Evidence Accumulation Models Research Articles

A method, framework, and tutorial for efficiently simulating models of decision-making

Evidence accumulation models (EAMs) have become the dominant models of rapid decision-making. Several variants of these models have been proposed, ranging from the simple linear ballistic accumulator (LBA) to the more complex leaky-competing accumulator (LCA), and further extensions that include time-varying rates of evidence accumulation or decision thresholds. Although applications of the simpler variants have been widespread, applications of the more complex models have been fewer, largely due to their intractable likelihood function and the computational cost of mass simulation. Here, I present a framework for efficiently fitting complex EAMs, which uses a new, efficient method of simulating these models. I find that the majority of simulation time is taken up by random number generation (RNG) from the normal distribution, needed for the stochastic noise of the differential equation. To reduce this inefficiency, I propose using the well-known concept within computer science of “look-up tables” (LUTs) as an approximation to the inverse cumulative density function (iCDF) method of RNG, which I call “LUT-iCDF”. I show that when using an appropriately sized LUT, simulations using LUT-iCDF closely match those from the standard RNG method in R. My framework, which I provide a detailed tutorial on how to implement, includes C code for 12 different variants of EAMs using the LUT-iCDF method, and should make the implementation of complex EAMs easier and faster.

A parameter recovery assessment of time-variant models of decision-making.

Evidence accumulation models have been one of the most dominant modeling frameworks used to study rapid decision-making over the past several decades. These models propose that evidence accumulates from the environment until the evidence for one alternative reaches some threshold, typically associated with caution, triggering a response. However, researchers have recently begun to reconsider the fundamental assumptions of how caution varies with time. In the past, it was typically assumed that levels of caution are independent of time. Recent investigations have however suggested the possibility that levels of caution decrease over time and that this strategy provides more efficient performance under certain conditions. Our study provides the first comprehensive assessment of this newer class of models accounting for time-varying caution to determine how robustly their parameters can be estimated. We assess five overall variants of collapsing threshold/urgency signal models based on the diffusion decision model, linear ballistic accumulator model, and urgency gating model frameworks. We find that estimation of parameters, particularly those associated with caution/urgency modulation are most robust for the linearly collapsing threshold diffusion model followed by an urgency-gating model with a leakage process. All other models considered, particularly those with ballistic accumulation or nonlinear thresholds, are unable to recover their own parameters adequately, making their usage in parameter estimation contexts questionable.

Ensemble clustering based on evidence extracted from the co-association matrix

The evidence accumulation model is an approach for collecting the information of base partitions in a clustering ensemble method, and can be viewed as a kernel transformation from the original data space to a co-association matrix. However, cluster structure information may be partially lost in this transformation; hence, some methods proposed in the literature try to find the lost information and return it to the ensemble process. In this paper, an interesting phenomenon is introduced: remove some evidences from the co-association matrix, which can result in more accurate clustering results. The intuitive explanation for this is that some evidences in the original co-association matrix could be noise, with negative effects on the final clustering. However, it is difficult to detect those evidences practically, let alone remove them from the matrix. To remedy this problem, we remove multiple level evidences having low occurrence frequencies, because negative evidences do not normally occur regularly in the base partitions. Subsequently, we use normalized cut to achieve multiple clustering results. To discriminate the optimal ensemble result, an internal validity index, which uses only the co-association matrix, is specially designed for the clustering ensemble. The experimental results on 16 datasets demonstrate that the proposed scheme outperforms some state-of-the-art clustering ensemble approaches.

Scene categorization in the presence of a distractor.

Humans display a very good understanding of the content in briefly presented photographs. To achieve this understanding, humans rely on information from both high-acuity central vision and peripheral vision. Previous studies have investigated the relative contribution of central and peripheral vision. However, the role of attention in this task remains unclear. In this study, we presented composite images with a scene in the center and another scene in the periphery. The two channels conveyed different information, and the participants were asked to focus on one channel while ignoring the other. In two experiments, we showed that (a) people are better at recognizing the central part, (b) the conflicting signal in the ignored part hinders performance, and (c) this effect is true for both parts (focusing on the central or peripheral part). We conclude that scene recognition is based on both central and peripheral information, even when participants are instructed to focus only on one part of the image and ignore the other. In contrast to the zoom-out hypothesis, we propose that the gist recognition process should be interpreted in terms of the evidence accumulation model in which information from the to-be-ignored parts is also included.

