Wigner Research Centre for Physics,
Hungarian Academy of Sciences
Computational Cognitive Science
The general aim of computational cognitive science is to understand human cognition in engineering terms . How can we formally describe what is learned by the human brain? Can we provide a mathematical formalisation of knowledge? In this sense, understanding the human brain means that we can replicate its computations, creating a machine with human-like cognitive abilities, at least in some aspects.
In machine learning, recently there has been great progress in achieving human-level performance in various tasks, for example in object recognition or the game of Go . However, humans utilize data at hand much more efficiently than current machine learning approaches. Children as young as three years old can readily learn names of categories (e.g. horses) and generalise these concepts to new objects from very few examples . It is this key observation that renders humans an essential subject for artificial intelligence research as well (at least for now).
Perception and Learning
The world around us is inherently ambiguous. That is due to the fact that the information reaching our sensors is never satisfactory to logically deduce the true state of the world (Figure 2). For example different objects may look the same from certain angles or you may not know about objects that are occluded. Unsurprisingly, even the number of models that could describe our world is infinite.
How can we then select a model if there are infinitely many? A key to how humans could still learn meaningful models of the world is that they look for parsimonious explanations. Therefore, in order to build machines that can learn like humans, we need to find out what is a good measure for model complexity that enables efficient learning in this universe that we live in (in which we have the physical laws that we do).
Inference, that is guessing the likely causes of our sensory inputs can be formalised in probability theory, in particular, Bayesian inference. The basic idea is that, as cognitive beings, we weigh our hypotheses by how probable our observations are conditioning on (assuming) that hypothesis. Human perception has been well characterised using ideal observer models that integrate incoming information with previously acquired knowledge according to the rules of Bayesian inference [5,6]. In this framework, learned structure and regularities about the surrounding world, internal representations used to make inferences about the world around us are formalised as probabilistic generative models. You can think of generative models like physics and graphics engines in a video game. They provide a (possibly stochastic) algorithm of how your observations (video game frames) are generated from the underlying causes (scenes, objects, object interactions etc). Inference arises from inverting these forward models.
It has been demonstrated that these internal representations of individuals can be uncovered from behavioural data using newly developed machine learning techniques .
The work presented here was conducted in the Computational Systems Neuroscience Lab at the Wigner Research Centre for Physics, an institute part of the Hungarian Academy of Sciences in Budapest, Hungary. At the lab we focus on understanding neural computations from a normative learning perspective. We test these theories by analysing behavioural and electrophysiological measurements. Our work is a joint project with the Brain, Memory and Language Lab at Eötvös Loránd University. I am currently a PhD student enrolled in the Psychology Doctoral School in the Cognitive Science Department at the Budapest University of Technology and Economics under the supervision of Gergő Orbán and Dezső Németh.
History and context of the research
Sequential predictions are ubiquitous in a learning agent’s existence. In order to devise efficient responses in a dynamic environment, one needs to build an internal representation of the latent dynamics of the environment. Humans have been shown to create dynamical models such as intuitive physics  that approximate the laws of Newtonian physics and are able to reason about their model in terms of formulating new predictions or imagining hypothetical situations (e.g. what would happen to objects on a table if I kicked the table from a certain angle?). However, the structure of their internal model, namely Newtonian physics is assumed. In another study, humans are demonstrated to make probabilistic inferences about noisy stimuli using a two-state dynamical model in a simple task . It remains elusive how a learning agent can acquire such complex knowledge about the dynamics of the environment.
Research goals, open questions
Subject-by-subject differences in temporal predictions resulting from variations in subjective internal models and individual learning paths have remained unexplored due to the immense difficulty related to inferring subjective dynamical representations. Cognitive Tomography  has been proposed to discover static internal representations from discrete choices. We extend this method in two critical ways: 1, We aim to infer internal representations from a richer set of behavioral measures, specifically we use response times; 2, Our goal is to infer a dynamical representation. We demonstrate its utility by predicting response times and choices of human participants in a probabilistic learning task on a trial-by-trial basis. Inferred behaviour-based trial-specific subjective predictions can be directly used to test theories of neural underpinnings of computations in physiological and imaging data. A peer-reviewed 4-page conference abstract on this project submitted to the Cognitive Computational Neuroscience Conference is available here.
