TensorFlow Probability 0.10 : ガイド : 概要 – TensorFlow Probability のツアー (翻訳/解説)

翻訳 : (株)クラスキャット セールスインフォメーション
作成日時 : 06/17/2020 (0.10.0)

* 本ページは、TensorFlow Probability の以下のドキュメントを翻訳した上で適宜、補足説明したものです：

• A Tour of TensorFlow Probability

* サンプルコードの動作確認はしておりますが、必要な場合には適宜、追加改変しています。
* ご自由にリンクを張って頂いてかまいませんが、sales-info@classcat.com までご一報いただけると嬉しいです。

 

ガイド : 概要 – TensorFlow Probability のツアー

依存性 & 要件

!pip install -q --upgrade seaborn

#@title Import { display-mode: "form" }


from pprint import pprint
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns

import tensorflow.compat.v2 as tf
tf.enable_v2_behavior()

import tensorflow_probability as tfp

sns.reset_defaults()
sns.set_context(context='talk',font_scale=0.7)
plt.rcParams['image.cmap'] = 'viridis'

%matplotlib inline

tfd = tfp.distributions
tfb = tfp.bijectors

#@title Utils { display-mode: "form" }
def print_subclasses_from_module(module, base_class, maxwidth=80):
  import functools, inspect, sys
  subclasses = [name for name, obj in inspect.getmembers(module)
                if inspect.isclass(obj) and issubclass(obj, base_class)]
  def red(acc, x):
    if not acc or len(acc[-1]) + len(x) + 2 > maxwidth:
      acc.append(x)
    else:
      acc[-1] += ", " + x
    return acc
  print('\n'.join(functools.reduce(red, subclasses, [])))

 

アウトライン

• TensorFlow
• TensorFlow Probability
  • Distributions
  • Bijectors
  • MCMC
  • …and more!

 

序: TensorFlow

TensorFlow は科学計算ライブラリです。

それは以下をサポートします :

• 多くの数学的演算
• 効率的なベクトル化計算
• 容易なハードウェア・アクセラレーション
• 自動微分

 

ベクトル化

• ベクトル化は物事を高速にします！
• それはまた shape について多く考えることも意味します。

mats = tf.random.uniform(shape=[1000, 10, 10])
vecs = tf.random.uniform(shape=[1000, 10, 1])

def for_loop_solve():
  return np.array(
    [tf.linalg.solve(mats[i, ...], vecs[i, ...]) for i in range(1000)])

def vectorized_solve():
  return tf.linalg.solve(mats, vecs)

# Vectorization for the win!
%timeit for_loop_solve()
%timeit vectorized_solve()

1 loops, best of 3: 2.49 s per loop
1000 loops, best of 3: 836 µs per loop

 

ハードウェア・アクセラレーション

# Code can run seamlessly on a GPU, just change Colab runtime type
# in the 'Runtime' menu.
if tf.test.gpu_device_name() == '/device:GPU:0':
  print("Using a GPU")
else:
  print("Using a CPU")

Using a CPU

 

自動微分

a = tf.constant(np.pi)
b = tf.constant(np.e)
with tf.GradientTape() as tape:
  tape.watch([a, b])
  c = .5 * (a**2 + b**2)
grads = tape.gradient(c, [a, b])
print(grads[0])
print(grads[1])

tf.Tensor(3.1415927, shape=(), dtype=float32)
tf.Tensor(2.7182817, shape=(), dtype=float32)

 

TensorFlow Probability

TensorFlow Probability は TensorFlow で確率的推論と統計的分析のためのライブラリです。

低位モジュールコンポーネントの構成を通してモデル化、推論と評価をサポートします。

 

低位ビルディングブロック

• Distributions
• Bijectors

 

高位構成概念

• マルコフ連鎖モンテカルロ法
• 確率的層
• 構造時系列
• 一般化線形モデル
• Edward2
• Optimizers

 

Distributions

tfp.distributions.Distribution は 2 つの中心的なメソッド: sample と log_prob を持つクラスです。

TFP has a lot of distributions!

