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Bloc-notes Notes de version TFP (0.12.1)

L'objectif de ce bloc-notes est d'aider TFP 0.12.1 à « prendre vie » via quelques petits extraits - de petites démonstrations de choses que vous pouvez réaliser avec TFP.

Voir sur TensorFlow.org

Exécuter dans Google Colab

Voir la source sur GitHub

Télécharger le cahier

Installe et importe

!pip3 install -qU tensorflow==2.4.0 tensorflow_probability==0.12.1 tensorflow-datasets inference_gym

import tensorflow as tf
import tensorflow_probability as tfp
assert '0.12' in tfp.__version__, tfp.__version__
assert '2.4' in tf.__version__, tf.__version__

physical_devices = tf.config.list_physical_devices('CPU')
tf.config.set_logical_device_configuration(
    physical_devices[0],
    [tf.config.LogicalDeviceConfiguration(),
     tf.config.LogicalDeviceConfiguration()])

tfd = tfp.distributions
tfb = tfp.bijectors
tfpk = tfp.math.psd_kernels

import matplotlib.pyplot as plt
import numpy as np
import scipy.interpolate
import IPython
import seaborn as sns
from inference_gym import using_tensorflow as gym
import logging

Bijecteurs

`Glow`

Un bijector du papier Glow: générative débit avec inversible 1x1 Convolutions , par Kingma et Dhariwal.

Voici comment dessiner une image à partir d'une distribution (notez que la distribution n'a rien "appris" ici).

image_shape = (32, 32, 4)  # 32 x 32 RGBA image

glow = tfb.Glow(output_shape=image_shape,
                coupling_bijector_fn=tfb.GlowDefaultNetwork,
                exit_bijector_fn=tfb.GlowDefaultExitNetwork)

pz = tfd.Sample(tfd.Normal(0., 1.), tf.reduce_prod(image_shape))

# Calling glow on distribution p(z) creates our glow distribution over images.
px = glow(pz)

# Take samples from the distribution to get images from your dataset.
image = px.sample(1)[0].numpy()

# Rescale to [0, 1].
image = (image - image.min()) / (image.max() - image.min())
plt.imshow(image);

png

`RayleighCDF`

Bijector pour la de distribution de Rayleigh CDF. Une utilisation est l'échantillonnage à partir de la distribution de Rayleigh, en prélevant des échantillons uniformes, puis en les passant par l'inverse de la CDF.

bij = tfb.RayleighCDF()
uniforms = tfd.Uniform().sample(10_000)
plt.hist(bij.inverse(uniforms), bins='auto');

png

`Ascending()` remplace `Invert(Ordered())`

x = tfd.Normal(0., 1.).sample(5)
print(tfb.Ascending()(x))
print(tfb.Invert(tfb.Ordered())(x))

tf.Tensor([1.9363368 2.650928  3.4936204 4.1817293 5.6920815], shape=(5,), dtype=float32)
WARNING:tensorflow:From <ipython-input-5-1406b9939c00>:3: Ordered.__init__ (from tensorflow_probability.python.bijectors.ordered) is deprecated and will be removed after 2021-01-09.
Instructions for updating:
`Ordered` bijector is deprecated; please use `tfb.Invert(tfb.Ascending())` instead.
tf.Tensor([1.9363368 2.650928  3.4936204 4.1817293 5.6920815], shape=(5,), dtype=float32)

Ajouter `low` arg: `Softplus(low=2.)`

x = tf.linspace(-4., 4., 100)

for low in (-1., 0., 1.):
  bij = tfb.Softplus(low=low)
  plt.plot(x, bij(x));

png

`tfb.ScaleMatvecLinearOperatorBlock` supporte par blocs `LinearOperator` , args plusieurs parties

op_1 = tf.linalg.LinearOperatorDiag(diag=[1., -1., 3.])
op_2 = tf.linalg.LinearOperatorFullMatrix([[12., 5.], [-1., 3.]])
scale = tf.linalg.LinearOperatorBlockDiag([op_1, op_2], is_non_singular=True)

bij = tfb.ScaleMatvecLinearOperatorBlock(scale)
bij([[1., 2., 3.], [0., 1.]])

