[Question] Make predicitions about derivatives (mean *and* variance)

The example: gpytorch/examples/08_Advanced_Usage/Simple_GP_Regression_Derivative_Information_1d.ipynb shows us how to make predictions about derivatives, and also to make inference based on derivative information.

What is the best way to go about making predictions about derivatives when derivative information isn’t available? In other words I would like to be able to not only draw samples from the trained GP but also the first derivatives.

I tried passing gpytorch.distributions.MultivariateNormal the first gradient block (K[..., :n1, n2:]) returned from gpytorch.kernels.RBFKernelGrad but without success.

[KeAWang]

Is #772 what you’re looking for?

[KeAWang]

The easiest way to get the variance is to do an empirical estimate from sampling the derivatives.

Modifying the same code as https://github.com/cornellius-gp/gpytorch/issues/772#issuecomment-508124687:

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

# Test points are regularly spaced along [0,1]
# Make predictions by feeding model through likelihood
with gpytorch.settings.fast_pred_var():
    test_x = torch.linspace(0, 1, 51, requires_grad=True)
    observed_pred = likelihood(model(test_x))
    dydtest_x = torch.autograd.grad(observed_pred.mean.sum(), test_x, retain_graph=True)[0]

    n_samples = 100
    sampled_pred = observed_pred.rsample(torch.Size([n_samples]))
    sampled_dydtest_x = torch.stack([torch.autograd.grad(pred.sum(), test_x, retain_graph=True)[0] for pred in sampled_pred])
    dydtest_x_sample_std = sampled_dydtest_x.std(0)
    lower = dydtest_x - 2 * dydtest_x_sample_std
    upper = dydtest_x + 2 * dydtest_x_sample_std
    

with torch.no_grad():
    # Initialize plot
    f, ax = plt.subplots(1, 1, figsize=(4, 3))

    # Plot training data as black stars
    ax.plot(train_x.numpy(), train_y.numpy(), 'k*')
    # Plot predictive means as blue line
    ax.plot(test_x.detach().numpy(), observed_pred.mean.detach().numpy(), 'b')
    # Plot expected derivative
    ax.plot(test_x.detach().numpy(), dydtest_x.numpy(), 'r')
    # Plot 2 times empirical standard deviations about expected derivative
    ax.fill_between(test_x.detach().numpy(), lower.detach().numpy(), upper.detach().numpy(), alpha=0.1, color="r")
    # Plot real derivative
    ax.plot(test_x.detach().numpy(), 2 * math.pi * torch.cos(2 * math.pi * test_x).detach().numpy(), 'g')
    ax.set_ylim([-7, 7])
    ax.legend(['Observed Data', 'Mean',  'Estimated Derivative', 'True Derivative'])

This gives