2012 Course: The Statistician Brain: the Bayesian Revolution in Cognitive Sciences

Transcription

1 2012 Course: The Statistician Brain: the Bayesian Revolution in Cognitive Sciences Stanislas Dehaene Chair of Experimental Cognitive Psychology Lecture n 5 Bayesian Decision-Making Lecture material translated from the French version by CG Traduction & Interprétation

2 Decision-Making: from Continuous to Discrete Making a decision consists in choosing one out of many possible interpretations or actions sensation S object O In the preceding lecture, we discussed how, even as regards perception, our nervous system let us perceive a single interpretation at a given moment in time. The fluctuation of these interpretations as a function of time (Pouget), and the fact that observers provide several successive responses (Vul), suggest that the nervous system appropriately computes and samples from the full distribution of Bayesian probabilities.

3 The All-or-none Nature of Change Perception K. O Regan, R. Rensink

4 Transitions from Continuous to Discrete in the Access to Consciousness Sergent, C., & Dehaene, S. (2004) Psychological Science In the attentional blink, access to consciousness follows an all-or-none distribution T2 Task: subjective visibility Not seen Seen all 90 Mean Subjective Visibility Lag T2 present, dual task T2 present, single task T2 absent, dual task T2 absent, single task Percent of trials Lag

5 All-or-none transition in access to consciousness Sergent, C., Baillet, S., & Dehaene, S. (2005). Timing of the brain events underlying access to consciousness during the attentional blink. Nat Neurosci, 8(10), Up to 200 ms, brain processing is indistinguishable, whether perception is conscious or not. Circa 270 ms however, the brain quickly shifts into one of two discrete states.

6 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Episodic memory tests suggest that the strength of memory traces varies continuously, but also that some traces are associated to one discrete mental state. - We need separate familiarity judgment from precise recollection (when, where). Similarly, perceptual judgment could be based on two different signal types: - a quantitative, analog signal, based on a global measure of similarity (strength) - a qualitative, discrete signal, providing consciousness with precise, high resolution information (state of knowing).

7 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Most psycho-physical trials use simple stimuli, close to the perception threshold, a fact which could minimize the contribution of the discrete component. The perception of more complex and realistic images may imply an all-or-none state of access to consciousness. In a series of experiments, subjects are asked to make same/different judgments on simple and complex images.

8 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Two processes: -all-or-none access to qualitative details - global measurement of perceptual similarity The all-or-none process allows the identification of differences but not of sameness.

9 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e If both processes are different, it must be possible to influence them in opposite directions. Two groups of subjects participated in trials with global or discrete changes. Results: double dissociation. Global distortion Discrete detail

10 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Time dynamics: ten repetitive exposures to image pairs, followed each time by same/different judgment (confidence judgment on a scale from 1 to 9). Step function, only in the case of discrete changes.

11 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Time dynamics: ten repetitive exposures to each pair of images, each time followed by same/different judgment (confidence judgment on a scale from 1 to 9). Step function, only in the case of discrete changes

12 Coexistence of Continuous and Discrete Perception Processes Aly, M., & Yonelinas, A. P. (2012). Bridging consciousness and cognition in memory and perception: evidence for both state and strength processes. PLoS ONE, 7(1), e Inspired by the remember-know paradigm in trials on memory, the authors asked the subjects if they perceived the difference or if they just had the feeling of knowing. These subjective reports correlate very closely with the corresponding objective parameters of the response curve. Conclusion: two processes contribute to perception, one continuous, the other discrete. Both could be consciously reported.

13 The Concept of the Probabilistic Sampling of Continuous Distributions also Applies to our Cognitive Inferences Vul, E., & Pashler, H. (2008). Measuring the Crowd Within: Probabilistic Representations Within Individuals. Psychological Science (Wiley-Blackwell), 19(7), what percentage of the world s airports is located in the US? Two successive samplings, made by one or several people, come close to the true value.

