I’m aware PyTorch has Pyro for Bayesian inference and I have a bit of experience with Bayesian regression using PyMC3.
I’ve also heard of people using noise injection as a better regularizer than dropout (e.g. add in some small amount of gaussian noise to each of the outputs of the layers of a neural network). So I tried doing this with a simple linear regression in PyTorch, trying to model it as if I were setting up a Bayesian linear model in pymc3.
def model(x):
    beta = torch.abs(b_sd) * torch.randn(x.shape) + b_mu
    alpha = torch.abs(a_sd) * torch.randn(x.shape) + a_mu
    mu = beta * x + alpha
    sigma = torch.abs(sig) * torch.abs(torch.randn(x.shape)) + sig_mean#
    y = torch.abs(sigma) * torch.randn(x.shape) + mu
    return y

b_mu = Variable(torch.Tensor([2.0]), requires_grad=True)
b_sd = Variable(torch.Tensor([1.0]), requires_grad=True)
a_mu = Variable(torch.Tensor([0.1]), requires_grad=True)
a_sd = Variable(torch.Tensor([1.0]), requires_grad=True)
sig = Variable(torch.Tensor([1.0]), requires_grad=True)
sig_mean = Variable(torch.Tensor([1.0]), requires_grad=True)

loss_fn = torch.nn.MSELoss(size_average=True)
lr = 0.0001
optimizer = torch.optim.SGD(params=[b_mu, b_sd, a_mu, a_sd, sig, sig_mean], lr=lr)


And then I train it like a ordinary linear regression using SGD and mini-batches on some synthetic noisy linear data. After training, the standard deviation variables actually do reflect the amount variance in the training data and in particular, the b_sd variable actually ends up very close to what pymc3 gives me using “exact” bayesian inference from MCMC. This noise-injection technique seems much simpler than most of the probabilistic programming languages/libraries and is just a few extra lines of code compared to a normal regression algorithm.
My question is, what exactly am I doing with this method? This noise injection technique w/ SGD seems to be a way of doing variational inference on a model with conjugate priors (i.e. the prior and posteriors are all gaussians), but I’m still learning the mechanics of variational inference so I’m not sure.
In any case, from a practical standpoint,  even if its not giving me exact variance/standard deviation values, it does seem to quantify uncertainty which is useful and is probably regularizing as well.