Nassar, M. R. and Gold, J. I.
PLoS Comput Biol, 9:e1003015, 2013
DOI, Google Scholar
Fitting models to behavior is commonly used to infer the latent computational factors responsible for generating behavior. However, the complexity of many behaviors can handicap the interpretation of such models. Here we provide perspectives on problems that can arise when interpreting parameter fits from models that provide incomplete descriptions of behavior. We illustrate these problems by fitting commonly used and neurophysiologically motivated reinforcement-learning models to simulated behavioral data sets from learning tasks. These model fits can pass a host of standard goodness-of-fit tests and other model-selection diagnostics even when the models do not provide a complete description of the behavioral data. We show that such incomplete models can be misleading by yielding biased estimates of the parameters explicitly included in the models. This problem is particularly pernicious when the neglected factors are unknown and therefore not easily identified by model comparisons and similar methods. An obvious conclusion is that a parsimonious description of behavioral data does not necessarily imply an accurate description of the underlying computations. Moreover, general goodness-of-fit measures are not a strong basis to support claims that a particular model can provide a generalized understanding of the computations that govern behavior. To help overcome these challenges, we advocate the design of tasks that provide direct reports of the computational variables of interest. Such direct reports complement model-fitting approaches by providing a more complete, albeit possibly more task-specific, representation of the factors that drive behavior. Computational models then provide a means to connect such task-specific results to a more general algorithmic understanding of the brain.
Nassar and Gold use tasks from their recent experiments (e.g. Nassar et al., 2012) to point to the difficulties of interpreting model fits of behavioural data. The background is that it has become more popular to explain experimental findings (often behaviour) using computational models. But how reliable are those computational interpretations and how to ensure that they are valid? I will briefly review what Nassar and Gold did and point out that researchers investigating reward learning using computational models should think about learning rate adaptation in their experiments, because, in the light of the present paper, their results may else not be interpretable. Further, I will argue that Nassar and Gold’s appeal to more interaction between modelling and task design is just how science should work in principle.
The considered tasks belong to the popular class of reward learning tasks in which a subject has to learn which choices are rewarded to maximise reward. These tasks may be modelled by a simple delta-rule mechanism which updates current (learnt) estimates of reward by an amount proportional to a prediction error where the exact amount of update is determined by a learning rate. This learning rate is one of the parameters that you want to fit to data. The second parameter Nassar and Gold consider is the ‘inverse temperature’ which tells how a subject trades off exploitation (choose to get reward) against exploration (choose randomly).
Nassar and Gold’s tasks are special, because at so-called change points during an experiment the underlying rewards may abruptly change (in addition to smaller variation of reward between single trials). The experimental subject then has to learn the new reward values. Importantly, Nassar and Gold have found that subjects use an adaptive learning rate, i.e., when subjects encounter small prediction errors they tend to reduce the learning rate while they tend to increase learning rate when experiencing large prediction errors. However, typical delta-rule learning models assume a fixed learning rate.
The issue discussed in the paper is that it will not be easily possible to detect a problem when fitting a fixed learning rate model to choices which were produced with an adaptive learning rate. As shown in the present paper, this issue results from a redundancy between learning rate adaptiveness (a hyperparameter, or hidden factor) and the inverse temperature with respect to subject choices, i.e., a change in learning rate adaptiveness can equivalently be explained by a change in inverse temperature (with fixed learning rate adaptiveness) when such a change is only measured by the choices a subject makes. Statistically, this means that, if you were to fit learning rate adaptiveness with inverse temperature to subject choices, then you should find that the two parameters are highly correlated given the data. Even better, if you were to look at the posterior distribution of the two parameters given subject choices, you should observe a large variance of them together with a strong covariance between them. As a statistician you would then report this variance and acknowledge that interpretation may be difficult. But learning rate adaptiveness is not typically fitted to choices. Instead only learning rate itself is fitted given a particular adaptiveness. Then, the relation between adaptiveness and inverse temperature is hidden from the analysis and investigators may be fooled into thinking that the combination of fitted learning rate and inverse temperature comprehensively explains the data. Well, it does explain the data, but there are potentially many other explanations of this kind which become apparent when the hidden factor learning rate adaptiveness is taken into account.
What does it mean?
The discussed issue exemplifies a general problem of cognitive psychology: that you try to investigate (computational) mechanisms, e.g., decision making, by looking at quite impoverished data, e.g., decisions, which only represent the final product of the mechanisms. So what you do is to guess a mechanism (a model) and see whether it fits the data. In the case of Nassar and Gold there was a prevailing guess which fit the data reasonably well. By investigating decision making in a particular, new situation (environment with change points) they found that they needed to extend that mechanism to account for the new data. However, the extended mechanism now has many explanations for the old impoverished data, because the extended mechanism is more flexible than the old mechanism. To me, this is all just part of the normal progress in science and nothing to be alarmed about in principle. Yet, Nassar and Gold are right to point out that in the light of the extended mechanism fits of the old mechanism to old data may be misleading. Interpreting the parameters of the old mechanism may then be similar to saying that you find that the earth is a disk, because from your window it looks like the ground goes to the horizon in a straight line and then stops.
Essentially, Nassar and Gold try to convince us that when looking at reward learning we should now also take learning rate adaptiveness into account, i.e., that we should interpret subject choices within their extended mechanism. Two questions remain: 1) Do we trust that their extended mechanism is worth pursuing? 2) If yes, what can we do with the old data?
The present paper does not provide evidence that their extended mechanism is a useful model for subject choices (1), because they here assumed that the extended mechanism is true and investigated how you would interpret the new data using the old mechanism. However, their original study and others point to the importance of learning rate adaptiveness [see their refs. 9-11,26-28].
If the extended mechanism is correct, then the present paper shows that the old data is pretty much useless (2) unless learning rate adaptiveness has been, perhaps accidentally, controlled for in previous studies. This is because the old data from previous experiments (probably) does not allow to estimate learning rate adaptiveness. Of course, if you can safely assume that the learning rate of subjects stayed roughly fixed in your experiment, for example, because prediction errors were very similar during the whole experiment, then the old mechanism with fixed learning rate should still apply and your data is interpretable in the light of the extended mechanism. Perhaps it would be useful to investigate how robust fitted parameters are to varying learning rate adaptiveness in a typical experiment producing old data (here we only see results for experiments designed to induce changes in learning rate through large jumps in mean reward values).
Overall the paper has a very general tone. It tries to discuss the difficulties of fitting computational models to behaviour in general. In my opinion, these things should be clear to anyone in science as they just reflect how science progresses: you make models which need to fit an observed phenomenon and you need to refine models when new observations are made. You progress by seeking new observations. There is nothing special about fitting computational models to behaviour with respect to this.