WARNING:tensorflow:From /usr/local/lib/python3.6/dist-packages/tensorflow/python/ops/linalg/linear_operator_block_diag.py:223: LinearOperator.graph_parents (from tensorflow.python.ops.linalg.linear_operator) is deprecated and will be removed in a future version.
Instructions for updating:
Do not call `graph_parents`.
[<tf.Tensor: shape=(3,), dtype=float32, numpy=array([ 1., -2.,  9.], dtype=float32)>,
 <tf.Tensor: shape=(2,), dtype=float32, numpy=array([5., 3.], dtype=float32)>]

Répartition

`Skellam`

La distribution sur les différences de deux Poisson VR. Notez que les échantillons de cette distribution peuvent être négatifs.

x = tf.linspace(-5., 10., 10 - -5 + 1)

rates = (4, 2)
for i, rate in enumerate(rates):
  plt.bar(x - .3 * (1 - i), tfd.Poisson(rate).prob(x), label=f'Poisson({rate})', alpha=0.5, width=.3)
plt.bar(x.numpy() + .3, tfd.Skellam(*rates).prob(x).numpy(), color='k', alpha=0.25, width=.3,
        label=f'Skellam{rates}')

plt.legend();

png

`JointDistributionCoroutine[AutoBatched]` produits `namedtuple` -comme échantillons

Explicitement spécifier sample_dtype=[...] pour l'ancien tuple comportement.

@tfd.JointDistributionCoroutineAutoBatched
def model():
  x = yield tfd.Normal(0., 1., name='x')
  y = x + 4.
  yield tfd.Normal(y, 1., name='y')

draw = model.sample(10_000)

plt.hist(draw.x, bins='auto', alpha=0.5)
plt.hist(draw.y, bins='auto', alpha=0.5);

WARNING:tensorflow:Note that RandomStandardNormal inside pfor op may not give same output as inside a sequential loop.
WARNING:tensorflow:Note that RandomStandardNormal inside pfor op may not give same output as inside a sequential loop.

png

`VonMisesFisher` supports `dim > 5` , `entropy()`

La distribution de \(n-1\) \(\mathbb{R}^n\)von Mises-Fisher est une distribution sur le \(n-1\) sphère dimensionnelle \(\mathbb{R}^n\).

dist = tfd.VonMisesFisher([0., 1, 0, 1, 0, 1], concentration=1.)
draws = dist.sample(3)

print(dist.entropy())
tf.reduce_sum(draws ** 2, axis=1)  # each draw has length 1

tf.Tensor(3.3533673, shape=(), dtype=float32)
<tf.Tensor: shape=(3,), dtype=float32, numpy=array([1.0000002 , 0.99999994, 1.0000001 ], dtype=float32)>

`ExpGamma` , `ExpInverseGamma`

log_rate paramètre ajouté à Gamma . Améliorations numériques lors de l' échantillonnage à faible concentration Beta , Dirichlet et amis. Gradients de reparamétrage implicites dans tous les cas.

plt.figure(figsize=(10, 3))
plt.subplot(121)
plt.hist(tfd.Beta(.02, .02).sample(10_000), bins='auto')
plt.title('Beta(.02, .02)')
plt.subplot(122)
plt.title('GamX/(GamX+GamY) [the old way]')
g = tfd.Gamma(.02, 1); s0, s1 = g.sample(10_000), g.sample(10_000)
plt.hist(s0 / (s0 + s1), bins='auto')
plt.show()

plt.figure(figsize=(10, 3))
plt.subplot(121)
plt.hist(tfd.ExpGamma(.02, 1.).sample(10_000), bins='auto')
plt.title('ExpGamma(.02, 1)')
plt.subplot(122)
plt.hist(tfb.Log()(tfd.Gamma(.02, 1.)).sample(10_000), bins='auto')
plt.title('tfb.Log()(Gamma(.02, 1)) [the old way]');

png

`JointDistribution*AutoBatched` supporte échantillonnage reproductible (avec tuple / graines Tenseur longueur-2)

@tfd.JointDistributionCoroutineAutoBatched
def model():
  x = yield tfd.Normal(0, 1, name='x')
  y = yield tfd.Normal(x + 4, 1, name='y')

print(model.sample(seed=(1, 2)))
print(model.sample(seed=(1, 2)))