print_subclasses_from_module(tfp.distributions, tfp.distributions.Distribution)

Autoregressive, BatchReshape, Bernoulli, Beta, BetaBinomial, Binomial, Blockwise
Categorical, Cauchy, Chi, Chi2, CholeskyLKJ, ContinuousBernoulli, Deterministic
Dirichlet, DirichletMultinomial, Distribution, DoublesidedMaxwell, Empirical
ExpRelaxedOneHotCategorical, Exponential, FiniteDiscrete, Gamma, GammaGamma
GaussianProcess, GaussianProcessRegressionModel, GeneralizedPareto, Geometric
Gumbel, HalfCauchy, HalfNormal, HalfStudentT, HiddenMarkovModel, Horseshoe
Independent, InverseGamma, InverseGaussian, JointDistribution
JointDistributionCoroutine, JointDistributionCoroutineAutoBatched
JointDistributionNamed, JointDistributionNamedAutoBatched
JointDistributionSequential, JointDistributionSequentialAutoBatched, Kumaraswamy
LKJ, Laplace, LinearGaussianStateSpaceModel, LogNormal, Logistic, LogitNormal
Mixture, MixtureSameFamily, Multinomial, MultivariateNormalDiag
MultivariateNormalDiagPlusLowRank, MultivariateNormalFullCovariance
MultivariateNormalLinearOperator, MultivariateNormalTriL
MultivariateStudentTLinearOperator, NegativeBinomial, Normal, OneHotCategorical
OrderedLogistic, PERT, Pareto, PixelCNN, PlackettLuce, Poisson
PoissonLogNormalQuadratureCompound, ProbitBernoulli, QuantizedDistribution
RelaxedBernoulli, RelaxedOneHotCategorical, Sample, SinhArcsinh, StudentT
StudentTProcess, TransformedDistribution, Triangular, TruncatedNormal, Uniform
VariationalGaussianProcess, VectorDeterministic, VectorDiffeomixture
VectorExponentialDiag, VonMises, VonMisesFisher, WishartLinearOperator
WishartTriL, Zipf

 

単純なスカラー変量分布

# A standard normal
normal = tfd.Normal(loc=0., scale=1.)
print(normal)

tfp.distributions.Normal("Normal", batch_shape=[], event_shape=[], dtype=float32)

# Plot 1000 samples from a standard normal
samples = normal.sample(1000)
sns.distplot(samples)
plt.title("Samples from a standard Normal")
plt.show()

# Compute the log_prob of a point in the event space of `normal`
normal.log_prob(0.)

<tf.Tensor: shape=(), dtype=float32, numpy=-0.9189385>

# Compute the log_prob of a few points
normal.log_prob([-1., 0., 1.])

<tf.Tensor: shape=(3,), dtype=float32, numpy=array([-1.4189385, -0.9189385, -1.4189385], dtype=float32)>

 

Distributions と Shapes

Numpy ndarray と TensorFlow Tensor は shape を持ちます。

TensorFlow Probability Distributions は shape セマンティクスを持ちます — shapes を意味的に別個のピースに分割します、全体総てのためにメモリの同じチャンク (Tensor/ndarray) が利用されてさえも。

• バッチ shape は制限パラメータを持つ Distributions のコレクション を意味します。
• 事象 (= Event) shape は Distribution からの サンプリングの shape を意味します。

常にバッチ shape は「左側」にそして事象 shape は「右側」に置きます。

 

スカラー変量分布のバッチ

バッチは「ベクトル化」分布のようなものです : 独立インスタンスでその計算は並列に発生します。

# Create a batch of 3 normals, and plot 1000 samples from each
normals = tfd.Normal([-2.5, 0., 2.5], 1.)  # The scale parameter broadacasts!
print("Batch shape:", normals.batch_shape)
print("Event shape:", normals.event_shape)

Batch shape: (3,)
Event shape: ()

# Samples' shapes go on the left!
samples = normals.sample(1000)
print("Shape of samples:", samples.shape)

Shape of samples: (1000, 3)

