I’m trying to perform MCMC using a convoluted example of PCA where the posterior is known to be non-gaussian, using the GPLVM module, but I don’t seem to recover the correct posterior (for “X” / latent variable).

A minimal working example, where I code up the model and compare the results to the gplvm class:

```
import pyro, torch
import numpy as np
import pyro.contrib.gp as gp
import pyro.distributions as dist
import matplotlib.pyplot as plt
plt.ion(); plt.style.use('ggplot')
def mcmc(model):
nuts = pyro.infer.NUTS(model)
MCMC = pyro.infer.MCMC(
kernel=nuts,
num_samples=300,
warmup_steps=300,
num_chains=1)
MCMC.run()
X_mcmc = MCMC.get_samples()['X']
plt.scatter(X_mcmc[:, 0, 0], X_mcmc[:, 0, 1], alpha=0.1)
if __name__ == '__main__':
#########################################
# Create Synthetic Data
np.random.seed(42)
n = 2; q = 2; m = 10
X = np.random.normal(size = (n, q))
W = np.random.normal(size = (q, m))
Y = torch.tensor(X @ W).float()
#########################################
# MCMC with model
def model():
zeros = torch.zeros((n, q))
ones = torch.ones((n, q))
X = pyro.sample('X', dist.Normal(zeros, ones))
zeros = torch.tensor([0]).float()
sigma = X @ X.T + torch.eye(n)*1e-3
sigma = torch.cat([sigma[None, ...]]*m, axis=0)
pyro.sample('Y', dist.MultivariateNormal(zeros, sigma), obs=Y.T)
mcmc(model) # a ring
'''
MCMC with pyro's GPLVM doesn't generate the right latent distribution.
'''
X_init = torch.tensor(np.random.normal(size=(n, q))).float()
kernel = gp.kernels.Linear(q)
kernel.variance = pyro.sample('variance', dist.Delta(torch.ones(2)))
gpmodule = gp.models.GPRegression(X_init, Y.T, kernel, jitter=1e-6)
gplvm = gp.models.GPLVM(gpmodule)
mcmc(gplvm.model) # a blob
```