Calver-Travis et al. "Bayesian Confidence in Optimal Decisions"

Let's talk about this new preprint by Calder-Travis et al. It’s an interesting piece because it deals with the difficult question of the relation between confidence and response time. They do a very nice job explaining all the different options, and I have to say, the article is a model of clarity. It’d be great if all scientific articles were written as clearly as this one.

If you’re not interested in this question, I understand. It doesn’t seem so exciting at first. But it turns out the relation between confidence and response time is far from being straightforward. So, let me try to spark your interest.

Imagine that you have to determine whether a briefly flashed stimulus is a square or a diamond, and then provide a confidence judgment. You don’t always take the exact same time to answer. Sometimes you answer a bit more quickly, sometimes a bit more slowly. That’s just how things are.

Now let me ask you a question. When will you be more confident: when you take a lot of time to respond, or when you answer immediately after the stimulus is presented?

The first option seems right. If I don’t have enough time to decide, I won’t be confident in my decision. Confidence should increase as decision time increases.

But the other option could be true as well. After all, if I’m confident in my decision, there’s no need for me to wait before answering. So, we should expect confidence to decrease as response time increases.

See? Not so straightforward! Does confidence increase with response time? Or does it decrease with response time?

The sad truth is... it depends. It depends on whether the experimenter sets the response time, or whether you're free to set your own response time. Under time pressure set by the experimenter, confidence tends to increase as response time increases. But when subjects can set their own response time, confidence decreases as response time increases.

This phenomenon can be (approximately) explained by drift diffusion models, or, more precisely, by a variant developed by Pleskac & Busemeyer (2010): the 2-stage signal detection model (2DSD).

Drift diffusion models postulate that perceptual systems sequentially accumulate evidence favoring one alternative over the other, until a threshold is reached and a perceptual decision is triggered. The quality of the evidence determines how fast evidence is accumulated – also called the drift rate of the evidence. In turn, confidence judgments are based on the evidence accumulated in favor of the choice.

The 2DSD model can explain why confidence increases as decision time increases when subjects aren’t free to set their own response time. They do so in part by postulating that evidence continues to accumulate after the decision is reached to inform confidence judgments. More time means more accumulated evidence, which leads to higher confidence. So, as decision time increases, confidence increases.

To some extent, the 2DSD model can also explain why confidence decreases as decision time increases when participants are free to set their own response time. In their model, Pleskac & Busemeyer postulate that, all other things being equal, the quality of the evidence accumulated (i.e. the drift rate) can vary randomly from trial to trial. A high drift rate leads to fast decision times and high levels of confidence. A low drift rate leads to slow decision times and low levels of confidence. As a result, as drift rate decreases, decision time increases and confidence decreases.

That’s a convincing story, but it’s still somewhat incomplete. As Pleskac & Busemeyer recognize, “The model does, however, underestimate the relationship between observed decision time and confidence” (p.881).

Here come Calder-Travis et al. now. In a nutshell, their model keeps the basic features of the 2DSD model but also postulates that the system generating confidence judgments takes as input an estimate of the rate of accumulation of evidence.

That’s not an ad hoc hypothesis to explain a weird feature of confidence judgments. It really makes intuitive sense. If I accumulate evidence in favor of one decision (and not the other) at a crazy rate, it’s probably because this trial is really easy, so I should be more confident. On the other hand, if I struggle to accumulate evidence, it’s probably a difficult trial, and then I shouldn’t be confident in my response.

In sum, under time pressure, confidence increases as response time increases, because more evidence accumulates if you take more time to answer, and this should make you more confident. Without time pressure, confidence decreases as response time increases, because a long response time indicates a difficult trial, and if the trial is difficult you shouldn’t be confident. So, overall, the model nicely explains what we currently know about confidence and response time.

Still, it seems to me that Calder-Travis et al.’s model doesn’t account for the intuition that I answer faster because I’m more confident. There seems to be a causal relation between the fact that I’m more confident and the fact that I answer more quickly. Maybe it’s just an illusion, or a “strange inversion” as Dennett would say. But before saying that, we could search for models that do account for this.

In fact, several research groups have recently suggested that confidence could modulate the accumulation of evidence online, even before a perceptual decision is reached (See here, here, and here for some evidence that, in my opinion, goes in that direction). I really like this idea. Maybe confidence influences evidence accumulation, or can modulate the decision boundary. If that’s right, then we can make sense of the intuition that we take some perceptual decisions faster because we’re more confident.

But again, things aren’t so easy… One question is: where does confidence come from, if not from the accumulation of evidence that leads to the perceptual decision itself? If confidence is computed at least partly based on the evidence accumulated for the perceptual decision, it can’t, at the same time, modulate the accumulation of evidence for that very same perceptual decision.

As a result, these models have to postulate two accumulators: one for the perceptual decision, and one for confidence (as done here). This can be accomodated with some kind of dual channel model. However, as Calder-Travis et al. remark, it’s unclear why Mother Nature would have duplicated evidence accumulation in this way. In this respect, models like Calder-Travis et al.’s are much simpler for now.

In addition, these models have to explain exactly how the system that computes confidence influences the accumulation of evidence online. That’s a tough problem too! So, even if I like the idea that confidence modulates perceptual decisions online, I have to admit that it’s not so clear how things work at this point.

Who knows, maybe Mother Nature did partly duplicate evidence accumulation, and maybe things aren’t so simple. The relation between response time and confidence is still mysterious. But Calder-Travis et al.’s piece, among others, shows that we can make a lot of progress on this question.

This line of work also illustrates something about the field. People often seem puzzled by the place that the study of confidence has taken in recent years. Sure, studying confidence is interesting in itself. But there’s also a higher purpose to all this. Ultimately, analyzing the mechanisms of perceptual confidence can reveal a lot on the mechanisms of perceptual decision making.

I thank Joshua Calder-Travis for his comments on a previous version of this blogpost.

MM