Bayesian Deep Learning


Thomas Wiecki & Maxim Kochurov


June 1, 2016

Variational Inference: Bayesian Neural Networks

  1. 2016-2018 by Thomas Wiecki, updated by Maxim Kochurov

Original blog post:

Bayesian Neural Networks in PyMC3

Generating data

First, lets generate some toy data – a simple binary classification problem that’s not linearly separable.

%matplotlib inline
import theano
import pymc3 as pm
import sklearn
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from warnings import filterwarnings
from sklearn import datasets
from sklearn.preprocessing import scale
from sklearn.cross_validation import train_test_split
from sklearn.datasets import make_moons
/Users/twiecki/anaconda/lib/python3.6/site-packages/h5py/ FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  from ._conv import register_converters as _register_converters
X, Y = make_moons(noise=0.2, random_state=0, n_samples=1000)
X = scale(X)
X = X.astype(floatX)
Y = Y.astype(floatX)
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=.5)
fig, ax = plt.subplots(figsize=(12, 8))
ax.scatter(X[Y==0, 0], X[Y==0, 1], label='Class 0')
ax.scatter(X[Y==1, 0], X[Y==1, 1], color='r', label='Class 1')
sns.despine(); ax.legend()
ax.set(xlabel='X', ylabel='Y', title='Toy binary classification data set');

Model specification

A neural network is quite simple. The basic unit is a perceptron which is nothing more than logistic regression. We use many of these in parallel and then stack them up to get hidden layers. Here we will use 2 hidden layers with 5 neurons each which is sufficient for such a simple problem.

def construct_nn(ann_input, ann_output):
    n_hidden = 5
    # Initialize random weights between each layer
    init_1 = np.random.randn(X.shape[1], n_hidden).astype(floatX)
    init_2 = np.random.randn(n_hidden, n_hidden).astype(floatX)
    init_out = np.random.randn(n_hidden).astype(floatX)
    with pm.Model() as neural_network:
        # Weights from input to hidden layer
        weights_in_1 = pm.Normal('w_in_1', 0, sd=1, 
                                 shape=(X.shape[1], n_hidden), 
        # Weights from 1st to 2nd layer
        weights_1_2 = pm.Normal('w_1_2', 0, sd=1, 
                                shape=(n_hidden, n_hidden), 
        # Weights from hidden layer to output
        weights_2_out = pm.Normal('w_2_out', 0, sd=1, 
        # Build neural-network using tanh activation function
        act_1 = pm.math.tanh(, 
        act_2 = pm.math.tanh(, 
        act_out = pm.math.sigmoid(, 
        # Binary classification -> Bernoulli likelihood
        out = pm.Bernoulli('out', 
                           total_size=Y_train.shape[0] # IMPORTANT for minibatches
    return neural_network

# Trick: Turn inputs and outputs into shared variables. 
# It's still the same thing, but we can later change the values of the shared variable 
# (to switch in the test-data later) and pymc3 will just use the new data. 
# Kind-of like a pointer we can redirect.
# For more info, see:
ann_input = theano.shared(X_train)
ann_output = theano.shared(Y_train)
neural_network = construct_nn(ann_input, ann_output)

That’s not so bad. The Normal priors help regularize the weights. Usually we would add a constant b to the inputs but I omitted it here to keep the code cleaner.

Variational Inference: Scaling model complexity

We could now just run a MCMC sampler like NUTS which works pretty well in this case, but as I already mentioned, this will become very slow as we scale our model up to deeper architectures with more layers.

Instead, we will use ADVI variational inference algorithm which was recently added to PyMC3, and updated to use the operator variational inference (OPVI) framework. This is much faster and will scale better. Note, that this is a mean-field approximation so we ignore correlations in the posterior.

from pymc3.theanof import set_tt_rng, MRG_RandomStreams

with neural_network:
    inference = pm.ADVI()
    approx =, method=inference)
Average Loss = 120.67: 100%|██████████| 50000/50000 [00:47<00:00, 1055.45it/s]
Finished [100%]: Average Loss = 120.51
CPU times: user 40.6 s, sys: 17.8 s, total: 58.4 s
Wall time: 48.9 s

Performance wise that’s pretty good considering that NUTS is having a really hard time. Further below we make this even faster. To make it really fly, we probably want to run the Neural Network on the GPU.

As samples are more convenient to work with, we can very quickly draw samples from the variational approximation using the sample method (this is just sampling from Normal distributions, so not at all the same like MCMC):

trace = approx.sample(draws=5000)

Plotting the objective function (ELBO) we can see that the optimization slowly improves the fit over time.


Now that we trained our model, lets predict on the hold-out set using a posterior predictive check (PPC).