StructTuple(
  x=<tf.Tensor: shape=(), dtype=float32, numpy=-0.59835213>,
  y=<tf.Tensor: shape=(), dtype=float32, numpy=6.2380724>
)
StructTuple(
  x=<tf.Tensor: shape=(), dtype=float32, numpy=-0.59835213>,
  y=<tf.Tensor: shape=(), dtype=float32, numpy=6.2380724>
)

`KL(VonMisesFisher || SphericalUniform)`

# Build vMFs with the same mean direction, batch of increasing concentrations.
vmf = tfd.VonMisesFisher(tf.math.l2_normalize(tf.random.normal([10])),
                         concentration=[0., .1, 1., 10.])
# KL increases with concentration, since vMF(conc=0) == SphericalUniform.
print(tfd.kl_divergence(vmf, tfd.SphericalUniform(10)))

tf.Tensor([4.7683716e-07 4.9877167e-04 4.9384594e-02 2.4844694e+00], shape=(4,), dtype=float32)

`parameter_properties`

Cours de distribution exposent maintenant un parameter_properties(dtype=tf.float32, num_classes=None) méthode de classe, ce qui peut permettre la construction automatisée de nombreuses classes de distributions.

print('Gamma:', tfd.Gamma.parameter_properties())
print('Categorical:', tfd.Categorical.parameter_properties(dtype=tf.float64, num_classes=7))

Gamma: {'concentration': ParameterProperties(event_ndims=0, shape_fn=<function ParameterProperties.<lambda> at 0x7ff6bbfcdd90>, default_constraining_bijector_fn=<function Gamma._parameter_properties.<locals>.<lambda> at 0x7ff6afd95510>, is_preferred=True), 'rate': ParameterProperties(event_ndims=0, shape_fn=<function ParameterProperties.<lambda> at 0x7ff6bbfcdd90>, default_constraining_bijector_fn=<function Gamma._parameter_properties.<locals>.<lambda> at 0x7ff6afd95ea0>, is_preferred=False), 'log_rate': ParameterProperties(event_ndims=0, shape_fn=<function ParameterProperties.<lambda> at 0x7ff6bbfcdd90>, default_constraining_bijector_fn=<class 'tensorflow_probability.python.bijectors.identity.Identity'>, is_preferred=True)}
Categorical: {'logits': ParameterProperties(event_ndims=1, shape_fn=<function Categorical._parameter_properties.<locals>.<lambda> at 0x7ff6afd95510>, default_constraining_bijector_fn=<class 'tensorflow_probability.python.bijectors.identity.Identity'>, is_preferred=True), 'probs': ParameterProperties(event_ndims=1, shape_fn=<function Categorical._parameter_properties.<locals>.<lambda> at 0x7ff6afdc91e0>, default_constraining_bijector_fn=<class 'tensorflow_probability.python.bijectors.softmax_centered.SoftmaxCentered'>, is_preferred=False)}

`experimental_default_event_space_bijector`

Accepte désormais des arguments supplémentaires épinglant certaines parties de la distribution.

@tfd.JointDistributionCoroutineAutoBatched
def model():
  scale = yield tfd.Gamma(1, 1, name='scale')
  obs = yield tfd.Normal(0, scale, name='obs')

model.experimental_default_event_space_bijector(obs=.2).forward(
    [tf.random.uniform([3], -2, 2.)])

StructTuple(
  scale=<tf.Tensor: shape=(3,), dtype=float32, numpy=array([0.6630705, 1.5401832, 1.0777743], dtype=float32)>
)

`JointDistribution.experimental_pin`

Pins quelques pièces de distribution communes, retour JointDistributionPinned objet représentant la densité non normalisée commune.