# Sample shapes can themselves be more complicated
print("Shape of samples:", normals.sample([10, 10, 10]).shape)

Shape of samples: (10, 10, 10, 3)

# A batch of normals gives a batch of log_probs.
print(normals.log_prob([-2.5, 0., 2.5]))

tf.Tensor([-0.9189385 -0.9189385 -0.9189385], shape=(3,), dtype=float32)

# The computation broadcasts, so a batch of normals applied to a scalar
# also gives a batch of log_probs.
print(normals.log_prob(0.))

tf.Tensor([-4.0439386 -0.9189385 -4.0439386], shape=(3,), dtype=float32)

# Normal numpy-like broadcasting rules apply!
xs = np.linspace(-6, 6, 200)
try:
  normals.log_prob(xs)
except Exception as e:
  print("TFP error:", e.message)

TFP error: Incompatible shapes: [200] vs. [3] [Op:SquaredDifference]

# That fails for the same reason this does:
try:
  np.zeros(200) + np.zeros(3)
except Exception as e:
  print("Numpy error:", e)

Numpy error: operands could not be broadcast together with shapes (200,) (3,) 

# But this would work:
a = np.zeros([200, 1]) + np.zeros(3)
print("Broadcast shape:", a.shape)

Broadcast shape: (200, 3)

# And so will this!
xs = np.linspace(-6, 6, 200)[..., np.newaxis]
# => shape = [200, 1]

lps = normals.log_prob(xs)
print("Broadcast log_prob shape:", lps.shape)

Broadcast log_prob shape: (200, 3)

# Summarizing visually
for i in range(3):
  sns.distplot(samples[:, i], kde=False, norm_hist=True)
plt.plot(np.tile(xs, 3), normals.prob(xs), c='k', alpha=.5)
plt.title("Samples from 3 Normals, and their PDF's")
plt.show()

 

ベクトル変量分布

mvn = tfd.MultivariateNormalDiag(loc=[0., 0.], scale_diag = [1., 1.])
print("Batch shape:", mvn.batch_shape)
print("Event shape:", mvn.event_shape)

Batch shape: ()
Event shape: (2,)

samples = mvn.sample(1000)
print("Samples shape:", samples.shape)

Samples shape: (1000, 2)

 

行列変量分布

lkj = tfd.LKJ(dimension=10, concentration=[1.5, 3.0])
print("Batch shape: ", lkj.batch_shape)
print("Event shape: ", lkj.event_shape)

Batch shape:  (2,)
Event shape:  (10, 10)

samples = lkj.sample()
print("Samples shape: ", samples.shape)

Samples shape:  (2, 10, 10)

fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(6, 3))
sns.heatmap(samples[0, ...], ax=axes[0], cbar=False)
sns.heatmap(samples[1, ...], ax=axes[1], cbar=False)
fig.tight_layout()
plt.show()

 

ガウス過程

kernel = tfp.math.psd_kernels.ExponentiatedQuadratic()
xs = np.linspace(-5., 5., 200).reshape([-1, 1])
gp = tfd.GaussianProcess(kernel, index_points=xs)
print("Batch shape:", gp.batch_shape)
print("Event shape:", gp.event_shape)

Batch shape: ()
Event shape: (200,)

upper, lower = gp.mean() + [2 * gp.stddev(), -2 * gp.stddev()]
plt.plot(xs, gp.mean())
plt.fill_between(xs[..., 0], upper, lower, color='k', alpha=.1)
for _ in range(5):
  plt.plot(xs, gp.sample(), c='r', alpha=.3)
plt.title(r"GP prior mean, $2\sigma$ intervals, and samples")
plt.show()

#    *** Bonus question ***
# Why do so many of these functions lie outside the 95% intervals?

 

