When applying a Bayesian inference method such as Gaussian Process Regression (GPR), the assumption of a prior and likelihood function following a normal distribution is inherent. One can use an arbitrary kernel for the covariance function, but nevertheless, a Gaussian is the underlying distribution for the model (as the name implies).
But what is the significance of using a Gaussian distribution besides mathematical convenience? Under what scenarios would using the assumption of Gaussianity be invalid? For example, if a detector measures a time series with error following a Poisson distribution, should GPR be avoided when fitting this data? Namely, I am trying to understand when GPR is applicable and when it breaks down. As far as I can tell, there is no general property or extension of the Central Limit Theorem for why GPR should generalize well when performing functional regression on arbitrary data even with arbitrary kernels. Thus I am trying to gain insight and motivation on when GPR does (and does not) work.