Identifying expectations about the strength of causal relationships

When we try to identify causal relationships, how strong do we expect that relationship to be? Bayesian models of causal induction rely on assumptions regarding people’s a priori beliefs about causal systems, with recent research focusing on people’s expectations about the strength of causes. These expectations are expressed in terms of prior probability distributions. While proposals about the form of such prior distributions have been made previously, many different distributions are possible, making it difficult to test such proposals exhaustively. In Experiment 1 we used iterated learning—a method in which participants make inferences about data generated based on their own responses in previous trials—to estimate participants’ prior beliefs about the strengths of causes. This method produced estimated prior distributions that were quite different from those previously proposed in the literature. Experiment 2 collected a large set of human judgments on the strength of causal relationships to be used as a benchmark for evaluating different models, using stimuli that cover a wider and more systematic set of contingencies than previous research. Using these judgments, we evaluated the predictions of various Bayesian models. The Bayesian model with priors estimated via iterated learning compared favorably against the others. Experiment 3 estimated participants’ prior beliefs concerning different causal systems, revealing key similarities in their expectations across diverse scenarios.     

Prior probability modulates anticipatory activity in category-specific areas

Bayesian models are currently a dominant framework for describing human information processing. However, it is not clear yet how major tenets of this framework can be translated to brain processes. In this study, we addressed the neural underpinning of prior probability and its effect on anticipatory activity in category-specific areas. Before fMRI scanning, participants were trained in two behavioral sessions to learn the prior probability and correct order of visual events within a sequence. The events of each sequence included two different presentations of a geometric shape and one picture of either a house or a face, which appeared with either a high or a low likelihood. Each sequence was preceded by a cue that gave participants probabilistic information about which items to expect next. This allowed examining cue-related anticipatory modulation of activity as a function of prior probability in category-specific areas (fusiform face area and parahippocampal place area). Our findings show that activity in the fusiform face area was higher when faces had a higher prior probability. The finding of a difference between levels of expectations is consistent with graded, probabilistically modulated activity, but the data do not rule out the alternative explanation of a categorical neural response. Importantly, these differences were only visible during anticipation, and vanished at the time of stimulus presentation, calling for a functional distinction when considering the effects of prior probability. Finally, there were no anticipatory effects for houses in the parahippocampal place area, suggesting sensitivity to stimulus material when looking at effects of prediction.     

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.     

Physician Bayesian updating from personal beliefs about the base rate and likelihood ratio

Whether humans can accurately make decisions in line with Bayes’ rule has been one of the most important yet contentious topics in cognitive psychology. Though a number of paradigms have been used for studying Bayesian updating, rarely have subjects been allowed to use their own preexisting beliefs about the prior and the likelihood. A study is reported in which physicians judged the posttest probability of a diagnosis for a patient vignette after receiving a test result, and the physicians’ posttest judgments were compared to the normative posttest calculated from their own beliefs in the sensitivity and false positive rate of the test (likelihood ratio) and prior probability of the diagnosis. On the one hand, the posttest judgments were strongly related to the physicians’ beliefs about both the prior probability as well as the likelihood ratio, and the priors were used considerably more strongly than in previous research. On the other hand, both the prior and the likelihoods were still not used quite as much as they should have been, and there was evidence of other nonnormative aspects to the updating, such as updating independent of the likelihood beliefs. By focusing on how physicians use their own prior beliefs for Bayesian updating, this study provides insight into how well experts perform probabilistic inference in settings in which they rely upon their own prior beliefs rather than experimenter-provided cues. It suggests that there is reason to be optimistic about experts’ abilities, but that there is still considerable need for improvement.     

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.     

Optimal designs for the generalized partial credit model

Analysing ordinal data is becoming increasingly important in psychology, especially in the context of item response theory. The generalized partial credit model (GPCM) is probably the most widely used ordinal model and has found application in many large-scale educational assessment studies such as PISA. In the present paper, optimal test designs are investigated for estimating persons’ abilities with the GPCM for calibrated tests when item parameters are known from previous studies. We find that local optimality may be achieved by assigning non-zero probability only to the first and last categories independently of a person's ability. That is, when using such a design, the GPCM reduces to the dichotomous two-parameter logistic (2PL) model. Since locally optimal designs require the true ability to be known, we consider alternative Bayesian design criteria using weight distributions over the ability parameter space. For symmetric weight distributions, we derive necessary conditions for the optimal one-point design of two response categories to be Bayes optimal. Furthermore, we discuss examples of common symmetric weight distributions and investigate under what circumstances the necessary conditions are also sufficient. Since the 2PL model is a special case of the GPCM, all of these results hold for the 2PL model as well.     

