20101208

Bayesian hallucination

Have you ever felt your phone buzz (when it didn't), or saw an email notification in the corner of the screen (when there was none)? Don't worry—you're not loosing your mind. 
 
This happens because the brain performs value-weighted predictive coding of unreliable sensory inputs. It can be explained in terms of optimizing costs and benefits using unreliable information from peripheral vision:
  • let $u$ be the utility ( benefit ) of responding to a notification,
  • let $c$ be the cost of verifying whether a notification is real or imagined
  • let $\Pr(\mathrm{present})$ be the probability that a notification is really there
Optimally, you should check a notification if the expected benefit of responding to the notification outweighs the cost : check notification if and only if $\mathbb E(u)>c$
[0] $\mathbb E(u) = u \cdot \Pr(\mathrm{present})$

[1] check notification if and only if : $u \cdot \Pr(\mathrm{present}) > c$
How does one know $\Pr(\mathrm{present})$ given some unreliable observation $\theta$ in peripheral vision, that is $\Pr(\mathrm{present}|\theta)$ ? This can be computed using Bayes' theorem : [2]
[2] $\Pr(\mathrm{present}|\theta)=\Pr(\theta|\mathrm{present})\cdot\Pr(\mathrm{present})/\Pr(\theta)$
So, $\Pr(\mathrm{present}|\theta)$ is the probability of observing $\theta$ when the notification is really there, $\Pr(\theta)$ is the probability of observing $\theta$ overall, and $\Pr(\mathrm{present})$ is the background probability of the notification being present. Plugging in expression [2] for $\Pr(\mathrm{present}|\theta)$ into equation [1] :
[3] check if and only if : $u \cdot \Pr(\theta|\mathrm{present}) \cdot \Pr(\mathrm{present}) / \Pr(\theta) > c$
Peripheral observations $\theta$ are noisy, and $\Pr(\theta|\mathrm{present})$ has different but overlapping distributions depending on whether or nor a stimulus is present. If the expected benefit from checking a notification is high, this can lower the threshold for checking a notification. The sensory system automatically optimizes unreliable perception into a (possibly inaccurate) high-level report for the parts of the brain that deal with behavior and attention.


2 comments:

  1. I recently did a lab involving similar psycho-physics. Detecting signal in noise. Sadly it was much less implemented than your pseudo coded example, but nevertheless, a fun problem in perception.

    Here's the model we used
    http://psych.hanover.edu/JavaTest/Media/Chapter2/MedFig.ROC.html

    I only have a cursory knowledge of Signal Detection Theory, but I'd be willing to say u*p(θ|present) relates to d' and criterion is cost.

    Then again, criterion is a measure of action, whether you actually check to confirm if its real. Its assumed you will only check if you're sure you've perceived a signal...

    ...and from the way you described your hallucinations it sounds like you're always sure the signal is there and would always check. I'm not that up on the lit regarding this subject so I'm don't have a super stringent definition of criterion. I really want to say that criterion only affects the action in a perception action loop, but I'll play it safe and wont.

    Back to seeing shit that isn't there.

    So for whatever reason you've managed to lower d' so that you can't distinguish signal from noise. I guess the best experiment to test this would be to increase cost/criterion and see if the hallucinations persist.

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  2. "So for whatever reason you've managed to lower d' so that you can't distinguish signal from noise. I guess the best experiment to test this would be to increase cost/criterion and see if the hallucinations persist."

    good call.

    So, its not really surprising that the brain is optimizing to determine whether or not to check into some noisy signal. What is surprising is that, subjectively, I don't experience "that looks a bit like a notification, lets go check it", I experience "that definitely is a notification, lets go investigate", even when there's nothing there. The net behaviors are identical, except that in the fist case I am aware of the degree of noise in the signal. In reality, I think my visual cortex is making decisions for me and not telling me about how reliable that choice is. Which means non-conscious mechanisms have trained visual cortex to understand the cost/benefit trade of of investigating computer notifications in the periphery. It also means that I've been exposed to these user interface components so much that their appearance has been stored in visual systems, I think, which is a little weird.

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