Travailler avec le experimental_default_event_space_bijector , cela fait faire l' inférence variationnelle ou MCCM avec bon sens par défaut beaucoup plus simple. Dans le exemple ci - dessous, les deux premières lignes de l' sample font courir MCMC un jeu d' enfant.

dist = tfd.JointDistributionSequential([
    tfd.HalfNormal(1.),
    lambda scale: tfd.Normal(0., scale, name='observed')])

@tf.function
def sample():
  bij = dist.experimental_default_event_space_bijector(observed=1.)
  target_log_prob = dist.experimental_pin(observed=1.).unnormalized_log_prob

  kernel = tfp.mcmc.TransformedTransitionKernel(
      tfp.mcmc.HamiltonianMonteCarlo(target_log_prob,
                                     step_size=0.6,
                                     num_leapfrog_steps=16),
      bijector=bij)
  return tfp.mcmc.sample_chain(500, 
                               current_state=tf.ones([8]),  # multiple chains
                               kernel=kernel,
                               trace_fn=None)

draws = sample()

fig, (hist, trace) = plt.subplots(ncols=2, figsize=(16, 3))
trace.plot(draws, alpha=0.5)
for col in tf.transpose(draws):
  sns.kdeplot(col, ax=hist);

png

`tfd.NegativeBinomial.experimental_from_mean_dispersion`

Paramétrage alternatif. Envoyez un e-mail à tfprobability@tensorflow.org ou envoyez-nous un PR pour ajouter des méthodes de classe similaires pour d'autres distributions.

nb = tfd.NegativeBinomial.experimental_from_mean_dispersion(30., .01)
plt.hist(nb.sample(10_000), bins='auto');

png

`tfp.experimental.distribute`

DistributionStrategy -Aware distributions communes, ce qui permet pour les calculs de probabilité multi-appareils. Fragmentées Independent et Sample distributions.

# Note: 2-logical devices are configured in the install/import cell at top.
strategy = tf.distribute.MirroredStrategy()
assert strategy.num_replicas_in_sync == 2

@tfp.experimental.distribute.JointDistributionCoroutine
def model():
  root = tfp.experimental.distribute.JointDistributionCoroutine.Root
  group_scale = yield root(tfd.Sample(tfd.Exponential(1), 3, name='group_scale'))
  _ = yield tfp.experimental.distribute.ShardedSample(tfd.Independent(tfd.Normal(0, group_scale), 1),
                                                      sample_shape=[4], name='x')

seed1, seed2 = tfp.random.split_seed((1, 2))

@tf.function
def sample(seed):
  return model.sample(seed=seed)
xs = strategy.run(sample, (seed1,))
print("""
Note that the global latent `group_scale` is shared across devices, whereas
the local `x` is sampled independently on each device.
""")
print('sample:', xs)
print('another sample:', strategy.run(sample, (seed2,)))

@tf.function
def log_prob(x):
  return model.log_prob(x)
print("""
Note that each device observes the same log_prob (local latent log_probs are
summed across devices).
""")
print('log_prob:', strategy.run(log_prob, (xs,)))

@tf.function
def grad_log_prob(x):
  return tfp.math.value_and_gradient(model.log_prob, x)[1]

print("""
Note that each device observes the same log_prob gradient (local latents have
independent gradients, global latents have gradients aggregated across devices).
""")
print('grad_log_prob:', strategy.run(grad_log_prob, (xs,)))

WARNING:tensorflow:There are non-GPU devices in `tf.distribute.Strategy`, not using nccl allreduce.
INFO:tensorflow:Using MirroredStrategy with devices ('/job:localhost/replica:0/task:0/device:CPU:0', '/job:localhost/replica:0/task:0/device:CPU:1')

Note that the global latent `group_scale` is shared across devices, whereas 
the local `x` is sampled independently on each device.

sample: StructTuple(
  group_scale=PerReplica:{
      0: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([2.6355493, 1.1805456, 1.245112 ], dtype=float32)>,
      1: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([2.6355493, 1.1805456, 1.245112 ], dtype=float32)>
    },
  x=PerReplica:{
      0: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[-0.90548456,  0.7675636 ,  0.27627748],
           [-0.3475989 ,  2.0194046 , -1.2531326 ]], dtype=float32)>,
      1: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[ 3.251305  , -0.5790973 ,  0.42745453],
           [-1.562331  ,  0.3006323 ,  0.635732  ]], dtype=float32)>
    }
)
another sample: StructTuple(
  group_scale=PerReplica:{
      0: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([2.41133   , 0.10307606, 0.5236566 ], dtype=float32)>,
      1: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([2.41133   , 0.10307606, 0.5236566 ], dtype=float32)>
    },
  x=PerReplica:{
      0: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[-3.2476294 ,  0.07213175, -0.39536062],
           [-1.2319602 , -0.05505352,  0.06356457]], dtype=float32)>,
      1: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[ 5.6028705 ,  0.11919801, -0.48446828],
           [-1.5938259 ,  0.21123725,  0.28979057]], dtype=float32)>
    }
)

Note that each device observes the same log_prob (local latent log_probs are
summed across devices).