# Replace arrays our NN references with the test data

with neural_network:
    ppc = pm.sample_ppc(trace, samples=500, progressbar=False)

# Use probability of > 0.5 to assume prediction of class 1
pred = ppc['out'].mean(axis=0) > 0.5

Let’s look at our predictions:

fig, ax = plt.subplots()
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
ax.set(title='Predicted labels in testing set', xlabel='X', ylabel='Y');

print('Accuracy = {}%'.format((Y_test == pred).mean() * 100))
Accuracy = 95.0%

Hey, our neural network did all right!

Lets look at what the classifier has learned

For this, we evaluate the class probability predictions on a grid over the whole input space.

grid = pm.floatX(np.mgrid[-3:3:100j,-3:3:100j])
grid_2d = grid.reshape(2, -1).T
dummy_out = np.ones(grid.shape[1], dtype=np.int8)

with neural_network:
    ppc = pm.sample_ppc(trace, samples=500, progressbar=False)

Probability surface

cmap = sns.diverging_palette(250, 12, s=85, l=25, as_cmap=True)
fig, ax = plt.subplots(figsize=(14, 8))
contour = ax.contourf(grid[0], grid[1], ppc['out'].mean(axis=0).reshape(100, 100), cmap=cmap)
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
cbar = plt.colorbar(contour, ax=ax)
_ = ax.set(xlim=(-3, 3), ylim=(-3, 3), xlabel='X', ylabel='Y');'Posterior predictive mean probability of class label = 0');

Uncertainty in predicted value

So far, everything I showed we could have done with a non-Bayesian Neural Network. The mean of the posterior predictive for each class-label should be identical to maximum likelihood predicted values. However, we can also look at the standard deviation of the posterior predictive to get a sense for the uncertainty in our predictions. Here is what that looks like:

cmap = sns.cubehelix_palette(light=1, as_cmap=True)
fig, ax = plt.subplots(figsize=(14, 8))
contour = ax.contourf(grid[0], grid[1], ppc['out'].std(axis=0).reshape(100, 100), cmap=cmap)
ax.scatter(X_test[pred==0, 0], X_test[pred==0, 1])
ax.scatter(X_test[pred==1, 0], X_test[pred==1, 1], color='r')
cbar = plt.colorbar(contour, ax=ax)
_ = ax.set(xlim=(-3, 3), ylim=(-3, 3), xlabel='X', ylabel='Y');'Uncertainty (posterior predictive standard deviation)');

We can see that very close to the decision boundary, our uncertainty as to which label to predict is highest. You can imagine that associating predictions with uncertainty is a critical property for many applications like health care. To further maximize accuracy, we might want to train the model primarily on samples from that high-uncertainty region.

Mini-batch ADVI

So far, we have trained our model on all data at once. Obviously this won’t scale to something like ImageNet. Moreover, training on mini-batches of data (stochastic gradient descent) avoids local minima and can lead to faster convergence.

Fortunately, ADVI can be run on mini-batches as well. It just requires some setting up:

minibatch_x = pm.Minibatch(X_train, batch_size=32)
minibatch_y = pm.Minibatch(Y_train, batch_size=32)

neural_network_minibatch = construct_nn(minibatch_x, minibatch_y)
with neural_network_minibatch:
    inference = pm.ADVI()
    approx =, method=inference)
Average Loss = 141.38: 100%|██████████| 40000/40000 [00:33<00:00, 1210.30it/s]
Finished [100%]: Average Loss = 141.36

As you can see, mini-batch ADVI’s running time is much lower. It also seems to converge faster.

For fun, we can also look at the trace. The point is that we also get uncertainty of our Neural Network weights.



Hopefully this blog post demonstrated a very powerful new inference algorithm available in PyMC3: ADVI. I also think bridging the gap between Probabilistic Programming and Deep Learning can open up many new avenues for innovation in this space, as discussed above. Specifically, a hierarchical neural network sounds pretty bad-ass. These are really exciting times.

Next steps

Theano, which is used by PyMC3 as its computational backend, was mainly developed for estimating neural networks and there are great libraries like Lasagne that build on top of Theano to make construction of the most common neural network architectures easy. See my follow-up blog post on how to use Lasagne together with PyMC3.

You can also run this example on the GPU by setting device = gpu and floatX = float32 in your .theanorc.

You might argue that the above network isn’t really deep, but note that we could easily extend it to have more layers, including convolutional ones to train on more challenging data sets, as demonstrated [here](follow-up blog post on how to use Lasagne together with PyMC3.

I also presented some of this work at PyData London, view the video below:

Finally, you can download this NB here. Leave a comment below, and follow me on twitter.


Taku Yoshioka did a lot of work on the original ADVI implementation in PyMC3. I’d also like to the thank the Stan guys (specifically Alp Kucukelbir and Daniel Lee) for deriving ADVI and teaching us about it. Thanks also to Chris Fonnesbeck, Andrew Campbell, Taku Yoshioka, and Peadar Coyle for useful comments on an earlier draft.