GP 回帰

# Suppose we have some observed data
obs_x = [[-3.], [0.], [2.]]  # Shape 3x1 (3 1-D vectors)
obs_y = [3., -2., 2.]        # Shape 3   (3 scalars)

gprm = tfd.GaussianProcessRegressionModel(kernel, xs, obs_x, obs_y)

upper, lower = gprm.mean() + [2 * gprm.stddev(), -2 * gprm.stddev()]
plt.plot(xs, gprm.mean())
plt.fill_between(xs[..., 0], upper, lower, color='k', alpha=.1)
for _ in range(5):
  plt.plot(xs, gprm.sample(), c='r', alpha=.3)
plt.scatter(obs_x, obs_y, c='k', zorder=3)
plt.title(r"GP posterior mean, $2\sigma$ intervals, and samples")
plt.show()

 

Bijector

Bijector は (殆どの場合) 可逆 (= invertible)、滑らかな関数を表します。それらは分布を変換するために使用できて、サンプリングして log_probs を計算する能力を維持します。それらは tfp.bijectors モジュールに在り得ます。

各 bijector は少なくとも 3 つのメソッドを実装します :

• forward,
• inverse そして
• (少なくとも) forward_log_det_jacobian と inverse_log_det_jacobian の一つ。

これらの構成要素で、分布を変換して結果から依然としてサンプルと log prob を得ることができます！

数学では、幾分雑です :

• $X$ は pdf $p(x)$ を持つ確率変数です。
• $g$ は $X$ の空間上の滑らかで、可逆な関数です。
• $Y = g(X)$ は新しい、変換された確率変数です。
• $p(Y=y) = p(X=g^{-1}(y)) \cdot |\nabla g^{-1}(y)|$

 

キャッシング

Bijector はまた forward と inverse 計算、そして log-det-Jacobians をキャッシュもします、これは潜在的に非常に高価な演算を繰り返すことを節約します！

print_subclasses_from_module(tfp.bijectors, tfp.bijectors.Bijector)

AbsoluteValue, Affine, AffineLinearOperator, AffineScalar, BatchNormalization
Bijector, Blockwise, Chain, CholeskyOuterProduct, CholeskyToInvCholesky
CorrelationCholesky, Cumsum, DiscreteCosineTransform, Exp, Expm1, FFJORD
FillScaleTriL, FillTriangular, FrechetCDF, GeneralizedPareto, GumbelCDF
Identity, Inline, Invert, IteratedSigmoidCentered, KumaraswamyCDF, LambertWTail
Log, Log1p, MaskedAutoregressiveFlow, MatrixInverseTriL, MatvecLU, NormalCDF
Ordered, Pad, Permute, PowerTransform, RationalQuadraticSpline, RealNVP
Reciprocal, Reshape, Scale, ScaleMatvecDiag, ScaleMatvecLU
ScaleMatvecLinearOperator, ScaleMatvecTriL, ScaleTriL, Shift, Sigmoid
SinhArcsinh, SoftClip, Softfloor, SoftmaxCentered, Softplus, Softsign, Square
Tanh, TransformDiagonal, Transpose, WeibullCDF

 

単純な Bijector

normal_cdf = tfp.bijectors.NormalCDF()
xs = np.linspace(-4., 4., 200)
plt.plot(xs, normal_cdf.forward(xs))
plt.show()

plt.plot(xs, normal_cdf.forward_log_det_jacobian(xs, event_ndims=0))
plt.show()

 

分布を変換する Bijector

exp_bijector = tfp.bijectors.Exp()
log_normal = exp_bijector(tfd.Normal(0., .5))

samples = log_normal.sample(1000)
xs = np.linspace(1e-10, np.max(samples), 200)
sns.distplot(samples, norm_hist=True, kde=False)
plt.plot(xs, log_normal.prob(xs), c='k', alpha=.75)
plt.show()

 

Bijector をバッチ化する

# Create a batch of bijectors of shape [3,]
softplus = tfp.bijectors.Softplus(
  hinge_softness=[1., .5, .1])
print("Hinge softness shape:", softplus.hinge_softness.shape)

Hinge softness shape: (3,)

# For broadcasting, we want this to be shape [200, 1]
xs = np.linspace(-4., 4., 200)[..., np.newaxis]
ys = softplus.forward(xs)
print("Forward shape:", ys.shape)

Forward shape: (200, 3)

