Noise estimate for training points

For my gaussian I assign an arbitrary noise function to it. I do have the standard deviation of each training point though. When sampling the MAP function uses a value which is actually the mean of my actual standard deviation and the trace is weighted there also. Is this just a coincidence or is this expected. Below is the noise function I use and the line of code which includes it in the gaussian process

s = pm.HalfCauchy("s",beta=5)
ym = gp.marginal_likelihood("ym",X=x,y=y, noise=s)

Additionally I am trying to work in my actual standard deviation, I currently have them as a numpy array for each y value used to train the gaussian process. I was going to use a deterministic function to use that as a floor for the half cauchy prior to learn from but I can’t seem to get that to work, so if anybody has any suggestions that would be very useful

Also when using the MAP function will it pretty much produce quite similar results no matter the initial prior put in (if same function)