Variance of Derivative GP (found through the Function GP)

[shikhar-saxena]

I am fitting a GP on the function values and want to make derivative predictions.

The derivative is also a Gaussian Process with the following parameters:

I followed this blogpost to get the mean function for the derivative.

Code from the blogpost:

# Get into evaluation (predictive posterior) mode
model.eval()
likelihood.eval()

X = torch.autograd.Variable(torch.Tensor(xtest), requires_grad=True)

def mean_f(X):
    return likelihood(model(X)).mean.sum()

nabla_mu = torch.autograd.functional.jacobian(mean_f, X) # Derivative Mean

I tried to extend it to get the variance for the derivative GP like this:

def var_f(X):
    return likelihood(model(X)).variance.sum()

# Is this correct??
nabla_variance = torch.autograd.functional.jacobian(var_f, X) # Derivative Variance

I get negative variances with this approach; so this seems incorrect.

In GPy there's a kernel.dK2_dXdX2 to compute the terms in the expression for derivative-variance. Do we have something similar in gpytorch? It would be very helpful if someone could let me know how to compute the derivative variance.