#StackBounty: #distributions #logistic #normal-distribution #pdf #model Relationship between a logistic decision function and Gaussian …

Bounty: 50

Imagine an experiment, in which an observer has to discriminate between two stimulus categories at different contrast levels $$|x|$$. As $$|x|$$ becomes lower, the observer will be more prone to making perceptual mistakes. The stimulus category is coded in the sign of $$x$$. I’m interested in the relationship between two different ways of modeling the observer’s "perceptual noise" based on their choices in a series of stimulus presentations.

The first way would be to fit a logistic function

$$p_1(x) = frac{1}{1+e^{-betacdot x}}$$

where $$p_1(x)$$ is the probability to choose the stimulus category with positive signs ($$S^+$$). Here, $$beta$$ would reflect the degree of perceptual noise.

A second way would be to assume that the observer has Gaussian Noise $$mathcal{N}(0,sigma)$$ around each observation of $$x$$ and then compute the probability to choose $$S^+$$ by means of the cumulative probability density function as follows:

$$p_2(x) = frac{1}{sigmasqrt{2pi}}intlimits_{z=0}^{infty}e^{-frac{(z-x)^2}{2sigma^2}}$$

In this case, $$sigma$$ would be an estimate of the perceptual noise.

I have a hunch that both these approaches are intimately related, but I’m not sure how. Is it an underlying assumption of the logistic function that the noise is normally distributed? Is there a formula that describes the relationship between $$beta$$ of $$p_1(x)$$ and $$sigma$$ of $$p_2(x)$$? Are, in the end, $$p_1(x)$$ and $$p_2(x)$$ essentially identical and could $$p_1$$ be derived from $$p_2$$?

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