INFO:tensorflow:Reduce to /job:localhost/replica:0/task:0/device:CPU:0 then broadcast to ('/job:localhost/replica:0/task:0/device:CPU:0', '/job:localhost/replica:0/task:0/device:CPU:1').
log_prob: PerReplica:{
  0: tf.Tensor(-25.05747, shape=(), dtype=float32),
  1: tf.Tensor(-25.05747, shape=(), dtype=float32)
}

Note that each device observes the same log_prob gradient (local latents have
independent gradients, global latents are aggregated across devices).

INFO:tensorflow:Reduce to /job:localhost/replica:0/task:0/device:CPU:0 then broadcast to ('/job:localhost/replica:0/task:0/device:CPU:0', '/job:localhost/replica:0/task:0/device:CPU:1').
INFO:tensorflow:Reduce to /job:localhost/replica:0/task:0/device:CPU:0 then broadcast to ('/job:localhost/replica:0/task:0/device:CPU:0', '/job:localhost/replica:0/task:0/device:CPU:1').
grad_log_prob: StructTuple(
  group_scale=PerReplica:{
      0: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([-1.7555585, -1.2928739, -3.0554674], dtype=float32)>,
      1: <tf.Tensor: shape=(3,), dtype=float32, numpy=array([-1.7555585, -1.2928739, -3.0554674], dtype=float32)>
    },
  x=PerReplica:{
      0: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[ 0.13035832, -0.5507428 , -0.17820862],
           [ 0.05004217, -1.4489648 ,  0.80831426]], dtype=float32)>,
      1: <tf.Tensor: shape=(2, 3), dtype=float32, numpy=
    array([[-0.46807498,  0.41551432, -0.27572307],
           [ 0.22492138, -0.21570992, -0.41006932]], dtype=float32)>
    }
)

Noyaux PSD

`GeneralizedMatern`

Le GeneralizedMatern positif semidéfinie noyau généralise MaternOneHalf , MAterhThreeHalves et MaternFiveHalves .

gm = tfpk.GeneralizedMatern(df=[0.5, 1.5, 2.5], length_scale=1., amplitude=0.5)
m1 = tfpk.MaternOneHalf(length_scale=1., amplitude=0.5)
m2 = tfpk.MaternThreeHalves(length_scale=1., amplitude=0.5)
m3 = tfpk.MaternFiveHalves(length_scale=1., amplitude=0.5)
xs = tf.linspace(-1.5, 1.5, 100)

gm_matrix = gm.matrix([[0.]], xs[..., tf.newaxis])
plt.plot(xs, gm_matrix[0][0])
plt.plot(xs, m1.matrix([[0.]], xs[..., tf.newaxis])[0])
plt.show()
plt.plot(xs, gm_matrix[1][0])
plt.plot(xs, m2.matrix([[0.]], xs[..., tf.newaxis])[0])
plt.show()
plt.plot(xs, gm_matrix[2][0])
plt.plot(xs, m3.matrix([[0.]], xs[..., tf.newaxis])[0])
plt.show()

png

`Parabolic` (Epanechnikov)

epa = tfpk.Parabolic()
xs = tf.linspace(-1.05, 1.05, 100)
plt.plot(xs, epa.matrix([[0.]], xs[..., tf.newaxis])[0]);

png

VI

`build_asvi_surrogate_posterior`

Construisez automatiquement une postérieure de substitution structurée pour VI d'une manière qui incorpore la structure graphique de la distribution antérieure. Ceci utilise la méthode décrite dans le document automatique structuré Variational Inference ( https://arxiv.org/abs/2002.00643 ).

