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I was reading about gaussian process for classification and there is something that is unclear.

My understanding is the inverse probit is used instead of the sigmoid and this is because it allows for the analytical integration.

Is this correct? And what practical effect does this have vs using a sigmoid?

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I am uncertain that this presented rationale holds strongly.

I think it is mostly because that "assuming Gaussian noise (for the latent Gaussian process) and a step-function likelihood is exactly equivalent to a noise-free latent process and probit likelihood". (Rasmussen & Williams (2006) Gaussian Process for Machine Learning, Chapt. 3.3) We use numerical integration coupled with an Laplace approximation to the (binary) GP classifier likelihood anyway, so I am unclear if "allowing for the analytical integration" is of substantial benefit.

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