Grid search provides the roughest characterization of the model fit across the parameter space, as shown in Fig. As the samples are spread out evenly all across the parameter space, the optimum is not identified very accurately. Furthermore, as no statistical model is used to interpolate the surface and filter out noise, it is not easy to accurately visualize the shape of the surface.
In comparison to the grid visualization, the GP regression model constructed during BO is able to provide a better characterization of the model surface with fewer resources, as shown in Fig. As the sample locations have been optimized by BO, the optimum of the function is estimated with significantly higher precision. Furthermore, the general shape of the model fit function is much easier to interpret from the visualizations, thanks to the statistical interpolation.
However, as the optimum is relatively flat in this case, it is still challenging to identify which precise regions of the parameter space are the most likely. The overall resolution of the visualization is the same as for the GP model, but the posterior better visualizes the parameter regions that both lead to a good model fit and are probable given our prior understanding of reasonable parameter values. Thus, for example, although different values of the ANS parameter seem to lead to equally good predictions, the posterior shows that based on our prior understanding larger values of ANS should be more likely.
The difference to ground truth is visualized in Fig. This improvement is most notable in learning phase 1. With the manually tuned parameter values, the durations of the solving stages with heights 4 and 5 were over 1 standard deviation away from the observed means, visible in the top left panel of Fig. By automatically tuning the parameter values, these durations are visibly closer to the observation data, as shown in the bottom left panel.
For height 3, the predictions are not yet close to the observation data. However, through extensive searching, it is certain that the model is unable to replicate this behavior just by tuning these four parameters within sensible limits. Thus, if one was to continue development of this model, there is now clear evidence of a particular type of behavior that is not reproduced by this model.
Probabilities and Optimal Inference to Understand the Brain - olagynulehyb.gq
This would allow one to focus their efforts better, as it is clear that the issue cannot be remedied simply by further tuning of these four parameters. However, as there is significant variability in the observation data as well at this point, the predictions are still credible. Computational rationality is a framework for cognitive modeling that is based on the idea that cognitive behaviors are generated by behavioral policies 9 9 By behavioral policy, we mean a decision rule which chooses what cognitive action is to be executed in each possible cognitive state.
A key problem in computational rationality is finding the optimal behavioral policy. The MDP is often designed in such a way that these parameters correspond to interesting psychometric quantities, such as the level of motivation or alertness of the subject. An interesting property of computational rationality, that is, pertinent to the current paper, is that it effectively reduces the number of free parameters in the model. Although in other modeling frameworks the parameters of the behavioral policy can be adjusted freely, computational rationality demands that these parameter values are derived through optimization, given the limits imposed by the cognitive architecture.
In models based on computational rationality, the free parameters generally relate to these cognitive limits only. Thus, the number of free parameters that remain to be inferred is often much smaller compared to the number of parameters needed to fully define the behavioral policy. While the parameters of the behavioral policy are derived through optimization and are, therefore, not fitted to data, the parameters that define the limitations of the cognitive architecture remain to be inferred. However, a key challenge with this inference process is the fact that evaluating the model fit using new parameter values takes significant time, as the parameters of the behavioral policy need to be solved before predictions can be made.
Likely for this reason, the majority of parameters in existing models have been set manually. We use here a slightly simplified version of the model, in order to somewhat reduce the time required for solving the optimal policy. This allowed us to run studies with more model evaluations for better demonstrating the convergence properties of the algorithms. The model structure is as follows. The model contains a menu composed of eight items. The agent has multiple possible actions at each step. She can either fixate on any of the eight items or declare that the item is not present in the menu i.
If the agent fixates on the target item, it is automatically selected. In the original model, the agent had to manually choose to select the item, which now happens automatically as it is clearly the optimal option at that point. Fixating on the target item or quitting ends the episode. The cognitive state s consists of the semantic relevances and lengths of the observed menu items. In the original model, the state also included the location of the previous fixation, but this was determined to have little effect on the policy and was thus left out. The agent receives a reward after each action.