Sequential prediction in general
In order to study the internal models entertained by human subjects, we first need to look at how we can solve sequential prediction in general. The question is, what is the general form of a model, with which we can predict upcoming observation given our previous observations from the sequence? The key is that we may assume that the system has a latent (not directly observable) state that captures all the effects of past events. Just like in the case of Newtonian physics: when judging the trajectory of a projectile, if we know its position and impulse in any given moment, we can predict its movement (using our intuitive physics model) without requiring its past trajectory. In statistical terms, the history becomes stochastically independent from the future given the current state. Interestingly, this idea generalises to any sequence with temporal dependencies. We need to learn the structure over these latent states and how our observations depend on the latent state.
Ideal Observer Model and Cognitive Tomography
The participant assumes a model for the latent dynamics and a model relating their observations to the latent states. They use these components to update their beliefs over the current state of the observed system and then generate predictions for the upcoming stimulus. The median and standard deviation of response times decreases linearly with log subjective probability . Cognitive Tomography is the method of inverting this generative model. We inferred the internal representations of individuals from the stimulus sequences and the response times.
We tested our model in a probabilistic sequence learning task. In each trial, the face of a dog appeared
in one of four possible positions. Participants had to manually respond to where the dog appeared using their
middle and index fingers. We conducted 25 blocks of 80 trials each with all participants and recorded their response times.
Figure 4. Experimental design. Example stimulus sequence. Participants formulate predictions about the upcoming stimulus. Response times are stochastic but they depend on the participants' predictions. The more the participant expects the stimulus, the smaller the response time.
Since we defined the complete generative model of response times, we can create synthetic data (Figure 5, left) to test whether our method can recover the true internal model. The great advantage here, in contrast to human data, is that we know the ground truth and can check whether our method works correctly. We are also able to explore how well we can recover the model depending on the noisiness and other response time parameters of the participant. Perfect recovery would mean the inferred log predicted probability and the ground truth are equal, i.e. on the middle panel of Figure 5 all points fall on the x=y line. However, a high correlation between the recovered predictions and the ground truth predicted probabilities means the internal model inferred is mainly correct but the inferred response time parameters are at the wrong scale. Importantly, even if the internal model is correctly recovered, response times cannot be predicted perfectly due to their inherent noisiness (Figure 5, right).
Figure 5. (Interactive.) Results on synthetic data for three different internal models and different response time parameters. Recovery of the ground truth model depends on the parameters of the response time model. Sensitivity: the measure of how sensitive response times are to the predictions of the internal model. Baseline RT: a parameter setting the baseline response time. Noise: noisiness of the response times. Increasing noise and decreasing sensitivity prohibits accurate recovery of the ground truth internal model. You can see that by increasing noise, the correlation in the middle panel drops.
Predicting response times
First, we contrasted our model's (iHMM, see ) performance with that of a classical measure developed to assess learning in a probabilistic sequence learning task [11-12] (Figure 6 left: hover on data points to see our model's results). We inferred the internal representation of the participant on one part of the experiment and tested the inferred model on a later part. The performance of the model is measured by the correlation between the log predicted probabilities and the reaction times: the smaller the better (see reaction time model on Figure 3.) Our model shows substantially better performance for all but one participants than the classical trigram model which predicts the most likely element based on the previous two elements (the most likely final element of a triplet).
Figure 6. (Interactive.) Individual results. Left: our model's performance vs. the trigram model (smaller correlation is better). Our model captures the variability in the response times substantially better than the earlier model for all but one participant. Right: you can see our model's performance on each individual's data. You can compare these graphs to Figure 5, right.
Next, we aimed at testing the inferred dynamical internal model in a substantially different way: our goal was to demonstrate that the inferred model is indeed an internal model used by the participant for predictions rather than a phenomenological model capturing idiosyncratic effects. Importantly, we fitted our model only on response times of correctly executed trials. Hence, testing our model on mistake prediction can be used to verify that we indeed inferred the internal models used for prediction by the participants. As you can see on Figure 7 (right) we can predict the mistakenly pressed buttons above chance (and it is statistically significant).
Figure 7. Predicting mistakes. Left: the inferred internal model gives lower predicted probabilities on those trials that were eventually missed. Dots are individual participant's averages with two standard errors of the mean. Right: rank of the erronous response among the incorrect alternatives given by the internal model. Our model predicts the mistakenly pressed button above chance.
Expected impact and future directions
The current work was first presented as a poster at the X. Dubrovnik Conference on Cognitive Science in May. It will also be presented as a poster at the Cognitive Computational Neuroscience conference in Philadelphia in September. The 4-page refereed abstract can be accessed here. It will also be submitted to a peer review journal in the coming months. The main novelty of our project is providing trial-by-trial predictions for response time measurements in a sequential learning task. Such quantitative measures may help elucidate neurophysiological and imaging data. We hope to further explore the rich internal representations of dynamical models entertained by individuals in future research.
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