Combining error-driven models of associative learning with evidence accumulation models of decision-making.

As people learn a new skill, performance changes along two fundamental dimensions: Responses become progressively faster and more accurate. In cognitive psychology, these facets of improvement have typically been addressed by separate classes of theories. Reductions in response time (RT) have usually been addressed by theories of skill acquisition, whereas increases in accuracy have been explained by associative learning theories. To date, relatively little work has examined how changes in RT relate to changes in response accuracy, and whether these changes can be accounted for quantitatively within a single theoretical framework. The current work examines joint changes in accuracy and RT in a probabilistic category learning task. We report a model-based analysis of changes in the shapes of RT distributions for different category responses at the level of individual stimuli over the course of learning. We show that changes in performance are determined solely by changes in the quality of information entering the decision process. We then develop a new model that combines an associative learning front end with a sequential sampling model of the decision process, showing that the model provides a good account of all aspects of the learning data. We conclude by discussing potential extensions of the model and future directions for theoretical development that are opened up by our findings.

Optimizing sequential decisions in the drift–diffusion model

To make decisions organisms often accumulate information across multiple timescales. However, most experimental and modeling studies of decision-making focus on sequences of independent trials. On the other hand, natural environments are characterized by long temporal correlations, and evidence used to make a present choice is often relevant to future decisions. To understand decision-making under these conditions we analyze how a model ideal observer accumulates evidence to freely make choices across a sequence of correlated trials. We use principles of probabilistic inference to show that an ideal observer incorporates information obtained on one trial as an initial bias on the next. This bias decreases the time, but not the accuracy of the next decision. Furthermore, in finite sequences of trials the rate of reward is maximized when the observer deliberates longer for early decisions, but responds more quickly towards the end of the sequence. Our model also explains experimentally observed patterns in decision times and choices, thus providing a mathematically principled foundation for evidence-accumulation models of sequential decisions.

Evidence accumulation in a Laplace domain decision space.

Evidence accumulation models of simple decision-making have long assumed that the brain estimates a scalar decision variable corresponding to the log-likelihood ratio of the two alternatives. Typical neural implementations of this algorithmic cognitive model assume that large numbers of neurons are each noisy exemplars of the scalar decision variable. Here we propose a neural implementation of the diffusion model in which many neurons construct and maintain the Laplace transform of the distance to each of the decision bounds. As in classic findings from brain regions including LIP, the firing rate of neurons coding for the Laplace transform of net accumulated evidence grows to a bound during random dot motion tasks. However, rather than noisy exemplars of a single mean value, this approach makes the novel prediction that firing rates grow to the bound exponentially; across neurons there should be a distribution of different rates. A second set of neurons records an approximate inversion of the Laplace transform; these neurons directly estimate net accumulated evidence. In analogy to time cells and place cells observed in the hippocampus and other brain regions, the neurons in this second set have receptive fields along a "decision axis." This finding is consistent with recent findings from rodent recordings. This theoretical approach places simple evidence accumulation models in the same mathematical language as recent proposals for representing time and space in cognitive models for memory.

Low and variable correlation between reaction time costs and accuracy costs explained by accumulation models: Meta-analysis and simulations.

The underpinning assumption of much research on cognitive individual differences (or group differences) is that task performance indexes cognitive ability in that domain. In many tasks performance is measured by differences (costs) between conditions, which are widely assumed to index a psychological process of interest rather than extraneous factors such as speed–accuracy trade-offs (e.g., Stroop, implicit association task, lexical decision, antisaccade, Simon, Navon, flanker, and task switching). Relatedly, reaction time (RT) costs or error costs are interpreted similarly and used interchangeably in the literature. All of this assumes a strong correlation between RT-costs and error-costs from the same psychological effect. We conducted a meta-analysis to test this, with 114 effects across a range of well-known tasks. Counterintuitively, we found a general pattern of weak, and often no, association between RT and error costs (mean r = .17, range −.45 to .78). This general problem is accounted for by the theoretical framework of evidence accumulation models, which capture individual differences in (at least) 2 distinct ways. Differences affecting accumulation rate produce positive correlation. But this is cancelled out if individuals also differ in response threshold, which produces negative correlations. In the models, subtractions between conditions do not isolate processing costs from caution. To demonstrate the explanatory power of synthesizing the traditional subtraction method within a broader decision model framework, we confirm 2 predictions with new data. Thus, using error costs or RT costs is more than a pragmatic choice; the decision carries theoretical consequence that can be understood through the accumulation model framework.