14 «Two minds are better than one» Bahrami, B., Olsen, K., Latham, P. E., Roepstorff, A., Rees, G., & Frith, C. D. (2010). Optimally interacting minds. Science, 329(5995), Question: does decision-making improve when several people are requested to come to an agreement? -Psycho-physical task (deciding which interval contains a target) -Task carried out simultaneously by two subjects - In the case of disagreement, the experiment is stopped and both people exchange until they come to an agreement.

15 «Two minds are better than one» Bahrami, B., Olsen, K., Latham, P. E., Roepstorff, A., Rees, G., & Frith, C. D. (2010). Optimally interacting minds. Science, 329(5995), The joint performance exceeds in sensitivity (d-prime) that of each individual, their average or even the best of both. Less sensitive observer Joint performance Most sensitive observer Adjustment via the Gaussian or bell curve This can only be explained through the assumption that subjects share their level of confidence on what they have seen during a given trial (weighted confidence sharing model). Assumption: The social human brain may have evolved to the point where our internal probability distributions have become accessible and communicable to others so as to enable collective Bayesian inferences.

16 The Limits of Collective Inference Lorenz, J., Rauhut, H., Schweitzer, F., & Helbing, D. (2011). How social influence can undermine the wisdom of crowd effect. Proc Natl Acad Sci U S A, 108(22), Groups of 12 subjects answered questions such as how many murders were committed in Switzerland in 2006?. Some groups were only asked to answer the question 5 times without additional information. Others were given -Either the arithmetic average of group responses during a preceding experiment -Or the full details of the responses of 12 subjects. To prevent possible manipulation by the group, the subjects are only remunerated as a function of their response accuracy (at 10, 20 or 40% intervals of the true value). Results: - In this case, the wisdom of the crowd is only expressed via a logarithmic curve probably because subjects must determine the appropriate response scale.

17 The Limits of Collective Inference Lorenz, J., Rauhut, H., Schweitzer, F., & Helbing, D. (2011). How social influence can undermine the wisdom of crowd effect. Proc Natl Acad Sci U S A, 108(22), Observations during subject interaction reveal three major limits to collective inference: 1. The diversity of opinions quickly narrows but the percentage of errors does not decrease as a result. 2. Consequently, the correct response becomes more and more peripheral within the subject s response distribution 3. Subjects confidence increases but the precision of responses does not. Conclusion: The «wisdom of the crowd» effect is genuine, but the social influences it is submitted to considerably reduce its usefulness. Adverse conditions often arise: in all cases where a subjective opinion is shared, without indication of the level of confidence which can be attributed to it.

18 Why does the brain sample distributions? Why does it (at least partially) compute with discrete states? Sampling is not necessarily the best solution: it may be more appropriate to work with complete distributions or to consistently choose the most probable solution (maximum a posteriori, MAP). In this case, why sample? Several assumptions: -Above a given level of complexity, sampling may be the only calculation tool (distributions would then not be actually represented but simply used implicitly). -The nervous system could be capable of representing continuous distributions, but would need to reduce them to a single sample before making a motor decision. Our perceptual awareness may have evolved from a brain architecture originally essentially dedicated to action.

19 The Perception-action loop seen from a Bayesian perspective

20 Simplified decision-making diagram Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Res, 50(23), Each state of the world (w) is translated into sensory state distributions (x). The problem of decision-making consists in choosing action (a) as a function of sensory states (x). The consequences of actions can be positive or negative depending on the actual states of the world, according to a Gain (or Cost) function G (a,w). In this action-oriented perspective, the goal is to choose action a=d(x) which leads to higher expected gain:

21 The classic signal detection theory stems from this Bayesian vision of the decision to act Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Res, 50(23), According to classical signal detection theory, there are only two possible states of the world: w = S or w = non-s And a very simple Gain function

22 The classic signal detection theory stems from this Bayesian vision of the decision to act Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Res, 50(23), Let us consider several decision rules: -d1 = always choose action S -d2 = always choose action non-s -d3 = choose S if x>0.5, non-s otherwise -d4 = the reverse -d5 = d3 with probability p, d2 with 1-p The concept of dominance : d3 always leads to higher expected gain than d4, regardless of the state of the world. Only those decision rules for which expected gain is plotted on the blue curve are not dominated and are therefore admissible. Rule d3 falls under the minimax criterion: it maximizes the minimum gain, under the worst conditions.