If the agent found the target item, or quit when the target was absent from the menu, a large reward is given. If the agent quits when the target is present, an equally large penalty is given. Otherwise, the agent receives a small penalty, which is equal to the time spent for the action sum of the durations of the saccade and the fixation. After choosing values for the free model parameters, it takes roughly two hours to estimate a policy which is close to optimal. For example, the parameters that determine the duration of saccades were set based on a study by Baloh, Sills, Kumley, and Honrubia , and the duration of eye fixations was set based on a study by Brumby, Cox, Chung, and Fernandes The sensitivity of the model predictions to variation in parameter values was not reported.
The inference used a dataset collected by Bailly, Oulasvirta, Brumby, and Howes Our study extends their analysis in multiple ways. First, the full posterior distribution of the model is estimated, instead of just the maximum of the posterior. This provides a rigorous characterization of the remaining uncertainty in the parameter values, which was not discussed in the earlier study. Second, the mean of the posterior is estimated, which is often a more robust point estimate compared to the maximum.
Third, the efficiency of the method is rigorously compared to alternative methods, which was not done previously. These parameters were selected as they were judged to have the largest effect to the predicted behavior. Further, it would be very challenging to estimate the selection delay or recall probability based only on earlier literature, as they may be largely affected by the precise setup used to collect the data.
In addition to these parameters, the probability of observing the semantic similarity of neighboring items with peripheral vision was inferred to be 0. In this study, we use this constant value for this parameter as we assumed it would have the smallest effect on the performance, and as the model is expensive to evaluate, using fewer parameters allowed us to run more a extensive comparison study. The data from the first group user IDs 4, 18, 19, 21, 23, 37, 38, 39, 40, and 42 were used only for parameter inference, while the data from the second group user IDs 5, 6, 7, 8, 20, 22, 24, 36, and 41 were used only for estimating the prediction error.
Although Nelder—Mead is not a parallelizable method, it is possible to run multiple instances of the optimization algorithm in parallel, and select the best overall result. In the simulations, for each datapoint, we sampled five experiments without replacement from the corresponding 10 independent experiments and selected the parameter location with smallest error on training data.
Comparing the results, we observe that here is more overall disagreement regarding the optimal parameters between the methods than in Example 1. One contributing factor is likely the fact that none of the methods has properly converged within the available computation time caused by the expensive model evaluations.
This means that there is likely considerable remaining uncertainty about the location of the best possible parameter values with all of these methods. Out of the compared methods, ABC has the most visually intuitive quantification of this uncertainty Fig. This is demonstrated by the fact that prediction error starts increasing after a certain amount of optimization has been performed.
This leads to the method making overly optimistic assumptions about the model fit, based on chance occurrences where the model fit happened to be lower than the average fit achieved with those parameter values. In this case, as the prior is more restrictive, the method is able to achieve good model fit even at the beginning of the inference process.
As the method optimizes a balance between model fit and credibility of parameter values, the final model fit is higher compared to BO or Nelder—Mead. The ability of the methods to quantify model fit in different parts of the parameter space is visualized in Figs. As Nelder—Mead only provides a point estimate, no such visualization is possible with it. And again, the ABC posterior, shown in Fig. In this example, we notice a clear difference between the optima of the BO model fit surface and the ABC posterior.
This is explained by the more restrictive prior distribution used. In this case, the feature that is most restricted by the prior is the fixation duration. We omitted users that had a low number of observations from either menu condition. We were left with seven subjects subject IDs 5, 18, 19, 24, 37, 39, and We repeated the above inference procedure for the data collected of each of these subjects individually, with the restriction of model evaluations in batches of The estimated posterior distributions are visualized in Fig. We observe that most posteriors are similar in nature as was the population level posterior estimated before, shown in Fig.
Especially, subjects 18, 19, and 37 have posteriors that are very similar to the population mean, indicating that a model fit with population level data would be a good approximation for these individuals. However, we also observe clear individual variation in the posteriors, which indicates that the model offers different explanations for the behavior of each individual subject. For example, subjects 5 and 39 had a relatively low selection delay, but in contrast, a slightly longer fixation duration compared to other subjects.