# Visualization
lines = plt.plot(np.tile(xs, 3), ys)
for line, hs in zip(lines, softplus.hinge_softness):
  line.set_label("Softness: %1.1f" % hs)
plt.legend()
plt.show()

 

キャッシング

# This bijector represents a matrix outer product on the forward pass,
# and a cholesky decomposition on the inverse pass. The latter costs O(N^3)!
bij = tfb.CholeskyOuterProduct()

size = 2500
# Make a big, lower-triangular matrix
big_lower_triangular = tf.eye(size)
# Squaring it gives us a positive-definite matrix
big_positive_definite = bij.forward(big_lower_triangular)

# Caching for the win!
%timeit bij.inverse(big_positive_definite)
%timeit tf.linalg.cholesky(big_positive_definite)

10000 loops, best of 3: 112 µs per loop
1 loops, best of 3: 437 ms per loop

 

MCMC

TFP はハミルトニアン・モンテカルロを含む、幾つかの標準的なマルコフ連鎖モンテカルロ・アルゴリズムのための組込みサポートを持ちます。

 

データセットを生成する

# Generate some data
def f(x, w):
  # Pad x with 1's so we can add bias via matmul
  x = tf.pad(x, [[1, 0], [0, 0]], constant_values=1)
  linop = tf.linalg.LinearOperatorFullMatrix(w[..., np.newaxis])
  result = linop.matmul(x, adjoint=True)
  return result[..., 0, :]

num_features = 2
num_examples = 50
noise_scale = .5
true_w = np.array([-1., 2., 3.])

xs = np.random.uniform(-1., 1., [num_features, num_examples])
ys = f(xs, true_w) + np.random.normal(0., noise_scale, size=num_examples)

# Visualize the data set
plt.scatter(*xs, c=ys, s=100, linewidths=0)

grid = np.meshgrid(*([np.linspace(-1, 1, 100)] * 2))
xs_grid = np.stack(grid, axis=0)
fs_grid = f(xs_grid.reshape([num_features, -1]), true_w)
fs_grid = np.reshape(fs_grid, [100, 100])
plt.colorbar()
plt.contour(xs_grid[0, ...], xs_grid[1, ...], fs_grid, 20, linewidths=1)
plt.show()

 

結合 log-prob 関数を定義する

非正規化事後 (分布) は結合 log prob の部分適用を形成するためデータに渡り閉じた結果です。

# Define the joint_log_prob function, and our unnormalized posterior.
def joint_log_prob(w, x, y):
  # Our model in maths is
  #   w ~ MVN([0, 0, 0], diag([1, 1, 1]))
  #   y_i ~ Normal(w @ x_i, noise_scale),  i=1..N

  rv_w = tfd.MultivariateNormalDiag(
    loc=np.zeros(num_features + 1),
    scale_diag=np.ones(num_features + 1))

  rv_y = tfd.Normal(f(x, w), noise_scale)
  return (rv_w.log_prob(w) +
          tf.reduce_sum(rv_y.log_prob(y), axis=-1))

# Create our unnormalized target density by currying x and y from the joint.
def unnormalized_posterior(w):
  return joint_log_prob(w, xs, ys)

 

HMC TransitionKernel を構築して sample_chain を呼び出す

# Create an HMC TransitionKernel
hmc_kernel = tfp.mcmc.HamiltonianMonteCarlo(
  target_log_prob_fn=unnormalized_posterior,
  step_size=np.float64(.1),
  num_leapfrog_steps=2)

# We wrap sample_chain in tf.function, telling TF to precompile a reusable
# computation graph, which will dramatically improve performance.
@tf.function
def run_chain(initial_state, num_results=1000, num_burnin_steps=500):
  return tfp.mcmc.sample_chain(
    num_results=num_results,
    num_burnin_steps=num_burnin_steps,
    current_state=initial_state,
    kernel=hmc_kernel,
    trace_fn=lambda current_state, kernel_results: kernel_results)

initial_state = np.zeros(num_features + 1)
samples, kernel_results = run_chain(initial_state)
print("Acceptance rate:", kernel_results.is_accepted.numpy().mean())

Acceptance rate: 0.913

That’s not great! We’d like an acceptance rate closer to .65.

