Decision Making; Evidence Matters

Everybody makes a decision when subjected to certain issues in everyday life. Each day we are making a whole lot of decisions with the expectation of getting a better outcome. You may have heard about decision making in its broad sense, about how important it and all. But, let’s dig deeper to get something new. What is the mechanism driving these? What determines our decision? How confident are we about it? How are we making a decision when the time is constrained? What are the priorities that we have?  

First of all, we can categorize decisions into perceptual, economic, physical, emotional, etc. Among these perceptual decisions are well studied in neurosciences. It is the act of selecting an option from a group based on the available sensory information. If we plot the reaction time, then it is intuitive that the reaction time(mean) would be lower and the standard deviation when the choice is easy. What if the choice is hard? You could easily tell that the mean of the response time distribution would be higher, and the standard deviation is also higher. That is to say, the standard deviation being a measure of uncertainty, the decision of an agent would be most uncertain when the choice is hard. But, when the hard choice becomes shorter and narrower, the response time curve shifts towards the lower mean and standard deviation. One would ask, “why”?

But, what is the neural mechanism behind this? It is said that there is a threshold, and when the rising neural activity hits the threshold, it is exactly the point when you make the decision. But, it would help if you asked what is crucial here. It is indeed the slope of the curve which is determined by the strength of the evidence. But, if that is so, we would get only a single time point, which is the mean. What about the spread? The answer is simply noise. When a decision becomes more common, the starting point is elevated, and even if the slope is identical, it will hit the threshold earlier. If you are confused at this point, think about a board having numerous squares having only two colors. If someone is asking you to identify the color more in that board, if both colors are almost equally distributed, you would initially take a lot more time. Sometimes you would not be able to decide. But, what if you are repeatedly subjected to this same task? Then you would be able to recognize a pattern or something which would make your decision-making faster. So, the distence from threshold and slope gives the reaction time.

One of the models that explain the decision-making process is the drift-diffusion model. Suppose an agent has to select either of the two options. The model suggests that some neural variable undergoes random drift, which is biased by the sensory information we receive. In this article, it’s mathematical formulation won’t be covered(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2474742/) even though it suffices to say that at each point of time, the direction and magnitude could be given by the weight of the evidence. And the decision making happens when the curve hits either of the two threshold. Also, the prior bias determine the starting point.

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Wait, what about the threshold? Can we reduce it for getting a speedy response as the time is determined by the height of the threshold from the starting point? We can reduce the threshold, but sometimes even noise could cause the curve to touch the threshold. That is higher the threshold higher the accuracy. Interestingly, the model predicts that during the process, the brain is integrating the evidence in favor of one option over the other.

But why the brain has to integrate the evidence? One could easily say that integration filters out noise from the distribution. It also enables us to do a statistical test on the sample. Let our evidences be:

evidences= { e₂, e₂, e₃,… e_n}

And the options that we have are A and B such that we want error rate to be less than a particular limit(a)Then we should choose A when:

P(A|evidences)/P(B|evidences) > 1-a/a

And choose B when:

P(B/evidences)/P(A/evidences) > 1-a/a

If neither of this holds then we should wait more time for getting more evidence to take an action. The sequential probability ratio test is exactly using the integration for reaching the threshold with smallest possible steps.

But, at this point, one may be doubtful about calculating the probabilities. How to calculate the probabilities? But, we need to initialize the process before getting any evidences. In otherwise if the criteria is satisfied without any evidences, we don’t have to look for evidences. That is:

P(A)/P(B) > 1-a/a

But, after getting the first piece of evidence e₁, we are no longer interested in looking at whether the prior satisfies the criteria. So, we would update the criteria as follows.

P(A|e₁)/P(B|e₁) > 1-a/a

But how are we going to calculate P(A|e₁) or P(B|e₁)? We can simply use the Bayes rule which is as follows:

P(A and B)= P(B)* P(A|B) —-(1)

also

P(A and B)=P(A)*P(B|A) —-(2)

Equating (1) and (2) we get:

P(A|B)= [P(A)*P(B|A)]/P(B)

From this we get that:

P(A|e₁)= [P(A)*P(e₁|A)]/P(e₁)

P(B|e₁)= [P(B)*P(e₁|B)]/P(e₁)

So, the criterion becomes

{[P(A)*P(e₁|A)]/P(e₁)}/{[P(B)*P(e₁|B)]/P(e₁)} = 1-a/a= [P(A)*P(e₁|A)]/[P(B)*P(e₁|B)]

Where P(e₁|A) is the probability of the piece of evidence given the action was to choose option A, which can be calculated. After n updations the criterion becomes:

[P(A)*P(e₁|A)*P(e₂|A)*P(e₃|A)*…*P(e_n|A)]/[P(B)*P(e₁|B)*P(e₂|B)*P(e₃|B)*…*P(e_n|B)]

Note that P(A)/P(B)is the prior probability. But, you would think of taking log as it would help us to convert this product to sum as follows:

log(1-a)/log(a)= log{P(A)/P(B)} + [ log(P(e₁|A)/P(e₁|B))+log(P(e₂|A)/P(e₂|B))+ log(P(e₃|A)/P(e₃|B))+…+log(P(e_n|A)/P(e_n|B))]

But, if you are looking carefully at the equations, you can see that this is actually what we had seen in the case of the drift-diffusion model. The first term is the logarithm of prior, and the rest is the accumulated evidence. One could easily point out that what we had done till now is building a mathematical model for the decision-making process. But, the crucial aspect is about testing it. How can we test it?

Here monkeys are trained to perform random-dot motion discrimination tasks. It is simply a simulation in which there are two fixed points on the screen’s right and left. And a population of dots are moving either left or right. Monkeys have to report whether they are moving right or left. They recorded different brain regions, MT, LIP, prefrontal cortex, frontal eye field, superior colliculus, etc. It is proposed that the decision is unfolded due to competition between actions, which is biased by evidence. In that, we can see that the neural activity is growing following the strength of the evidence. That is more the number of dots moving in the same direction, the more the neural activity. It is also found that neural activity in LIP is growing following the motion’s coherence, and it reaches a constant. And also, the rate of activity growth predicts reaction time. Neural recordings and fMRI data support this.

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Thus we are confident enough to say that the decisions, especially the perceptual decisions, are backed by evidence. This is how the decision-making process may be happening in the brain. So, evidence indeed matters.

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References:

[1] Response in LIP during Decision Formation, Roitman and Shadlen, J. Neurosci., November 1, 2002,22(21):9475–9489

[2] The Diffusion Decision Model: Theory and Data for Two-ChoiceDecision Tasks, Roger Ratcliff and Gail McKoon, Neural Comput. 2008 April ; 20(4): 873–922.