# What is the posterior kernel lengthscale of a Gaussian process?

If I have access to multiple samples from a Gaussian process with known covariance kernel but unknown parameters (i.e. unknown lengthscale), it is straightforward to estimate the lengthscale using maximum likelihood methods. What I would like to know is how to calculate the full posterior of the lengthscale parameter.

There are various approximations I could use, such as approximate Bayesian computation. But I feel sure there must be a way to do it in closed form.

• I guess you could try to look for a conjugate prior for the kernel hyperparameters that would result in a multivariate normal posterior but I suspect that you'd be severely limiting the range of kernels for which such a conjugate would exist. You could always use MCMC to sample from a posterior though, because no matter what the prior on the lengthscale, the posterior is known up to a constant of proportionality. – InfProbSciX Nov 27 '18 at 14:11