A computational analysis of the neural bases of Bayesian inference

Highlights

• We apply Bayesian model comparison to neural correlates of Bayesian inference.

• We analyze single-trial EEG dynamics during performance on an urn–ball task.

• We find dissociable anterior (P3a) and posterior (P3b, Slow Wave) correlates.

• Our data are consistent with nonlinear probability weighting.

• Our results are consistent with the Bayesian brain hypothesis.

Abstract

Empirical support for the Bayesian brain hypothesis, although of major theoretical importance for cognitive neuroscience, is surprisingly scarce. This hypothesis posits simply that neural activities code and compute Bayesian probabilities. Here, we introduce an urn–ball paradigm to relate event-related potentials (ERPs) such as the P300 wave to Bayesian inference. Bayesian model comparison is conducted to compare various models in terms of their ability to explain trial-by-trial variation in ERP responses at different points in time and over different regions of the scalp. Specifically, we are interested in dissociating specific ERP responses in terms of Bayesian updating and predictive surprise. Bayesian updating refers to changes in probability distributions given new observations, while predictive surprise equals the surprise about observations under current probability distributions. Components of the late positive complex (P3a, P3b, Slow Wave) provide dissociable measures of Bayesian updating and predictive surprise. Specifically, the updating of beliefs about hidden states yields the best fit for the anteriorly distributed P3a, whereas the updating of predictions of observations accounts best for the posteriorly distributed Slow Wave. In addition, parietally distributed P3b responses are best fit by predictive surprise. These results indicate that the three components of the late positive complex reflect distinct neural computations. As such they are consistent with the Bayesian brain hypothesis, but these neural computations seem to be subject to nonlinear probability weighting. We integrate these findings with the free-energy principle that instantiates the Bayesian brain hypothesis.

Introduction

How can the brain make reliable and valid inferences about the external world based on variable sensory information? Bayesian decision theory offers a useful theoretical framework for explaining this inference process (Jaynes, 2003, Robert, 2007). Bayesian inference constantly updates prior beliefs to posterior beliefs in light of observed data according to probability rules (Bayes' theorem; Baldi and Itti (2010)). Thus, it can hardly surprise that a hypothesis has been proposed according to which the brain codes and computes Bayesian probabilities (Knill and Pouget, 2004, Friston, 2005, Doya et al., 2007, Gold and Shadlen, 2007, Kopp, 2008). While earlier research provided results that are consistent with the Bayesian brain hypothesis (Hampton et al., 2006, Ostwald et al., 2012, Vilares et al., 2012, Lieder et al., 2013), no agreed-upon conclusion about the utility of the Bayesian brain hypothesis as a theoretical framework for explaining cognitive functions of the brain has been achieved in the field (Clark, 2013).

In order to test the Bayesian brain hypothesis, we tried to explain event related brain potentials (ERPs) during successive trials in an urn–ball task (Phillips and Edwards, 1966) in terms of underlying probability distributions. Specifically, we were interested in dissociating temporally and regionally specific cortical responses in terms of Bayesian updating and predictive surprise. To introduce experimental variance in terms of surprise-related responses, we manipulated the task's probabilistic contingencies at two levels. First, we introduced uncertainty about the sort of urn containing balls (by sampling urns from two distributions). Second, we manipulated the proportion of ball colors within each urn. We will refer to these probabilistic contingencies as prior probabilities and likelihoods, respectively. Notice that the subject inferred the nature of the urn based on small samples of balls that were drawn from the sampled urn.

Cognitive processes related to Bayesian updating can be split into two sub-processes: (1) Bayesian surprise represents the change in beliefs about hidden states given current observations. Bayesian surprise is computationally expressed as the divergence between the prior probability distribution over states and the posterior probability distribution over states given current observations; we henceforth label probability distributions over states as belief distributions. (2) Postdictive surprise represents the change in predictions of future observations given current observations. Postdictive surprise is computationally expressed as the divergence between the prior prediction distribution over observations and the posterior prediction distribution over observations given current observations. Notice that the probability distributions over observations (or prediction distributions) are dependent on the belief distributions such that both, the belief distributions as well as the prediction distributions, are dependent on Bayes' theorem. Thus, postdictive surprise represents an aspect of Bayesian updating which is dependent on, yet computationally distinguishable from, Bayesian surprise. Bayesian surprise and postdictive surprise are both fundamentally different from predictive surprise that is simply the surprise about current observations under the current prediction distribution. Another way to grasp these distinctions is that Bayesian surprise refers to probability distributions (belief distributions) over unobservable random variables (here hidden states or urns), whereas postdictive surprise and predictive surprise refer – in different ways and as explained above – to probability distributions (prediction distributions) over observed random variables (here observed balls).

