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When we do a probit regression, we use the distribution of a standard normal to convert from the linear combination of the predictors to a probability value.

Why stop at the standard normal? Why not other normal distributions? Why not a Cauchy distribution? Looking at the CDFs, it seems like a Cauchy-based link function would have a much higher threshold for getting close to a certain outcome with probability near $0$ or $1$. Would this be reasonable to use in a low signal-to-noise setting where we would want to be "darn sure" before we made an assertion that something was $99\%$ likely to happen?

Dave
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There is no reason to stop at the normal! What you propose, is called the Cauchit link function, for an example at this site, see GARMA models for counts.

Here, at arXiv is a paper looking at and comparing various link functions, cauchit included, for binary regression.

kjetil b halvorsen
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