# Import a Brownian Motion model from TFP's inference gym.
model = gym.targets.BrownianMotionMissingMiddleObservations()
prior = model.prior_distribution()
ground_truth = ground_truth = model.sample_transformations['identity'].ground_truth_mean
target_log_prob = lambda *values: model.log_likelihood(values) + prior.log_prob(values)

Ceci modélise un processus de mouvement brownien avec un modèle d'observation gaussien. Il se compose de 30 pas de temps, mais les 10 pas de temps du milieu sont inobservables.

  locs[0] ~ Normal(loc=0, scale=innovation_noise_scale)
  for t in range(1, num_timesteps):
    locs[t] ~ Normal(loc=locs[t - 1], scale=innovation_noise_scale)

  for t in range(num_timesteps):
    observed_locs[t] ~ Normal(loc=locs[t], scale=observation_noise_scale)

L'objectif est de déduire les valeurs de locs à partir d' observations bruyantes ( observed_locs ). Depuis les 10 intermédiaires sont inobservables pas de temps, observed_locs sont NaN valeurs à pas de temps [10,19].

# The observed loc values in the Brownian Motion inference gym model
OBSERVED_LOC = np.array([
    0.21592641, 0.118771404, -0.07945447, 0.037677474, -0.27885845, -0.1484156,
    -0.3250906, -0.22957903, -0.44110894, -0.09830782, np.nan, np.nan, np.nan,
    np.nan, np.nan, np.nan, np.nan, np.nan, np.nan, np.nan, -0.8786016,
    -0.83736074, -0.7384849, -0.8939254, -0.7774566, -0.70238715, -0.87771565,
    -0.51853573, -0.6948214, -0.6202789
]).astype(dtype=np.float32)

# Plot the prior and the likelihood observations
plt.figure()
plt.title('Brownian Motion Prior Samples and Observations')

num_samples = 15
prior_samples = prior.sample(num_samples)

plt.plot(prior_samples, c='blue', alpha=0.1)
plt.plot(prior_samples[0][0], label="Prior Samples", c='blue', alpha=0.1)

plt.scatter(x=range(30),y=OBSERVED_LOC, c='black', alpha=0.5, label="Observations")

plt.legend(bbox_to_anchor=(1.05, 1), borderaxespad=0.);

png

logging.getLogger('tensorflow').setLevel(logging.ERROR)  # suppress pfor warnings

# Construct and train an ASVI Surrogate Posterior.
asvi_surrogate_posterior = tfp.experimental.vi.build_asvi_surrogate_posterior(prior)

asvi_losses = tfp.vi.fit_surrogate_posterior(target_log_prob,
                                        asvi_surrogate_posterior,
                                        optimizer=tf.optimizers.Adam(learning_rate=0.1),
                                        num_steps=500)
logging.getLogger('tensorflow').setLevel(logging.NOTSET)

# Construct and train a Mean-Field Surrogate Posterior.
factored_surrogate_posterior = tfp.experimental.vi.build_factored_surrogate_posterior(event_shape=prior.event_shape)

factored_losses = tfp.vi.fit_surrogate_posterior(target_log_prob,
                                        factored_surrogate_posterior,
                                        optimizer=tf.optimizers.Adam(learning_rate=0.1),
                                        num_steps=500)

logging.getLogger('tensorflow').setLevel(logging.ERROR)  # suppress pfor warnings

# Sample from the posteriors.
asvi_posterior_samples = asvi_surrogate_posterior.sample(num_samples)
factored_posterior_samples = factored_surrogate_posterior.sample(num_samples)

logging.getLogger('tensorflow').setLevel(logging.NOTSET)

Les distributions postérieures ASVI et de substitution à champ moyen ont toutes deux convergé, et la postérieure ASVI de substitution avait une perte finale plus faible (valeur ELBO négative).

# Plot the loss curves.
plt.figure()
plt.title('Loss Curves for ASVI vs Mean-Field Surrogate Posteriors')

plt.plot(asvi_losses, c='orange', label='ASVI', alpha = 0.4)
plt.plot(factored_losses, c='green', label='Mean-Field', alpha = 0.4)
plt.ylim(-50, 300)

plt.legend(bbox_to_anchor=(1.3, 1), borderaxespad=0.);

png

Les échantillons des postérieurs soulignent à quel point le postérieur de substitution ASVI capture bien l'incertitude pour les pas de temps sans observations. D'autre part, le champ moyen a posteriori a du mal à saisir la véritable incertitude.