We are also able to identify anomalous subjects, such as subject 24, for whom a very long selection delay was inferred, and on the other hand a very short fixation duration, and subject 40, for whom the posterior appears to be more complex and have a heavy tail, which places the posterior mean further away from the posterior maximum.
By examining the behavior of such anomalous users more carefully, it would be possible to either spot oddities in data collection procedures, identify completely new types of user strategies, or point out behaviors that the model is unable to reproduce. Computational cognitive models generally seek to explain aspects of human cognition.
However, arguably, the quality of these explanations has too often been undermined by misgivings regarding the parameter inference process. Many models have been published with parameter values that are difficult to justify; sometimes because the inference method is ad hoc or not reported, sometimes because no alternative parameter values have even been considered. Another reason might be that while a lot of progress has happened in computational statistics in terms of readily applicable inference methods, these have not yet been fully discovered in mainstream cognitive scientific computational modeling.
To remedy this situation, we reported an exploration of how principled and rigorous parameter inference can be performed for some of the most complex computational cognitive model families. One major benefit is that estimates of parameter values, along with their uncertainty, can be inferred efficiently for various types of computational cognitive models.
We note that neither of these models has a tractable likelihood function, which renders many traditional inference methods, such as gradient descent, infeasible. Our results also confirm the common observation that automated parameter estimation methods improve model fit over manual fitting.
We also note that the use of automated methods in general insists on explication of model fit functions, search methods, and search spaces, subjecting them to transparency and opening up the potential for scrutiny by the community. In order to get estimates quickly, it is not possible to estimate the model fit over the entire parameter space at the same time, and vice versa.
We give two suggestions for selecting inference methods, depending on the situation. While they are based on the two case studies presented herein, they are in line with previous applications. Example use cases include initial hypothesis testing and early model development. If the goal is to obtain robust parameter estimates accompanied by estimates of the sensitivity of parameters, we suggest that methods based on efficient global optimization should be used. These methods are able to estimate model fit across the entire parameter space, while also facilitating the search of optimal values.
Based on our experiments, we observed that ABC is an efficient and informative inference method. The method also allows prior information of reasonable parameter values to be taken into account in a principled way. There are multiple reasons why it is important to estimate the posterior distribution of parameters over just the point estimates.
The posterior probability distribution over the parameter values is a rigorously defined quantitative measure of our knowledge about the true parameter values. It grants proper, quantified estimates of uncertainties associated with parameter values, which is inherently valuable for understanding the models and the behavior that they describe. The posterior distribution is also a valuable diagnostic tool in modeling.
For example, the shape of the posterior can be informative of insignificant or poorly identified parameters. If the posterior of a certain parameter is flat, this means that either this parameter has no effect on the model predictions, or that there are insufficient observation data to infer the value of this parameter.
The posterior shape can also inform alternative explanations; if there are multiple modes in the posterior distribution, this indicates the existence of multiple alternative explanations to the data. Finally, accounting for the stochasticity of the predictions allows comparing their variance to the data using specific parameter values.
A complete check of model fit also takes the uncertainty from model fitting into account when estimating the quality of the predictions. Therefore, parameter inference plays a more decisive role in scientific modeling than just the determination of reasonable parameter values. It is via parameter inference that theories and models gain contact with reality, quantified by the observation data. In general, a good explanatory model should be such that given observation data, the model explicitly informs us about what we can and cannot tell about the unobserved quantities of the cognitive system based on the data, and how reliable these estimates are.
However, being able to access such information is only possible through principled parameter inference methods, such as those based on Bayesian statistics. While various point estimation methods, like Nelder—Mead, may quickly find parameters that allow replicating the observation data, probabilistic methods that consider the parameter space as a whole, like ABC, allow answering the above questions more robustly.
In conclusion, modern solutions to the parameter inference problem have the potential to transform the rigor, transparency, and efficiency of computational cognitive modeling. Recent statistical inference methods, such as BO and ABC, can be used for inferring the parameter values for some of the most complex simulation models developed in the field of cognitive science.