(“Optimal Scaling for Various Metropolis-Hastings Algorithms“, Roberts & Rosenthal, 2001 参照)

 

適応可能な (= adaptive) ステップサイズ

HMC TransitionKernel を SimpleStepSizeAdaptation “meta-kernel” にラップすることができます、これは burnin の間に HMC ステップサイズを適応させるために幾つかの (むしろ単純なヒューリスティックな) ロジックを適用します。ステップサイズを適応させるために burnin の 80% を割り当ててから残りの 20% を単に混合 (= mixing) に進ませます。

# Apply a simple step size adaptation during burnin
@tf.function
def run_chain(initial_state, num_results=1000, num_burnin_steps=500):
  adaptive_kernel = tfp.mcmc.SimpleStepSizeAdaptation(
      hmc_kernel,
      num_adaptation_steps=int(.8 * num_burnin_steps),
      target_accept_prob=np.float64(.65))

  return tfp.mcmc.sample_chain(
    num_results=num_results,
    num_burnin_steps=num_burnin_steps,
    current_state=initial_state,
    kernel=adaptive_kernel,
    trace_fn=lambda cs, kr: kr)

samples, kernel_results = run_chain(
  initial_state=np.zeros(num_features+1))
print("Acceptance rate:", kernel_results.inner_results.is_accepted.numpy().mean())

Acceptance rate: 0.431

# Trace plots
colors = ['b', 'g', 'r']
for i in range(3):
  plt.plot(samples[:, i], c=colors[i], alpha=.3)
  plt.hlines(true_w[i], 0, 1000, zorder=4, color=colors[i], label="$w_{}$".format(i))
plt.legend(loc='upper right')
plt.show()

# Histogram of samples
for i in range(3):
  sns.distplot(samples[:, i], color=colors[i])
ymax = plt.ylim()[1]
for i in range(3):
  plt.vlines(true_w[i], 0, ymax, color=colors[i])
plt.ylim(0, ymax)
plt.show()

 

診断

追跡プロットは素晴らしいですが、診断 (= diagnostics) は更に素晴らしいです！

最初に多重連鎖を実行する必要があります。これは initial_state tensor のバッチを与えるくらいに単純です。

# Instead of a single set of initial w's, we create a batch of 8.
num_chains = 8
initial_state = np.zeros([num_chains, num_features + 1])

chains, kernel_results = run_chain(initial_state)

r_hat = tfp.mcmc.potential_scale_reduction(chains)
print("Acceptance rate:", kernel_results.inner_results.is_accepted.numpy().mean())
print("R-hat diagnostic (per latent variable):", r_hat.numpy())

Acceptance rate: 0.6545
R-hat diagnostic (per latent variable): [1.00242245 1.00223462 1.00079678]

 

ノイズスケールのサンプリング

# Define the joint_log_prob function, and our unnormalized posterior.
def joint_log_prob(w, sigma, x, y):
  # Our model in maths is
  #   w ~ MVN([0, 0, 0], diag([1, 1, 1]))
  #   y_i ~ Normal(w @ x_i, noise_scale),  i=1..N

  rv_w = tfd.MultivariateNormalDiag(
    loc=np.zeros(num_features + 1),
    scale_diag=np.ones(num_features + 1))
  
  rv_sigma = tfd.LogNormal(np.float64(1.), np.float64(5.))

  rv_y = tfd.Normal(f(x, w), sigma[..., np.newaxis])
  return (rv_w.log_prob(w) +
          rv_sigma.log_prob(sigma) +
          tf.reduce_sum(rv_y.log_prob(y), axis=-1))

# Create our unnormalized target density by currying x and y from the joint.
def unnormalized_posterior(w, sigma):
  return joint_log_prob(w, sigma, xs, ys)


# Create an HMC TransitionKernel
hmc_kernel = tfp.mcmc.HamiltonianMonteCarlo(
  target_log_prob_fn=unnormalized_posterior,
  step_size=np.float64(.1),
  num_leapfrog_steps=4)



