the flexibility of the proposed approach in approximating distributions of very diverse shapes.     

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.