Confidence and varieties of bias

We test the proposition that response bias can have two different bases; reflecting either differing beliefs about the a priori likelihood of competing response alternatives, or their relative utilities. In evidence accumulation models, these two types of bias are thought to manifest as variations in the starting point for accumulation and threshold for responding, respectively. Although these two mechanisms are indistinguishable for linear accumulators in terms of accuracy and RT, Vickers’ (1979) balance-of-evidence hypothesis predicts they have dissociable effects on confidence. We derived ten ordinal predictions from these models and confirmed them at the level of group averages using traditional ANOVA analyses of results from a new experiment that manipulated the probability of correct responses and the rewards associated with them. However, individual effects were more variable, particularly with respect to the reward manipulation. We then used Bamber’s (1979) state–trace analysis to test the predicted dissociations using Bayes factors developed by Prince, Brown and Heathcote (2012) and Davis-Stober, Morey and Heathcote (2016). Once again, we found support at the aggregate level but more equivocal results for individuals. We discuss why individual results are to be preferred over aggregate results in state–trace analysis and draw the lesson that tailored designs are needed to obtain clear results from individual state–trace analyses.

Formation of abstract task representations: Exploring dosage and mechanisms of working memory training effects

Working memory is strongly involved in human reasoning, abstract thinking and decision making. Past studies have shown that working memory training generalizes to untrained working memory tasks with similar structure (near-transfer effect). Here, we focused on two questions: First, we ask how much training might be required in order to find a reliable near-transfer effect? Second, we ask which choice- mechanism might underlie training benefits? Participants were allocated to one of three groups: working-memory training (combined set-shifting and N-back task), active-control (visual search) and no-contact control. During pre/post testing, all participants completed tests tapping procedural and declarative working memory as well as reasoning. We found improved performance only in the procedural working-memory transfer tasks, a transfer task that shared a similar structure to that of the training task. Intermediate testing throughout the training period suggest that this effect emerged as soon as after 2 training sessions. We applied evidence accumulation modeling to investigate the choice process responsible for this near-transfer effect and found that trained participants, compared with active-controls had quicker retrieval of the action rules, and more efficient classification of the target. We conclude that participants were able to form abstract representations of the task procedure (i.e., stimulus-response rules) that was then ~applied to novel stimuli and responses.

Dynamic models of choice.

Parameter estimation in evidence-accumulation models of choice response times is demanding of both the data and the user. We outline how to fit evidence-accumulation models using the flexible, open-source, R-based Dynamic Models of Choice (DMC) software. DMC provides a hands-on introduction to the Bayesian implementation of two popular evidence-accumulation models: the diffusion decision model (DDM) and the linear ballistic accumulator (LBA). It enables individual and hierarchical estimation, as well as assessment of the quality of a model's parameter estimates and descriptive accuracy. First, we introduce the basic concepts of Bayesian parameter estimation, guiding the reader through a simple DDM analysis. We then illustrate the challenges of fitting evidence-accumulation models using a set of LBA analyses. We emphasize best practices in modeling and discuss the importance of parameter- and model-recovery simulations, exploring the strengths and weaknesses of models in different experimental designs and parameter regions. We also demonstrate how DMC can be used to model complex cognitive processes, using as an example a race model of the stop-signal paradigm, which is used to measure inhibitory ability. We illustrate the flexibility of DMC by extending this model to account for mixtures of cognitive processes resulting from attention failures. We then guide the reader through the practical details of a Bayesian hierarchical analysis, from specifying priors to obtaining posterior distributions that encapsulate what has been learned from the data. Finally, we illustrate how the Bayesian approach leads to a quantitatively cumulative science, showing how to use posterior distributions to specify priors that can be used to inform the analysis of future experiments.

A drift diffusion model account of the semantic congruity effect in a classification paradigm

The semantic congruity effect refers to the facilitation of judgements (i) when the direction of the comparison of two items coincides with the relative position of the items along the dimension comparison or (ii) when the relative size of a standard and a target stimulus coincides. For example, people are faster in judging 'which is bigger?' for two large items, than judging 'which is smaller?' for two large items (selection paradigm). Also, people are faster in judging a target stimulus as smaller when compared to a small standard, than when compared to a large standard, and vice versa (classification paradigm). We use the Drift Diffusion Model (DDM) to explain the time course of a semantic congruity effect in a classification paradigm. Formal modelling of semantic congruity allows the time course of the decision process to be described, using an established model of decision making. Moreover, although there have been attempts to explain the semantic congruity effect within evidence accumulation models, two possible accounts for the congruity effect have been proposed but their specific predictions have not been compared directly, using a model that could quantitatively account for both; a shift in the starting point of evidence accumulation or a change in the rate at which evidence is accumulated. With our computational investigation we provide evidence for the latter, while controlling for other possible explanations such as a variation in non-decision time or boundary separation, that have not been taken into account in the explanation of this phenomenon.