# Create a TransformedTransitionKernl
transformed_kernel = tfp.mcmc.TransformedTransitionKernel(
    inner_kernel=hmc_kernel,
    bijector=[tfb.Identity(),    # w
              tfb.Invert(tfb.Softplus())])   # sigma


# Apply a simple step size adaptation during burnin
@tf.function
def run_chain(initial_state, num_results=1000, num_burnin_steps=500):
  adaptive_kernel = tfp.mcmc.SimpleStepSizeAdaptation(
      transformed_kernel,
      num_adaptation_steps=int(.8 * num_burnin_steps),
      target_accept_prob=np.float64(.75))

  return tfp.mcmc.sample_chain(
    num_results=num_results,
    num_burnin_steps=num_burnin_steps,
    current_state=initial_state,
    kernel=adaptive_kernel,
    trace_fn=lambda cs, kr: kr)


# Instead of a single set of initial w's, we create a batch of 8.
num_chains = 8
initial_state = [np.zeros([num_chains, num_features + 1]),
                 .54 * np.ones([num_chains], dtype=np.float64)]

chains, kernel_results = run_chain(initial_state)

r_hat = tfp.mcmc.potential_scale_reduction(chains)
print("Acceptance rate:", kernel_results.inner_results.inner_results.is_accepted.numpy().mean())
print("R-hat diagnostic (per w variable):", r_hat[0].numpy())
print("R-hat diagnostic (sigma):", r_hat[1].numpy())

Acceptance rate: 0.75925
R-hat diagnostic (per w variable): [1.0009577  1.00296745 1.00102734]
R-hat diagnostic (sigma): 1.0042382564283778

w_chains, sigma_chains = chains

# Trace plots of w (one of 8 chains)
colors = ['b', 'g', 'r', 'teal']
fig, axes = plt.subplots(4, num_chains, figsize=(4 * num_chains, 8))
for j in range(num_chains):
  for i in range(3):
    ax = axes[i][j]
    ax.plot(w_chains[:, j, i], c=colors[i], alpha=.3)
    ax.hlines(true_w[i], 0, 1000, zorder=4, color=colors[i], label="$w_{}$".format(i))
    ax.legend(loc='upper right')
  ax = axes[3][j]
  ax.plot(sigma_chains[:, j], alpha=.3, c=colors[3])
  ax.hlines(noise_scale, 0, 1000, zorder=4, color=colors[3], label=r"$\sigma$".format(i))
  ax.legend(loc='upper right')
fig.tight_layout()
plt.show()

# Histogram of samples of w
fig, axes = plt.subplots(4, num_chains, figsize=(4 * num_chains, 8))
for j in range(num_chains):
  for i in range(3):
    ax = axes[i][j]
    sns.distplot(w_chains[:, j, i], color=colors[i], norm_hist=True, ax=ax, hist_kws={'alpha': .3})
  for i in range(3):
    ax = axes[i][j]
    ymax = ax.get_ylim()[1]
    ax.vlines(true_w[i], 0, ymax, color=colors[i], label="$w_{}$".format(i), linewidth=3)
    ax.set_ylim(0, ymax)
    ax.legend(loc='upper right')


  ax = axes[3][j]
  sns.distplot(sigma_chains[:, j], color=colors[3], norm_hist=True, ax=ax, hist_kws={'alpha': .3})
  ymax = ax.get_ylim()[1]
  ax.vlines(noise_scale, 0, ymax, color=colors[3], label=r"$\sigma$".format(i), linewidth=3)
  ax.set_ylim(0, ymax)
  ax.legend(loc='upper right')
fig.tight_layout()
plt.show()

 

There’s a lot more!

これらのクールなブログ投稿とサンプルを調べてください :

• 構造時系列サポート ブログ colab
• 確率的 Keras 層 (入力: Tensor, 出力: Distribution!) ブログ colab
  • 他の層サンプル : VAE ブログ colab
• ガウス過程回帰 colab と潜在変数モデリング colab

GitHub 上のより多くのサンプルとノートブックは ここ です！

以上