Bayesian latent variable models for the analysis of experimental psychology data

In this paper, we address the use of Bayesian factor analysis and structural equation models to draw inferences from experimental psychology data. While such application is non-standard, the models are generally useful for the unified analysis of multivariate data that stem from, e.g., subjects’ responses to multiple experimental stimuli. We first review the models and the parameter identification issues inherent in the models. We then provide details on model estimation via JAGS and on Bayes factor estimation. Finally, we use the models to re-analyze experimental data on risky choice, comparing the approach to simpler, alternative methods.     

Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data

Missing data are very common in behavioural and psychological research. In this paper, we develop a Bayesian approach in the context of a general nonlinear structural equation model with missing continuous and ordinal categorical data. In the development, the missing data are treated as latent quantities, and provision for the incompleteness of the data is made by a hybrid algorithm that combines the Gibbs sampler and the Metropolis‐Hastings algorithm. We show by means of a simulation study that the Bayesian estimates are accurate. A Bayesian model comparison procedure based on the Bayes factor and path sampling is proposed. The required observations from the posterior distribution for computing the Bayes factor are simulated by the hybrid algorithm in Bayesian estimation. Our simulation results indicate that the correct model is selected more frequently when the incomplete records are used in the analysis than when they are ignored. The methodology is further illustrated with a real data set from a study concerned with an AIDS preventative intervention for Filipina sex workers.     

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.     

Constraining bridges between levels of analysis: A computational justification for locally Bayesian learning

Different levels of analysis provide different insights into behavior: computational-level analyses determine the problem an organism must solve and algorithmic-level analyses determine the mechanisms that drive behavior. However, many attempts to model behavior are pitched at a single level of analysis. Research into human and animal learning provides a prime example, with some researchers using computational-level models to understand the sensitivity organisms display to environmental statistics but other researchers using algorithmic-level models to understand organisms’ trial order effects, including effects of primacy and recency. Recently, attempts have been made to bridge these two levels of analysis. Locally Bayesian Learning (LBL) creates a bridge by taking a view inspired by evolutionary psychology: Our minds are composed of modules that are each individually Bayesian but communicate with restricted messages. A different inspiration comes from computer science and statistics: Our brains are implementing the algorithms developed for approximating complex probability distributions. We show that these different inspirations for how to bridge levels of analysis are not necessarily in conflict by developing a computational justification for LBL. We demonstrate that a scheme that maximizes computational fidelity while using a restricted factorized representation produces the trial order effects that motivated the development of LBL. This scheme uses the same modular motivation as LBL, passing messages about the attended cues between modules, but does not use the rapid shifts of attention considered key for the LBL approximation. This work illustrates a new way of tying together psychological and computational constraints.     

Computational models of inductive reasoning using a statistical analysis of a Japanese corpus

Existing computational models of human inductive reasoning have been constructed based on psychological evaluations concerning the similarities or relationships between entities. However, the costs involved in collecting psychological evaluations for the sheer number of entities that exist mean that they are prohibitively impractical. In order to avoid this problem, the present article examines three types of models: a category-based neural network model, a category-based Bayesian model, and a feature-based neural network model. These models utilize the results of a statistical analysis of a Japanese corpus computing co-occurrence probabilities for word pairs, rather than using psychological evaluations. Argument strength ratings collected by a psychological experiment were found to correlate well with simulations for the category-based neural network model.     

From conditioning to category learning: an adaptive network model

We used adaptive network theory to extend the Rescorla-Wagner (1972) least mean squares (LMS) model of associative learning to phenomena of human learning and judgment. In three experiments subjects learned to categorize hypothetical patients with particular symptom patterns as having certain diseases. When one disease is far more likely than another, the model predicts that subjects will substantially overestimate the diagnosticity of the more valid symptom for the rare disease. The results of Experiments 1 and 2 provide clear support for this prediction in contradistinction to predictions from probability matching, exemplar retrieval, or simple prototype learning models. Experiment 3 contrasted the adaptive network model with one predicting pattern-probability matching when patients always had four symptoms (chosen from four opponent pairs) rather than the presence or absence of each of four symptoms, as in Experiment 1. The results again support the Rescorla-Wagner LMS learning rule as embedded within an adaptive network model.     

The frequentist implications of optional stopping on Bayesian hypothesis tests

Null hypothesis significance testing (NHST) is the most commonly used statistical methodology in psychology. The probability of achieving a value as extreme or more extreme than the statistic obtained from the data is evaluated, and if it is low enough, the null hypothesis is rejected. However, because common experimental practice often clashes with the assumptions underlying NHST, these calculated probabilities are often incorrect. Most commonly, experimenters use tests that assume that sample sizes are fixed in advance of data collection but then use the data to determine when to stop; in the limit, experimenters can use data monitoring to guarantee that the null hypothesis will be rejected. Bayesian hypothesis testing (BHT) provides a solution to these ills because the stopping rule used is irrelevant to the calculation of a Bayes factor. In addition, there are strong mathematical guarantees on the frequentist properties of BHT that are comforting for researchers concerned that stopping rules could influence the Bayes factors produced. Here, we show that these guaranteed bounds have limited scope and often do not apply in psychological research. Specifically, we quantitatively demonstrate the impact of optional stopping on the resulting Bayes factors in two common situations: (1) when the truth is a combination of the hypotheses, such as in a heterogeneous population, and (2) when a hypothesis is composite—taking multiple parameter values—such as the alternative hypothesis in a t-test. We found that, for these situations, while the Bayesian interpretation remains correct regardless of the stopping rule used, the choice of stopping rule can, in some situations, greatly increase the chance of experimenters finding evidence in the direction they desire. We suggest ways to control these frequentist implications of stopping rules on BHT.     

