Robust averaging during perceptual judgment.

de Gardelle, V. and Summerfield, C.
Proc Natl Acad Sci U S A, 108:13341–13346, 2011
DOI, Google Scholar


An optimal agent will base judgments on the strength and reliability of decision-relevant evidence. However, previous investigations of the computational mechanisms of perceptual judgments have focused on integration of the evidence mean (i.e., strength), and overlooked the contribution of evidence variance (i.e., reliability). Here, using a multielement averaging task, we show that human observers process heterogeneous decision-relevant evidence more slowly and less accurately, even when signal strength, signal-to-noise ratio, category uncertainty, and low-level perceptual variability are controlled for. Moreover, observers tend to exclude or downweight extreme samples of perceptual evidence, as a statistician might exclude an outlying data point. These phenomena are captured by a probabilistic optimal model in which observers integrate the log odds of each choice option. Robust averaging may have evolved to mitigate the influence of untrustworthy evidence in perceptual judgments.


The authors investigate what influence the variance of evidence has on perceptual decisions. A bit counterintuitively, they implement varying evidence by simultaneously presenting elements with different feature values (e.g. color) to subjects instead of presenting only one element which changes its feature value over time (would be my naive approach). Perhaps they did this to be able to assume constant evidence over time such that the standard drift diffusion model applies. My intuition is that subjects anyway implement a more sequential sampling of the stimulus display by varying attention to individual elements.

The behavioural results show that subjects take both mean presented evidence as well as the variance of evidence into account when making a decision: For larger mean evidence and smaller variance of evidence subjects are faster and make less mistakes. The results are attention dependent: mean and variance in a task-irrelevant feature dimension had no effect on responses.

The behavioural results can be explained by a drift diffusion model with a drift rate which takes the variance of the evidence into account. The authors present two such drift rates. 1) SNR drift = mean / standard deviation (as computed from trial-specific feature values). 2) LPR drift = mean log posterior ratio (also computed from trial-specific feature values). The two cannot be differentiated based on the measured mean RTs and error rates in the different conditions. So the authors provide an additional analysis which estimates the influence of the different presented elements, that is, the influence of the different feature values presented by them, on the given responses. This is done via a generalised linear regression by fitting a model which predicts response probabilites from presented feature values for individual trials. The fitted linear weights suggest that extreme (outlying) feature values have little influence on the final responses compared to the influence that (inlying) feature values close to the categorisation boundary have. Only the LPR model (2) replicates this effect.

Why have inlying feature values greater influence on responses than outlying ones in the LPR model, but not in the other models? The LPR model alone would not predict this, because for more extreme posterior values you get more extreme LPR values which then have a greater influence on the mean LPR value, i.e., the drift rate. Therefore, It is not entirely clear to me yet why they find a greater importance of inlying feature values in the generalised linear regression from feature values to responses. The best explanation I currently have is the influence of the estimated posterior values: Fig. S5 shows that the posterior values are constant for sufficiently outlying feature values and only change for inlying feature values, where the greatest change is at the feature value defining the categorisation boundary. When mapped through the LPR the posterior values lead to LPR values following the same sigmoidal form setting low and high feature values to constants. These constant high and low values may cancel each other out when, on average, they are equally many. Then, only the inlying feature values may have a lasting contribution on the LPR mean; especially those close to the categorisation boundary, because they tend to lead to larger variation in LPR values which may tip the LPR mean (drift rate) towards one of the two responses. This explanation means that the results depend on the estimated posterior values. In particular, that these are set to values of about 0.2, or 0.8, respectively, for a large range of extreme feature values.

I am unsure what conclusions can be drawn from the results. Although, the basic behavioural results are clear, it is not surprising that the responses of subjects depend on the variance of the presented evidence. You can define the feature values varying around the mean as noise. More variance then just means more noise and it is a basic result that people become slower and more error prone when presented with more noise. Perhaps surprisingly, it is here shown that this also works when noisy features are presented simultaneously on the screen instead of sequentially over time.

The DDM analysis shows that the drift rate of subjects decreases with increasing variance of evidence. This makes sense and means that subjects become more cautious in their judgements when confronted with larger variance (more noise). But I find the LPR model rather strange. It’s like pressing a Bayesian model into a mechanistic corset. The posterior ratio is an ad-hoc construct. Ok, it’s equivalent to the log-likelihood ratio, but why making it to a posterior ratio then? The vagueness arises already because of how the task is defined: all information is presented at once, but you want to describe accumulation of evidence over time. Consequently, you have to define some approximate, ad-hoc construct (mean LPR) which you can use to define the temporal integration. That the model based on that construct replicates an aspect of the behavioural data may be an artefact of the particular approximation used (apparently it is important that the estimated posterior values are constant for extreme feature values). So, it remains unclear to me whether an LPR-DDM is a good explanation for the involved processes in this case.

Actually, a large part of the paper (cf. title) concerns the finding that extreme feature values appear to have smaller influence on subject responses than feature values close to the categorisation boundary. This is surprising to me. Although it makes intuitive sense in terms of ‘robust averaging’, I wouldn’t predict it for optimal probabilistic integration of evidence, at least not without making further assumptions. Such assumptions are also implicit in the LPR-DDM and I’m a bit skeptical about it anyway. Thus, a good explanation is still needed, in my opinion. Finally, I wonder how reliable the generalised linear regression analysis, which led to these results, is. On the one hand, the authors report using two different generalised linear models and obtaining equivalent results. On the other hand, they estimate 9 parameters from only one binary response variable and I wonder how the optimisation landscape looks in this case.

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