Correct way to register a parameter for model in Pytorch
I tried to define a simple model in Pytorch. The model computes negative log prob for a gaussian distribution:
import torch
import torch.nn as nn

class GaussianModel(nn.Module):

    def __init__(self):
        super(GaussianModel, self).__init__()

        self.register_parameter('mean', nn.Parameter(torch.zeros(1),
                                                     requires_grad=True))
        
        self.pdf = torch.distributions.Normal(self.state_dict()['mean'],
                                              torch.tensor([1.0]))
    def forward(self, x):
        return -self.pdf.log_prob(x)

model = GaussianModel()

Then I tried to optimize the mean parameter:
optimizer = torch.optim.SGD(model.parameters(), lr=0.002)
for _ in range(5):
  optimizer.zero_grad()
  nll = model(torch.tensor([3.0], requires_grad=True))
  nll.backward()
  optimizer.step()
  print('mean : ', model.state_dict()['mean'],
                 ' - Negative Loglikelihood : ', nll.item())

But it seems the gradient is zero and mean does not change:
mean :  tensor([0.])  - Negative Loglikelihood :  5.418938636779785
mean :  tensor([0.])  - Negative Loglikelihood :  5.418938636779785
mean :  tensor([0.])  - Negative Loglikelihood :  5.418938636779785
mean :  tensor([0.])  - Negative Loglikelihood :  5.418938636779785
mean :  tensor([0.])  - Negative Loglikelihood :  5.418938636779785

Did I register and use the mean parameter correctly? can autograd compute the gradient for torch.distributions.Normal.log_prob or I should implement the backward() for the model?

You're over complicating registering your parameter. You can just assign a new self.mean attribute to be an nn.Parameter then use it like a tensor for the most part.
nn.Module overrides the __setattr__ method which is called every time you assign a new class attribute. One of the things it does is check to see if you assigned an nn.Parameter type, and if so, it adds it to the modules dictionary of registered parameters.
Because of this, the easiest way to register your parameter is as follows:
import torch
import torch.nn as nn

class GaussianModel(nn.Module):
    def __init__(self):
        super(GaussianModel, self).__init__()
        self.mean = nn.Parameter(torch.zeros(1))
        self.pdf = torch.distributions.Normal(self.mean, torch.tensor([1.0]))

    def forward(self, x):
        return -self.pdf.log_prob(x)