A tutorial on Bayes factor estimation with the product space method

The Bayes factor is an intuitive and principled model selection tool from Bayesian statistics. The Bayes factor quantifies the relative likelihood of the observed data under two competing models, and as such, it measures the evidence that the data provides for one model versus the other. Unfortunately, computation of the Bayes factor often requires sampling-based procedures that are not trivial to implement. In this tutorial, we explain and illustrate the use of one such procedure, known as the product space method (Carlin & Chib, 1995). This is a transdimensional Markov chain Monte Carlo method requiring the construction of a “supermodel” encompassing the models under consideration. A model index measures the proportion of times that either model is visited to account for the observed data. This proportion can then be transformed to yield a Bayes factor. We discuss the theory behind the product space method and illustrate, by means of applied examples from psychological research, how the method can be implemented in practice.     

Linda问题：“齐当别”抉择模型的解释

大量有关人类归因判断的研究表明,人类经常违反理性概率公理。Tversky和Kahneman（1983）使用Linda问题等特定场景的研究发现,人们系统性地表现出违反理性推断标准,判断合取事件发生概率大于其组成事件发生概率,称之为合取谬误,并用人们使用代表性启发式判断概率来解释该现象产生的原因。然而使用启发式观点对合取谬误现象进行解释过于模糊不清。该文首先介绍了合取谬误现象及其解释模型,然后应用Li（1994,2004）提出的不确定情形下决策理论——“齐当别”抉择模型对Linda问题中合取谬误产生的原因进行了新的解释     

Introduction to Bayesian Inference for Psychology

We introduce the fundamental tenets of Bayesian inference, which derive from two basic laws of probability theory. We cover the interpretation of probabilities, discrete and continuous versions of Bayes’ rule, parameter estimation, and model comparison. Using seven worked examples, we illustrate these principles and set up some of the technical background for the rest of this special issue of Psychonomic Bulletin & Review. Supplemental material is available via https://osf.io/wskex/.     

Bayesian techniques for analyzing group differences in the Iowa Gambling Task: A case study of intuitive and deliberate decision-makers

The Iowa Gambling Task (IGT) is one of the most popular experimental paradigms for comparing complex decision-making across groups. Most commonly, IGT behavior is analyzed using frequentist tests to compare performance across groups, and to compare inferred parameters of cognitive models developed for the IGT. Here, we present a Bayesian alternative based on Bayesian repeated-measures ANOVA for comparing performance, and a suite of three complementary model-based methods for assessing the cognitive processes underlying IGT performance. The three model-based methods involve Bayesian hierarchical parameter estimation, Bayes factor model comparison, and Bayesian latent-mixture modeling. We illustrate these Bayesian methods by applying them to test the extent to which differences in intuitive versus deliberate decision style are associated with differences in IGT performance. The results show that intuitive and deliberate decision-makers behave similarly on the IGT, and the modeling analyses consistently suggest that both groups of decision-makers rely on similar cognitive processes. Our results challenge the notion that individual differences in intuitive and deliberate decision styles have a broad impact on decision-making. They also highlight the advantages of Bayesian methods, especially their ability to quantify evidence in favor of the null hypothesis, and that they allow model-based analyses to incorporate hierarchical and latent-mixture structures.     

The Modal Logic of Bayesian Belief Revision

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated efficiently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.     

Prior distributions for random choice structures

We study various axioms of discrete probabilistic choice, measuring how restrictive they are, both alone and in the presence of other axioms, given a specific class of prior distributions over a complete collection of finite choice probabilities. We do this by using Monte Carlo simulation to compute, for a range of prior distributions, probabilities that various simple and compound axioms hold. For example, the probability of the triangle inequality is usually many orders of magnitude higher than the probability of random utility. While neither the triangle inequality nor weak stochastic transitivity imply the other, the conditional probability that one holds given the other holds is greater than the marginal probability, for all priors in the class we consider. The reciprocal of the prior probability that an axiom holds is an upper bound on the Bayes factor in favor of a restricted model, in which the axiom holds, against an unrestricted model. The relatively high prior probability of the triangle inequality limits the degree of support that data from a single decision maker can provide in its favor. The much lower probability of random utility implies that the Bayes factor in favor of it can be much higher, for suitable data.