Visual Hallucinations Are Characterized by Impaired Sensory Evidence Accumulation: Insights From Hierarchical Drift Diffusion Modeling in Parkinson’s Disease

BackgroundModels of hallucinations emphasize imbalance between sensory input and top-down influences over perception, as false perceptual inference can arise when top-down predictions are afforded too much precision (certainty) relative to sensory evidence. Visual hallucinations in Parkinson’s disease (PD) are associated with lower-level visual and attentional impairments, accompanied by overactivity in higher-order association brain networks. PD therefore provides an attractive framework to explore contributions of bottom-up versus top-down disturbances in hallucinations. MethodsWe characterized sensory processing during perceptual decision making in patients with PD with (n = 20) and without (n = 25) visual hallucinations and control subjects (n = 12), by fitting a hierarchical drift diffusion model to an attentional task. The hierarchical drift diffusion model uses Bayesian estimates to decompose task performance into parameters reflecting drift rates of evidence accumulation, decision thresholds, and nondecision time. ResultsWe observed slower drift rates in patients with hallucinations, which were less sensitive to changes in task demand. In contrast, wider decision boundaries and shorter nondecision times relative to control subjects were found in patients with PD regardless of hallucinator status. Inefficient and less flexible sensory evidence accumulation emerges as a unique feature of PD hallucinators. ConclusionsWe integrate these results with evidence accumulation and predictive coding models of hallucinations, suggesting that in PD sensory evidence is less informative and may therefore be down-weighted, resulting in overreliance on top-down influences. Considering impaired drift rates as an approximation of reduced sensory precision, our findings provide a novel computational framework to specify impairments in sensory processing that contribute to development of visual hallucinations.

Bayes factors for the linear ballistic accumulator model of decision-making.

Evidence accumulation models of decision-making have led to advances in several different areas of psychology. These models provide a way to integrate response time and accuracy data, and to describe performance in terms of latent cognitive processes. Testing important psychological hypotheses using cognitive models requires a method to make inferences about different versions of the models which assume different parameters to cause observed effects. The task of model-based inference using noisy data is difficult, and has proven especially problematic with current model selection methods based on parameter estimation. We provide a method for computing Bayes factors through Monte-Carlo integration for the linear ballistic accumulator (LBA; Brown and Heathcote, 2008), a widely used evidence accumulation model. Bayes factors are used frequently for inference with simpler statistical models, and they do not require parameter estimation. In order to overcome the computational burden of estimating Bayes factors via brute force integration, we exploit general purpose graphical processing units; we provide free code for this. This approach allows estimation of Bayes factors via Monte-Carlo integration within a practical time frame. We demonstrate the method using both simulated and real data. We investigate the stability of the Monte-Carlo approximation, and the LBA's inferential properties, in simulation studies.

Examining procedural working memory processing in obsessive-compulsive disorder

Previous research has suggested that a deficit in working memory might underlie the difficulty of obsessive-compulsive disorder (OCD) patients to control their thoughts and actions. However, a recent meta-analyses found only small effect sizes for working memory deficits in OCD. Recently, a distinction has been made between declarative and procedural working memory. Working memory in OCD was tested mostly using declarative measurements. However, OCD symptoms typically concerns actions, making procedural working-memory more relevant. Here, we tested the operation of procedural working memory in OCD. Participants with OCD and healthy controls performed a battery of choice reaction tasks under high and low procedural working memory demands. Reaction-times (RT) were estimated using ex-Gaussian distribution fitting, revealing no group differences in the size of the RT distribution tail (i.e., τ parameter), known to be sensitive to procedural working memory manipulations. Group differences, unrelated to working memory manipulations, were found in the leading-edge of the RT distribution and analyzed using a two-stage evidence accumulation model. Modeling results suggested that perceptual difficulties might underlie the current group differences. In conclusion, our results suggest that procedural working-memory processing is most likely intact in OCD, and raise a novel, yet untested assumption regarding perceptual deficits in OCD.