As argued here, these methods have important advantages compared to the traditional methods.
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In the future, we hope that these inference methods make it feasible for researchers in the field to work on even more ambitious computational cognitive models. Volume 43 , Issue 6. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account.
If the address matches an existing account you will receive an email with instructions to retrieve your username. Cognitive Science Volume 43, Issue 6. Extended Article Open Access. Jussi P. Jokinen Corresponding Author E-mail address: jussi. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract This paper addresses a common challenge with computational cognitive models: identifying parameter values that are both theoretically plausible and generate predictions that match well with empirical data.
First, assume a function for computing the model fit, which in ABC in called the discrepancy function. Figure 1 Open in figure viewer PowerPoint. The shaded region indicates the area between the 5th and 95th percentiles.
Left: Overall behavior. Right: Detail of lower left corner. Each point is estimated using 40 independent experiments. Figure 2 Open in figure viewer PowerPoint. Linear interpolation and constant extrapolation is used between sampled values. The color map is such that black is 3. Contours are superimposed for additional clarity. This model assumes that the sensory cortex infers the most likely values of attributes or features of sensory stimuli from the noisy inputs encoding the stimuli.
Remarkably, the model describes how this inference could be implemented in a network of very simple computational elements, suggesting that this inference could be performed by biological networks of neurons.
Furthermore, learning about the parameters describing the features and their uncertainty is implemented in these networks by simple rules of synaptic plasticity based on Hebbian learning. The talk will also discuss how this model could be used to capture behavioural data. Principles and psychophysics of Active Inference in anticipating a dynamic probabilistic bias. The brain has to constantly adapt to changes in the environment, for instance when a contextual probabilistic variable switched its state.
For an agent interacting with such an environment, it is important to respond to such switches with the shortest delay. However, this operation has in general to be done with noisy sensory inputs and solely based on the information available at the present time. Experimental results were compared to those of a probabilistic agent optimized with respect to this switching model. We found a good fit of the behaviorally observed anticipatory response compared with other models such as the leaky-integrator model.
Moreover, we could also fit the level of confidence given by human observers with that provided by the model. Such results provide evidence that human observers may efficiently represent an anticipatory belief along with its precision and they support a novel approach to more generically test human cognitive abilities in uncertain and dynamic environments.
The work in the Basso laboratory is aimed at understanding how mechanisms of brain function give rise to higher mental experience and cognition. The primary focus in the lab is on understanding the role of basal ganglia and superior colliculus circuits in perceptual decision making and in the use of memory to guide decisions when sensory information is uncertain.
Measuring the sensitivity of visual confidence. Visual confidence refers to our ability to predict the correctness of our perceptual decisions. Knowing the limits of this ability, both in terms of biases e. The measurement of visual confidence with the classical method of confidence ratings presents both advantages and disadvantages. In recent years, we have explored an alternative paradigm based on confidence forced-choice. In this paradigm, observers have to choose which of two perceptual decisions is more likely to be correct.
I will review some behavioural results obtained with the confidence forced-choice paradigm. I will also present two ideal observers based on signal detection theory, one that uses the same information for perceptual and confidence decisions, and another one that has access to additional information for confidence.
These ideal observers help us quantify the limitations of human confidence estimation. Using Bayesian models to investigate attentional mechanisms in the human brain. The deployment of attention rests on predictions about the likelihood of events and there is now accumulating evidence that the generation of such predictions can plausibly be described by Bayesian computational models.
These models can be regarded as variants of predictive coding and provide a principled prescription of how observers update their predictions after new observations. Behavioural computational modelling results as well as data from neuroimaging and neuromodulation experiments will be presented to elucidate the brain mechanisms underlying the flexible control of attention by inferred predictability. Additionally, I will discuss the advantages and the limitations of the modelling approach applied in this work.
Optimizing scene decoding with "three-party" generative modelS. The active inference framework Friston, ; Friston et al, models sequential scene uncovered through movement. Stemming from the auto-encoding theory Hinton, , it introduces a new perspective for it formally links dictionary construction from data with optimal motor control. In particular, motor control is here considered as a particular implementation of a predictive process that actively participates in estimating a complex posterior distribution.