23 The classic signal detection theory stems from this Bayesian vision of the decision to act Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Res, 50(23), Bayesian decision-making: The states of the world w have a priori π(w) probabilities They must be taken into account to maximize gain. This equation assigns a single number to each admissible rule. Sorting them suffices to identify the optimal rule. When there are only two states S and non- S, with π and 1-π priors, a graphic solution is possible. - All rules on a red dashed line, perpendicular to vector [1-π, π ], have the same expected gain. The tangent to the blue line defines the optimal rule.

24 The classic signal detection theory stems from this Bayesian vision of the decision to act Maloney, L. T., & Zhang, H. (2010). Decision-theoretic models of visual perception and action. Vision Res, 50(23), Compensation for changes in the Gain function: -The scale of the axes changes if there are changes in Gain. - This change simply shifts the optimal rule. It must be noted that any observed behavior can be plotted on this graph and can therefore be explained with the same accuracy: - either through a distortion of the Gain function - or through the distortion of priors

25 The essential role of the Gain function in action selection: experiments by Trommershaüser et al. Simple visuo-motor task: - aim very quickly for a green circle (which induces errors) - Penalties apply if the subject touches the red circle. Trommershauser, J., Gepshtein, S., Maloney, L. T., Landy, M. S., & Banks, M. S. (2005). Optimal compensation for changes in task-relevant movement variability. J Neurosci, 25(31), Trommershauser, J., Landy, M. S., & Maloney, L. T. (2006). Humans rapidly estimate expected gain in movement planning. Psychol Sci, 17(11), Trommershauser, J., Maloney, L. T., & Landy, M. S. (2008). Decision making, movement planning and statistical decision theory. Trends Cogn Sci, 12(8),

26 The essential role of the Gain function in action selection: experiments by Trommershaüser et al. When the experiment starts, subjects practice hitting only the green circle. This teaches them to respond quickly and to assess their own response probability distribution. When the latter is known, it becomes possible to compute the equivalent lottery of each target point.

27 The essential role of the Gain function in action selection: experiments by Trommershaüser et al. Results: circles are presented at 6 different distances (randomly mixed within a single block), subjects move their target point quasi-optimally. They do this spontaneously, apparently without requiring prior training. Conclusions: -Subjects become aware of their own sensori-motor uncertainty function. -They can combine the function with a new gain function for near-optimal performance (contra Kahneman & Tversky?)

28 Introduction of the «time» variable Battaglia, P. W., & Schrater, P. R. (2007). Humans trade off viewing time and movement duration to improve visuomotor accuracy in a fast reaching task. J Neurosci, 27(26), The decision to act comprises a time component: How to trade off time dedicated to decision-making and time dedicated to motion? Battaglia and Schrater asked subjects to reach to a target whose position is only known via a dot scatter distribution, with a time constraint. According to the level of dot scatter, subjects adapt their decision time (t v ) and their movement duration (t M ) Their choice is close to the optimal gain computed via a model that factors in the variability of movement as a function of speed.

29 Conclusion During the decision-making process, the nervous system: - Takes into account event priors - Quasi-optimally combines multiple sources of sensory information - Taps into its knowledge of motor uncertainties - Adapts its objectives as a function of the expected cost of different options - Even adapts the speed of decision-making First level of Bayesian decision: intra-individual Even after an all-or-none decision (perceptive, cognitive or motor): - We can derive another sample which is not random - We often have information on our degree of confidence in our decision - We can share our confidence with others and use the information to make better choices Second level of Bayesian decision: inter-individual