An evidence accumulation model of acoustic cue weighting in vowel perception

Listeners rely on multiple acoustic cues to recognize any phoneme. The relative contribution of these cues to listeners׳ perception is typically inferred from listeners׳ categorization of sounds in a two-alternative forced-choice task. Here we advocate the use of an evidence accumulation model to analyze categorization as well as response time data from such cue weighting paradigms in terms of the processes that underlie the listeners׳ categorization. We tested 30 Dutch listeners on their categorization of speech sounds that varied between typical /ɑ/ and /aː/ in vowel quality (F1 and F2) and duration. Using the linear ballistic accumulator model, we found that the changes in spectral quality and duration lead to changes in the speed of information processing, and the effects were larger for spectral quality. In addition, for stimuli with atypical spectral information, listeners accumulate evidence faster for /ɑ/ compared to /aː/. Finally, longer durations of sounds did not produce longer estimates of perceptual encoding time. Our results demonstrate the utility of evidence accumulation models for learning about the latent processes that underlie phoneme categorization. The implications for current theory in speech perception as well as future directions for evidence accumulation models are discussed.

The influence of evidence volatility on choice, reaction time and confidence in a perceptual decision

Many decisions are thought to arise via the accumulation of noisy evidence to a threshold or bound. In perception, the mechanism explains the effect of stimulus strength, characterized by signal-to-noise ratio, on decision speed, accuracy and confidence. It also makes intriguing predictions about the noise itself. An increase in noise should lead to faster decisions, reduced accuracy and, paradoxically, higher confidence. To test these predictions, we introduce a novel sensory manipulation that mimics the addition of unbiased noise to motion-selective regions of visual cortex, which we verified with neuronal recordings from macaque areas MT/MST. For both humans and monkeys, increasing the noise induced faster decisions and greater confidence over a range of stimuli for which accuracy was minimally impaired. The magnitude of the effects was in agreement with predictions of a bounded evidence accumulation model.

Implications of Visual Attention Phenomena for Models of Preferential Choice.

We use computational modeling to examine the ability of evidence accumulation models to produce the reaction time (RT) distributions and attentional biases found in behavioral and eye-tracking research. We focus on simulating RTs and attention in binary choice with particular emphasis on whether different models can predict the late onset bias (LOB), commonly found in eye movements during choice (sometimes called the gaze cascade). The first finding is that this bias is predicted by models even when attention is entirely random and independent of the choice process. This shows that the LOB is not evidence of a feedback loop between evidence accumulation and attention. Second, we examine models with a relative evidence decision rule and an absolute evidence rule. In the relative models a decision is made once the difference in evidence accumulated for 2 items reaches a threshold. In the absolute models, a decision is made once 1 item accumulates a certain amount of evidence, independently of how much is accumulated for a competitor. Our core result is simple—the existence of the late onset gaze bias to the option ultimately chosen, together with a positively skewed RT distribution means that the stopping rule must be relative not absolute. A large scale grid search of parameter space shows that absolute threshold models struggle to predict these phenomena even when incorporating evidence decay and assumptions of either mutual inhibition or feedforward inhibition.

Neural Evidence for a Role of Urgency in the Speed-Accuracy Trade-off in Perceptual Decision-Making.

In the study of perceptual decision-making, the dominant theory holds that subjects accumulate evidence over time until a threshold level is reached and a response is executed ([Gold and Shadlen, 2007][1]; [Forstmann et al., 2016][2]). Formal evidence accumulation models have been exceptionally

Strength and weight: The determinants of choice and confidence

Evidence for different hypotheses is often treated as a singular construct, but it can be dissociated into two parts: its strength, the proportion of pieces of information favoring one hypothesis; and its weight, the total number of pieces of information available. However, cognitive and neural models of evidence accumulation often make a proportional representation assumption, implying that people take these two factors into account equally when making their decisions and judgments. We examine this assumption by directly manipulating the number of samples and the proportion favoring either of two alternatives in dynamic decision making and judgment tasks. The results suggest that people tend to over-emphasize the strength of evidence relative to its weight in both an optional-stopping decision task and a probability judgment task. In a drift-diffusion model, this is reflected by drift rates that are determined foremost by strength with a smaller influence of weight. This result challenges the proportional representation assumption made by existing models of judgment and decision-making, and calls into question modeling evidence accumulation as a Bayesian belief updating process.