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Dynamical activity patterns in the macaque posterior parietal cortex during path integration. Neural circuits evolved to deal with the complex demands of a dynamic and uncertain world. To understand dynamic neural processing underlying natural behaviour, we use a continuous-time foraging task in which humans and macaques use a joystick to steer and catch flashing fireflies in a virtual environment. We introduce a probabilistic framework to refute a popular account of path integration that attributes biases to forgetful integration.
We instead find that such biases are explained naturally by an optimal strategy that maximizes rewards while accounting for prior expectations about our own movements. We use multi-electrode array and laminar probes to sample the activity of a large number of neurons in the posterior parietal cortex and find that different neurons are active during different epochs of integration. Neurons exhibit rich temporal diversity such that the integration dynamics appear embedded in the dynamical pattern of population activity. We are currently applying statistical techniques to characterise the precise dynamics of population activity to understand the associated neural computations.
Adaptive coding in the dopaminergic system in health and disease. Recent theories have construed the brain as performing a specific form of hierarchical Bayesian inference, known as predictive coding. In these models, the brain predicts upcoming information by weighting violations in its expectations prediction errors relative to their precision reliability ; a process termed adaptive coding.
Although dopamine is hypothesised to play a key role in the adaptive coding of cortical unsigned absolute prediction errors, no experimental data has addressed this hypothesis in humans. We used dopaminergic pharmacological manipulations in conjunction with an associative learning fMRI task that required adaptive coding. A computational model that included precision-weighting of prediction errors, provided the best fit to participants behaviour. At the level of the brain, unsigned prediction errors were adaptively coded relative to their precision in the superior frontal cortex.
Decreases in neural adaptation significantly correlated with decreased task performance. A dopamine antagonist significantly attenuated adaptive coding, in line with predictive coding hypotheses. These findings are likely to have important implications for understanding altered behaviour in individuals with dopamine-perturbed states such as psychosis. Indeed, in a separate dataset we observed that decreases in adaptive coding were associated with an increase in positive psychotic symptoms. Learning the payoffs and costs of actions. To select the most appropriate behaviour, the brain circuits need to learn about the consequences of different actions.
Much evidence suggests that such learning takes place in a set of subcortical nuclei called the basal ganglia. The basal ganglia circuit is organized in two main pathways connected with initiation and inhibition of movements respectively. It has been proposed that the neurons in these two pathways separately learn about payoffs and costs of actions, which are then differentially weighted during decision making depending on the motivation state.
However, it has not been shown what plasticity rules would allow the basal ganglia neurons to learn about payoffs and costs of actions. This talk will show that the learning rules, which have been previously proposed to learn reward uncertainty in addition to mean reward, also allow estimating payoffs and costs associated with different actions.
The resulting model accounts for diverse experimental data ranging from properties of dopaminergic receptors to the effects of medications on behaviour. Brain circuits of urgent decisions for action. Animals, including humans, constantly interact with a dynamic and unpredictable environment through successions of decisions and actions. Where in the brain decisions between actions are determined? What is the computational mechanism that transforms relevant information into action? Last, an overview on probability weighting by humans is given, the role of which in probabilistic reasoning is investigated in this work.
When conducting interdisciplinary research, the employed methods may not be common knowledge in all involved fields. This chapter serves to make this work accessible to a wide audience by describing in detail the methods used formodel estimation and selection. This chapter reviews state-of-the-art observer models of the P event-related potential and introduces a new digital filtering DIF model.
It starts with a brief overview of the models known from literature and of the approach proposed in this work. Next, the parameter optimization schemes as well as the composition of the design matrices for model estimation and selection see Chap. Results and conclusions complete this chapter. The Bayesian observer model adjusts internal beliefs about hidden states in the environment and predictions about observable events.
The scope of the analyzed data is extended to the complete late positive complex P3a, P3b, Slow Wave and the N