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Natural-Parameter Networks (NPN) in PyTorch

This is the PyTorch code for the NIPS paper 'Natural-Parameter Networks: A Class of Probabilistic Neural Networks'.

It is a class of probabilistic neural networks that treat both weights and neurons as distributions rather than just points in high-dimensional space. Distributions are first-citizens in the networks. The design allows distributions to feedforward and backprop across the network. Given an input data point, NPN will output a predicted distribution with information on both the prediction and uncertainty.

NPN can be used either independently or as a building block for Bayesian Deep Learning (BDL).

Note that this is the code for Gaussian NPN to run on the MNIST and Boston Housing datasets. For Gamma NPN or Poisson NPN please go to the other repo.

Neural networks v.s. natural-parameter-networks in two figures:

Distributions as first-class citizens:

Closed-form operations to handle uncertainty:

Example results on uncertainty-aware prediction:

Output both prediction and uncertainty for regression:

Above is the predictive distribution for NPN. The shaded regions correspond to 3 standard deviations. The black curve is the data-generating function and blue curves show the mean of the predictive distributions. Red stars are the training data.

Accuracy versus uncertainty (variance):

Above is the classification accuracy for different variance (uncertainty). Note that ‘1’ in the x-axis means the variance is in the range [0, 0.04), ‘2’ means the variance is in the range [0.04, 0.08), etc.


Using only 100 training samples in the training set of MNIST:

Method Accuracy
NPN (ours) 74.58%
MLP 69.02%
CNN+NPN (ours) 86.87%
CNN+MLP 82.90%


Regression task on Boston Housing:

Method RMSE
NPN (ours) 3.2197
MLP 3.5748

How to run the code:

  • In general, to train the model, run the command: 'sh'
  • To train NPN (fully connected), run the command: 'sh'
  • To train MLP (fully connected), run the command: 'sh'
  • To train CNN+NPN, run the command: 'sh'
  • To train CNN+MLP, run the command: 'sh'
  • For regression tasks (Boston Housing) using NPN, run the command: 'sh'
  • For regression tasks (Boston Housing) using MLP, run the command: 'sh'

Short code example:

This is everything to implement a three-layer NPN on PyTorch (essentially only need to replace nn.Linear with NPNLinear):

from npn import NPNLinear
from npn import NPNSigmoid
class NPNNet(nn.Module):
    def __init__(self):
        super(NPNNet, self).__init__()

        # Last parameter of NPNLinear
        # True: input contains both the mean and variance
        # False: input contains only the mean
        self.fc1 = NPNLinear(784, 800, False)
        self.sigmoid1 = NPNSigmoid()
        self.fc2 = NPNLinear(800, 800)
        self.sigmoid2 = NPNSigmoid()
        self.fc3 = NPNLinear(800, 10)
        self.sigmoid3 = NPNSigmoid()

    def forward(self, x):
        x = self.sigmoid1(self.fc1(x))
        x = self.sigmoid2(self.fc2(x))
        # output mean (x) and variance (s) of Gaussian NPN
        x, s = self.sigmoid3(self.fc3(x))
        return x, s 


The code is tested under PyTorch 0.2.03 and Python 3.5.2.

Official Matlab implementation:

The official Matlab version (with GPU support) can be found here

Other implementations (third-party):

Another version of Pytorch/Python code (with extension to GRU) by sohamghosh121.


Natural-Parameter Networks: A Class of Probabilistic Neural Networks

  author    = {Hao Wang and
               Xingjian Shi and
               Dit{-}Yan Yeung},
  title     = {Natural-Parameter Networks: {A} Class of Probabilistic Neural Networks},
  booktitle = {Advances in Neural Information Processing Systems 29: Annual Conference
               on Neural Information Processing Systems 2016, December 5-10, 2016,
               Barcelona, Spain},
  pages     = {118--126},
  year      = {2016}


Officially unofficial PyTorch code for the NIPS paper 'Natural-Parameter Networks: A Class of Probabilistic Neural Networks'






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