Log_prob yields positive values during training

I tried to use normalizing flows to transfer some distribution to the normal distribution. As a transform I used

arn = AutoRegressiveNN(2, [40], param_dims=[count_bins,count_bins,count_bins-1,count_bins])
spline = SplineAutoregressive(2, arn, order='linear', count_bins=count_bins) 
base_dist = dist.Normal(torch.zeros(2), torch.ones(2)) flow_dist = dist.TransformedDistribution(base_dist, [spline])

However, during training the - flow_dist.log_prob(x).mean() command decreased below zero and did not stop decreasing as depicted in the following figure.

125655919-b4cc941a-074d-45e5-a2be-94bcbc2c7a29

I would have expected that - flow_dist.log_prob(x).mean() is strictly greater zero for all inputs. Is this an issue?

Below you find the full MWE. Thanks for any help.

import torch
import numpy as np
from tqdm import tqdm
import pylab as plt
import pyro.distributions as dist
from pyro.nn import AutoRegressiveNN
from pyro.distributions.transforms import SplineAutoregressive

flow_layers = 0
count_bins = 8

arn = AutoRegressiveNN(2, [40], param_dims=[count_bins,count_bins,count_bins-1,count_bins])
spline = SplineAutoregressive(2, arn, order='linear', count_bins=count_bins)

base_dist = dist.Normal(torch.zeros(2), torch.ones(2))
flow_dist = dist.TransformedDistribution(base_dist, [spline])



optimizer = torch.optim.Adam(spline.parameters(), lr=1e-3)
vals = np.zeros((1000, 1))

x = torch.Tensor([[1, 0],
                   [0, 1],
                   [-1, 1],
                   [1, 1],
                   [1, -1],
                   ])

pbar = tqdm(range(vals.shape[0]))
for i in pbar:
  optimizer.zero_grad()

  prob = - flow_dist.log_prob(x).mean()
  vals[i] = prob.item()

  pbar.set_postfix(prob=prob.item())

  prob.backward()
  optimizer.step()
  flow_dist.clear_cache()

Y = flow_dist.sample(torch.Size([1000,]))
plt.figure()
plt.scatter(Y[:,0], Y[:,1], color='firebrick', label='flow', alpha=0.5)

plt.figure()
plt.plot(vals)

plt.show()

UPDATE:

May be it is already a problem in PyTorch. Executing the following code

import torch
from torch import distributions

dist = distributions.Normal(torch.Tensor([0.]),torch.Tensor([0.1]))
print(dist.log_prob(torch.Tensor([0.15])))

results is tensor([0.2586]), which should be a negative number. Is this an issue or do I do something wrong?

UPDATE:
I found this post, which clarifies that log_prob is the log of the density function.

Thanks for asking on the forum! Just to clearly answer: my_dist.log_prob(x) is the log of the density function. While the density must be nonnegative, its log can be any real number.

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