I'd like to build a GP with marginalized hyperparameters.
I have seen that this is possible with the HMC sampler provided in gpflow from this notebook
However, when I tried to run the following code as a first step of this (NOTE this is on gpflow 0.5, an older version), the returned samples are negative, even though the lengthscale and variance need to be positive (negative values would be meaningless).
import numpy as np
from matplotlib import pyplot as plt
import gpflow
from gpflow import hmc
X = np.linspace(-3, 3, 20)
Y = np.random.exponential(np.sin(X) ** 2)
Y = (Y - np.mean(Y)) / np.std(Y)
k = gpflow.kernels.Matern32(1, lengthscales=.2, ARD=False)
m = gpflow.gpr.GPR(X[:, None], Y[:, None], k)
m.kern.lengthscales.prior = gpflow.priors.Gamma(1., 1.)
m.kern.variance.prior = gpflow.priors.Gamma(1., 1.)
# dont want likelihood be a hyperparam now so fixed
m.likelihood.variance = 1e-6
m.likelihood.variance.fixed = True
m.optimize(maxiter=1000)
samples = m.sample(500)
print(samples)
Output:
[[-0.43764571 -0.22753325]
[-0.50418501 -0.11070128]
[-0.5932655 0.00821438]
[-0.70217714 0.05077999]
[-0.77745654 0.09362291]
[-0.79404456 0.13649446]
[-0.83989415 0.27118385]
[-0.90355789 0.29589641]
...
I don't know too much in detail about HMC sampling but I would expect that the sampled posterior hyperparameters are positive, I've checked the code and it seems maybe related to the Log1pe transform, though I failed to figure it out myself.
Any hint on this?
It would be helpful if you specified which GPflow version you are using - especially given that from the output you posted it looks like you are using a really old version of GPflow (pre-1.0), and this is actually something that got improved since. What is happening here (in old GPflow) is that the sample() method returns a single array S x P, where S is the number of samples, and P is the number of free parameters [e.g. for a M x M matrix parameter with lower-triangular transform (such as the Cholesky of the covariance of the approximate posterior, q_sqrt), only M * (M - 1)/2 parameters are actually stored and optimised!]. These are the values in the unconstrained space, i.e. they can take any value whatsoever. Transforms (see gpflow.transforms module) provide the mapping between this value (between plus/minus infinity) and the constrained value (e.g. gpflow.transforms.positive for lengthscales and variances). In old GPflow, the model provides a get_samples_df() method that takes the S x P array returned by sample() and returns a pandas DataFrame with columns for all the trainable parameters which would be what you want. Or, ideally, you would just use a recent version of GPflow, in which the HMC sampler directly returns the DataFrame!