We also examined the possibility that electrophysiological substrates of Bayesian inference may be better explained when nonlinear weighting of probabilities is taken into account. Nonlinear probability weighting was originally conjectured by prospect theory which represents a successful descriptive theory of economic decision behavior (Kahneman and Tversky, 1979, Tversky and Kahneman, 1992, Fox and Poldrack, 2009). In addition, nonlinear likelihood weighting has been applied in technical systems for audio–visual speech recognition (Neti et al., 2000). We addressed the issue of nonlinear probability weighting by repeating the above comparisons (of Bayesian updating and predictive surprise) using Bayesian observer models with and without nonlinear weighting.

We recorded 20-channel electroencephalographic data from human participants who performed our urn–ball paradigm. Our analyses focused on the late positive complex that is known to be decomposable into three separable ERP components (Sutton and Ruchkin, 1984, Dien et al., 2004): First, the late positive complex incorporates the anteriorly distributed P3a component that usually occurs at latencies around 340 ms post-stimulus (Kopp and Lange, 2013). Second, the late positive complex comprises the parietally distributed P3b component at, in comparison to the P3a latency, variably delayed latencies (Kolossa et al., 2012). Notice that the functional significance of these two ERP components seems to be related to uncertainty (Sutton et al., 1965, Kopp and Lange, 2013), surprise (Donchin, 1981, Kolossa et al., 2012), decision-making (O'Connell et al., 2012, Kelly and O'Connell, 2013) and – putatively – Bayesian inference (Friston, 2005, Kopp, 2008). Third, a posteriorly distributed Slow Wave (SW) accomplishes the late positive complex whose functional significance is, however, comparably less well understood (but see Ruchkin et al. (1988); García-Larrea and Cézanne-Bert (1998); Spencer et al. (2001); Matsuda and Nittono (2014)).

The present study aims at contributing to the literature by applying Bayesian model comparison to discuss Bayesian updating and predictive surprise in terms of their ability to explain trial-by-trial ERP variability at different points in time and over different regions of the scalp. Notice that this is a meta-Bayesian analysis, in the sense that we are using Bayesian model comparison to select among various regressors that are all based on the assumption of an ideal Bayesian observer (Daunizeau et al., 2010, Lieder et al., 2013). Thus, the present study should be regarded as a challenge to the validity of the Bayesian brain hypothesis as a theoretical framework for cognitive functions of the brain. One of us had proposed – on purely theoretical grounds – close relationships between cortical P3 responses and Bayesian inference several years ago (Kopp, 2008). We were further interested whether a Bayesian observer model that incorporates nonlinear probability weighting would outperform an unweighted model when explaining the measured cortical responses.

Notice that our earlier empirical research forms the basis of more specific hypotheses that could be examined in the context of the urn–ball task. This urn–ball task was especially designed for this purpose. The first hypothesis proposes that ERP responses at fronto-central channels (P3a) are best explained by Bayesian surprise. This suggestion has been made by Kopp and Lange (2013) by referring to Sokolov's (1966) Bayesian model of the orienting response. Notice that the P3a component of the ERP is usually considered as indicating the brain's orienting response (e.g., Friedman et al., 2001, Barceló et al., 2002, Nieuwenhuis et al., 2011), and Kopp and Lange's (2013) P3a data were consistent with Sokolov's (1966) model of the orienting response. The second hypothesis postulates that ERP responses at centro-parietal channels (P3b) are best explained by predictive surprise. This suggestion was made by Kolossa et al. (2012) who showed that trial-by-trial P3b amplitude fluctuations in a simple two-choice response time task could be best explained by a computational model of predictive surprise. Assuming that these two hypotheses possess some validity, a novel hypothesis arises according to which ERP responses at occipito-parietal channels (SW, Matsuda and Nittono (2014)) would be best explained by postdictive surprise.