Is there a Gaussian Process Kernel that limits functions to sigmoids? I am modeling a large number of Dose-response curves. I have strong reason to believe that the generating function will be sigmoidal against the concentration of the assay (Michaelis-Menten kinetics). I also know that there will be no effect at 0 concentration (homeopathy is not real). I have been playing with using Gaussian Process regression because I really like the idea of being able to model my uncertainty and sometimes I only have noisy measurements at a couple of concentrations over a given curve.
Is there a Gaussian Process Kernel that limits functions to sigmoids? There is no example in Duvenaud's excellent kernel cookbook. 
I have seen people mention such a thing, but I cannot find any concrete expression of such a kernel. Also, I cannot think of a way to use a covariance matrix to express ideas like monotonicity or sigmoidalness, maybe I have a fundamental confusion about GPs
 A: No such kernel exists:


*

*There is no nontrivial kernel such that samples from the GP are (almost surely) bounded, i.e. $0 \le f(x) \le 1$ as required for a sigmoid. The marginal distribution of, say, $f(0)$ is  a normal distribution, and normal distributions are only almost surely bounded if they have zero variance.

*There is no nontrivial kernel such that samples from the GP are (almost surely) monotonic. Once you've sampled $f(0)$, the conditional distribution of $f(1) \mid f(0)$ is also Gaussian, and thus either must have zero variance or else have some probability of being either larger than or smaller than $f(0)$.
As discussed in the comments, the normal thing to do in GP-land is to model some transformation of the data, e.g. $z$ in Sycorax's comment. You can choose some suitable non-stationary kernel to avoid the issue you mentioned about the influence of the kernel, e.g. $k_z(z, z') = k_\mathrm{base}(\operatorname{z\_to\_x}(z), \operatorname{z\_to\_x}(z'))$.
Or just don't use a GP if you really think a parametric model is the right choice. Or you could consider a "semi-parametric" approach.
