if it still hasn’t happened yet, it’s likely to take a long time longer!

The Cauchy distribution is a unimodal distribution with fatter tails than a Gaussian. (Fig 1 at right)

Janssen & Shadlen (2005), Nature Neuroscience found that monkey LIP neuron activity followed the subjective hazard function of an objective bimodal probability density function, which goes up, down, then up again. With a Gaussian distribution (bell-shaped curve), the hazard function increases monotonically with time (Fig 2), in other words it is increasingly likely that the event will occur in the next moment if it has not occurred already, because the hazard function is proportional to the likelihood the event will occur in the next moment if it has not yet occurred.

But would the neurons successfully represent a Cauchy distribution for which the hazard rate actually decreases with time after the mean? (Fig 3)
This hazard function is surprising to many, because it seems that for a unimodal distribution, as time elapses and the event still has not occurred it should be increasingly likely that it will occur. But this won’t occur if the tails are fat enough, as pointed out by Nassim Taleb in his book The Black Swan. Hence the title of this post. This kind of hazard function applies to various real-world phenomena, like construction contractors! as time passes after when they said it would be done, every day they don’t finish it indicates the time they’ll finish is probably even further into the future. I think Taleb suggests that humans don’t usually represent this hazard function, but he’s probably referring to cognition. I don’t know if the same is true for a go/no-go learned response time task or the like, something more automatic than cognition. Probably noone has done this experiment. Maybe it is indeed very difficult to learn this.

Indeed I think someone has shown (maybe Taleb) that it is hard, takes a lot of data even in principle to learn the fatness of tails. Maybe our default hazard function is increasing. It might be easier to see this effect in a two-button experiment, where the task is to press one or the other button, and one has a Gaussian (increasing hazard) and the other a Cauchy distribution (decreasing hazard).


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