Bayesian Neural Networks

In this tutorial, we demonstrate how one can implement a Bayesian Neural Network using a combination of Turing and Flux, a suite of machine learning tools. We will use Flux to specify the neural network's layers and Turing to implement the probabilistic inference, with the goal of implementing a classification algorithm.

We will begin with importing the relevant libraries.

using Turing
using FillArrays
using Flux
using Plots
using ReverseDiff

using LinearAlgebra
using Random

# Use reverse_diff due to the number of parameters in neural networks.
Turing.setadbackend(:reversediff)

:reversediff

Our goal here is to use a Bayesian neural network to classify points in an artificial dataset. The code below generates data points arranged in a box-like pattern and displays a graph of the dataset we will be working with.

# Number of points to generate.
N = 80
M = round(Int, N / 4)
Random.seed!(1234)

# Generate artificial data.
x1s = rand(M) * 4.5;
x2s = rand(M) * 4.5;
xt1s = Array([[x1s[i] + 0.5; x2s[i] + 0.5] for i in 1:M])
x1s = rand(M) * 4.5;
x2s = rand(M) * 4.5;
append!(xt1s, Array([[x1s[i] - 5; x2s[i] - 5] for i in 1:M]))

x1s = rand(M) * 4.5;
x2s = rand(M) * 4.5;
xt0s = Array([[x1s[i] + 0.5; x2s[i] - 5] for i in 1:M])
x1s = rand(M) * 4.5;
x2s = rand(M) * 4.5;
append!(xt0s, Array([[x1s[i] - 5; x2s[i] + 0.5] for i in 1:M]))

# Store all the data for later.
xs = [xt1s; xt0s]
ts = [ones(2 * M); zeros(2 * M)]

# Plot data points.
function plot_data()
    x1 = map(e -> e[1], xt1s)
    y1 = map(e -> e[2], xt1s)
    x2 = map(e -> e[1], xt0s)
    y2 = map(e -> e[2], xt0s)

    Plots.scatter(x1, y1; color="red", clim=(0, 1))
    return Plots.scatter!(x2, y2; color="blue", clim=(0, 1))
end

plot_data()

Building a Neural Network

The next step is to define a feedforward neural network where we express our parameters as distributions, and not single points as with traditional neural networks. For this we will use Dense to define liner layers and compose them via Chain, both are neural network primitives from Flux. The network nn_initial we created has two hidden layers with tanh activations and one output layer with sigmoid (σ) activation, as shown below.

The nn_initial is an instance that acts as a function and can take data as inputs and output predictions. We will define distributions on the neural network parameters and use destructure from Flux to extract the parameters as parameters_initial. The function destructure also returns another function reconstruct that can take (new) parameters in and return us a neural network instance whose architecture is the same as nn_initial but with updated parameters.

# Construct a neural network using Flux
nn_initial = Chain(Dense(2, 3, tanh), Dense(3, 2, tanh), Dense(2, 1, σ))

# Extract weights and a helper function to reconstruct NN from weights
parameters_initial, reconstruct = Flux.destructure(nn_initial)

length(parameters_initial) # number of paraemters in NN

20

The probabilistic model specification below creates a parameters variable, which has IID normal variables. The parameters vector represents all parameters of our neural net (weights and biases).

@model function bayes_nn(xs, ts, nparameters, reconstruct; alpha=0.09)
    # Create the weight and bias vector.
    parameters ~ MvNormal(Zeros(nparameters), I / alpha)

    # Construct NN from parameters
    nn = reconstruct(parameters)
    # Forward NN to make predictions
    preds = nn(xs)

    # Observe each prediction.
    for i in 1:length(ts)
        ts[i] ~ Bernoulli(preds[i])
    end
end;

Inference can now be performed by calling sample. We use the NUTS Hamiltonian Monte Carlo sampler here.

# Perform inference.
N = 5000
ch = sample(bayes_nn(hcat(xs...), ts, length(parameters_initial), reconstruct), NUTS(), N);

Now we extract the parameter samples from the sampled chain as theta (this is of size 5000 x 20 where 5000 is the number of iterations and 20 is the number of parameters). We'll use these primarily to determine how good our model's classifier is.

# Extract all weight and bias parameters.
theta = MCMCChains.group(ch, :parameters).value;

Prediction Visualization

We can use MAP estimation to classify our population by using the set of weights that provided the highest log posterior.

# A helper to create NN from weights `theta` and run it through data `x`
nn_forward(x, theta) = reconstruct(theta)(x)

# Plot the data we have.
plot_data()

# Find the index that provided the highest log posterior in the chain.
_, i = findmax(ch[:lp])

# Extract the max row value from i.
i = i.I[1]

# Plot the posterior distribution with a contour plot
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_forward([x1, x2], theta[i, :])[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z)

The contour plot above shows that the MAP method is not too bad at classifying our data.

Now we can visualize our predictions.

$$ p(\tilde{x} | X, \alpha) = \int_{\theta} p(\tilde{x} | \theta) p(\theta | X, \alpha) \approx \sum_{\theta \sim p(\theta | X, \alpha)}f_{\theta}(\tilde{x}) $$

The nn_predict function takes the average predicted value from a network parameterized by weights drawn from the MCMC chain.

# Return the average predicted value across
# multiple weights.
function nn_predict(x, theta, num)
    return mean([nn_forward(x, theta[i, :])[1] for i in 1:10:num])
end;

Next, we use the nn_predict function to predict the value at a sample of points where the x1 and x2 coordinates range between -6 and 6. As we can see below, we still have a satisfactory fit to our data, and more importantly, we can also see where the neural network is uncertain about its predictions much easier---those regions between cluster boundaries.

# Plot the average prediction.
plot_data()

n_end = 1500
x1_range = collect(range(-6; stop=6, length=25))
x2_range = collect(range(-6; stop=6, length=25))
Z = [nn_predict([x1, x2], theta, n_end)[1] for x1 in x1_range, x2 in x2_range]
contour!(x1_range, x2_range, Z)

Suppose we are interested in how the predictive power of our Bayesian neural network evolved between samples. In that case, the following graph displays an animation of the contour plot generated from the network weights in samples 1 to 1,000.

# Number of iterations to plot.
n_end = 500

anim = @gif for i in 1:n_end
    plot_data()
    Z = [nn_forward([x1, x2], theta[i, :])[1] for x1 in x1_range, x2 in x2_range]
    contour!(x1_range, x2_range, Z; title="Iteration $i", clim=(0, 1))
end every 5

This has been an introduction to the applications of Turing and Flux in defining Bayesian neural networks.

Appendix

These tutorials are a part of the TuringTutorials repository, found at: https://github.com/TuringLang/TuringTutorials.

To locally run this tutorial, do the following commands:

using TuringTutorials
TuringTutorials.weave("03-bayesian-neural-network", "03_bayesian-neural-network.jmd")

Computer Information:

Julia Version 1.9.2
Commit e4ee485e909 (2023-07-05 09:39 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-14.0.6 (ORCJIT, znver2)
  Threads: 1 on 16 virtual cores
Environment:
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Package Information:

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  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.9.2
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics v1.9.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.5+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.2+0
  [14a3606d] MozillaCACerts_jll v2022.10.11
  [4536629a] OpenBLAS_jll v0.3.21+4
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.42.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+6
  [83775a58] Zlib_jll v1.2.13+0
  [8e850b90] libblastrampoline_jll v5.8.0+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`