using Turing
using FillArrays
-using Flux
+using Lux
using Plots
-using ReverseDiff
-
-using LinearAlgebra
-using RandomBuilding 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 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 Lux. The network nn_initial we created has two hidden layers with tanh activations and one output layer with sigmoid (σ) activation, as shown below.
@@ -811,280 +809,286 @@
Building a Neura
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.
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.
-
# Construct a neural network using Flux
-nn_initial = Chain(Dense(2, 3, tanh), Dense(3, 2, tanh), Dense(2, 1, σ)) |> f32
+# Construct a neural network using Lux
+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)
+# Initialize the model weights and state
+ps, st = Lux.setup(rng, nn_initial)
-length(parameters_initial) # number of paraemters in NN
+Lux.parameterlength(nn_initial) # number of paraemters in NN20The 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;# Create a regularization term and a Gaussian prior variance term.
+alpha = 0.09
+sigma = sqrt(1.0 / alpha)
+
-3.3333333333333335Inference can now be performed by calling sample. We use the NUTS Hamiltonian Monte Carlo sampler here.
Construct named tuple from a sampled parameter vector. We could also use ComponentArrays here and simply broadcast to avoid doing this. But let’s do it this way to avoid dependencies.
-
+
+
+setprogress!(false)function vector_to_parameters(ps_new::AbstractVector, ps::NamedTuple)
+ @assert length(ps_new) == Lux.parameterlength(ps)
+ i = 1
+ function get_ps(x)
+ z = reshape(view(ps_new, i:(i + length(x) - 1)), size(x))
+ i += length(x)
+ return z
+ end
+ return fmap(get_ps, ps)
+end
+
vector_to_parameters (generic function with 1 method)To interface with external libraries it is often desirable to use the StatefulLuxLayer to automatically handle the neural network states.
-
-
# Perform inference.
-N = 5000
-ch = sample(bayes_nn(xs, ts, length(parameters_initial), reconstruct), NUTS(;adtype=AutoReverseDiff()), N);
-
+
-┌ Info: Found initial step size
-└ ϵ = 0.2const nn = StatefulLuxLayer(nn_initial, st)
+
+# Specify the probabilistic model.
+@model function bayes_nn(xs, ts; sigma = sigma, ps = ps, nn = nn)
+ # Sample the parameters
+ nparameters = Lux.parameterlength(nn_initial)
+ parameters ~ MvNormal(zeros(nparameters), Diagonal(abs2.(sigma .* ones(nparameters))))
+
+ # Forward NN to make predictions
+ preds = Lux.apply(nn, xs, vector_to_parameters(parameters, ps))
+
+ # Observe each prediction.
+ for i in eachindex(ts)
+ ts[i] ~ Bernoulli(preds[i])
+ end
+end
+
bayes_nn (generic function with 2 methods)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.
Inference can now be performed by calling sample. We use the NUTS Hamiltonian Monte Carlo sampler here.
-
-
# Extract all weight and bias parameters.
-theta = convert(Array{Float32}, MCMCChains.group(ch, :parameters).value)
-
+
+5000×20×1 Array{Float32, 3}:
-[:, :, 1] =
- -2.76754 4.33585 0.202649 … -2.83911 5.0453 3.58483 3.63858
- -1.22286 2.81332 0.647483 -4.37165 4.60901 6.17425 5.1122
- -1.28501 3.35274 0.2071 -3.39033 5.12901 5.3266 5.09537
- -0.939225 3.84344 1.6232 -3.00103 5.19338 5.38463 5.00454
- -1.10654 7.12114 0.261793 -3.91847 4.43435 4.35116 2.63475
- -2.31501 6.4595 0.480459 … -3.29219 2.28614 3.00919 2.63038
- -0.188994 2.56895 -0.0450458 -4.24586 5.93678 6.58049 5.05256
- -2.7119 3.76934 -0.70893 -4.79915 4.10487 3.6214 3.52587
- -0.0057128 2.371 -0.251854 -5.65489 5.08035 6.32055 5.58202
- 0.176594 2.81241 -1.13014 -5.36425 5.5353 5.81991 5.45871
- ⋮ ⋱
- 0.614511 6.33012 -0.263359 -2.76436 5.37732 5.38015 3.49351
- 0.321701 1.60552 1.30103 -3.79229 3.78463 3.42431 3.463
- 0.468764 1.42767 1.4261 -3.73313 3.7815 3.49601 3.22722
- 2.85275 6.0128 0.156228 -3.18823 5.69763 7.10259 5.52247
- -6.92752 2.60902 0.356687 … -2.25084 6.97216 4.37522 4.86558
- -2.60802 5.23288 0.324784 -2.26749 4.97215 6.90851 5.91779
- -2.68243 6.44173 -0.888499 -2.83151 4.6619 6.69534 5.54244
- 0.248891 5.80957 2.21389 -3.42257 4.88962 6.57245 6.11376
- -1.16777 9.65887 -0.692307 -3.36406 5.24506 4.36318 4.32127setprogress!(false)
+
+
+# Perform inference.
+N = 2_000
+ch = sample(bayes_nn(reduce(hcat, xs), ts), NUTS(; adtype=AutoTracker()), N);
+
┌ Info: Found initial step size
+└ ϵ = 0.4Now we extract the parameter samples from the sampled chain as θ (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.
+θ = 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(Float32[x1, x2], theta[i, :])[1] for x1 in x1_range, x2 in x2_range]
-contour!(x1_range, x2_range, Z)
+
# A helper to run the nn through data `x` using parameters `θ`
+nn_forward(x, θ) = nn(x, vector_to_parameters(θ, ps))
+
+# Plot the data we have.
+fig = 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], θ[i, :])[1] for x1 in x1_range, x2 in x2_range]
+contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
+fig
+
The contour plot above shows that the MAP method is not too bad at classifying our data.
@@ -1093,224 +1097,224 @@Prediction Visual 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)
- x = convert(Vector{Float32}, x) # Ensure x is Float32
- return mean([nn_forward(x, theta[i, :])[1] for i in 1:10:num])
-end
+
# Return the average predicted value across
+# multiple weights.
+function nn_predict(x, θ, num)
+ num = min(num, size(θ, 1)) # make sure num does not exceed the number of samples
+ return mean([first(nn_forward(x, view(θ, i, :))) for i in 1:10:num])
+endnn_predict (generic function with 1 method)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))
-
-# Ensure x1, x2 are Float32 within the comprehension
-Z = [nn_predict(Float32[x1, x2], theta, n_end)[1] for x1 in x1_range, x2 in x2_range]
-contour!(x1_range, x2_range, Z)
+
# Plot the average prediction.
+fig = 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], θ, n_end)[1] for x1 in x1_range, x2 in x2_range]
+contour!(x1_range, x2_range, Z; linewidth=3, colormap=:seaborn_bright)
+fig
+
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(Float32[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
+
# Number of iterations to plot.
+n_end = 500
+
+anim = @gif for i in 1:n_end
+ plot_data()
+ Z = [nn_forward([x1, x2], θ[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
-
[ Info: Saved animation to /tmp/jl_8l4buVEbwv.gif[ Info: Saved animation to /tmp/jl_NAT2FUXkVe.gif
-
+
+
This has been an introduction to the applications of Turing and Flux in defining Bayesian neural networks.