# Plot samples from the ASVI and Mean-Field Surrogate Posteriors.
plt.figure()
plt.title('Posterior Samples from ASVI vs Mean-Field Surrogate Posterior')

plt.plot(asvi_posterior_samples, c='orange', alpha = 0.25)
plt.plot(asvi_posterior_samples[0][0], label='ASVI Surrogate Posterior', c='orange', alpha = 0.25)

plt.plot(factored_posterior_samples, c='green', alpha = 0.25)
plt.plot(factored_posterior_samples[0][0], label='Mean-Field Surrogate Posterior', c='green', alpha = 0.25)

plt.scatter(x=range(30),y=OBSERVED_LOC, c='black', alpha=0.5, label='Observations')

plt.plot(ground_truth, c='black', label='Ground Truth')

plt.legend(bbox_to_anchor=(1.585, 1), borderaxespad=0.);

png

MCMC

`ProgressBarReducer`

Visualisez la progression de l'échantillonneur. (Peut avoir une pénalité de performance nominale ; pas actuellement pris en charge par la compilation JIT.)

kernel = tfp.mcmc.HamiltonianMonteCarlo(lambda x: -x**2 / 2, .05, 20)
pbar = tfp.experimental.mcmc.ProgressBarReducer(100)
kernel = tfp.experimental.mcmc.WithReductions(kernel, pbar)
plt.hist(tf.reshape(tfp.mcmc.sample_chain(100, current_state=tf.ones([128]), kernel=kernel, trace_fn=None), [-1]), bins='auto')
pbar.bar.close()

99%|█████████▉| 99/100 [00:03<00:00, 27.37it/s]

png

`sample_sequential_monte_carlo` supporte échantillonnage reproductible

initial_state = tf.random.uniform([4096], -2., 2.)
def smc(seed):
  return tfp.experimental.mcmc.sample_sequential_monte_carlo(
    prior_log_prob_fn=lambda x: -x**2 / 2,
    likelihood_log_prob_fn=lambda x: -(x-1.)**2 / 2,
    current_state=initial_state,
    seed=seed)[1]
plt.hist(smc(seed=(12, 34)), bins='auto');plt.show()

print(smc(seed=(12, 34))[:10])
print('different:', smc(seed=(10, 20))[:10])
print('same:', smc(seed=(12, 34))[:10])

png

tf.Tensor(
[ 0.665834    0.9892149   0.7961128   1.0016634  -1.000767   -0.19461267
  1.3070581   1.127177    0.9940303   0.58239716], shape=(10,), dtype=float32)
different: tf.Tensor(
[ 1.3284367   0.4374407   1.1349089   0.4557473   0.06510283 -0.08954388
  1.1735026   0.8170528   0.12443061  0.34413314], shape=(10,), dtype=float32)
same: tf.Tensor(
[ 0.665834    0.9892149   0.7961128   1.0016634  -1.000767   -0.19461267
  1.3070581   1.127177    0.9940303   0.58239716], shape=(10,), dtype=float32)

Ajout de calculs en continu de variance, covariance, Rhat

Remarque, les interfaces de ceux - ci ont quelque peu changé tfp-nightly .

def cov_to_ellipse(t, cov, mean):
  """Draw a one standard deviation ellipse from the mean, according to cov."""
  diag = tf.linalg.diag_part(cov)
  a = 0.5 * tf.reduce_sum(diag)
  b = tf.sqrt(0.25 * (diag[0] - diag[1])**2 + cov[0, 1]**2)
  major = a + b
  minor = a - b
  theta = tf.math.atan2(major - cov[0, 0], cov[0, 1])
  x = (tf.sqrt(major) * tf.cos(theta) * tf.cos(t) -
       tf.sqrt(minor) * tf.sin(theta) * tf.sin(t))
  y = (tf.sqrt(major) * tf.sin(theta) * tf.cos(t) +
       tf.sqrt(minor) * tf.cos(theta) * tf.sin(t))
  return x + mean[0], y + mean[1]

fig, axes = plt.subplots(nrows=4, ncols=5, figsize=(14, 8), 
                         sharex=True, sharey=True, constrained_layout=True)
t = tf.linspace(0., 2 * np.pi, 200)
tot = 10
cov = 0.1 * tf.eye(2) + 0.9 * tf.ones([2, 2])
mvn = tfd.MultivariateNormalTriL(loc=[1., 2.],
                                 scale_tril=tf.linalg.cholesky(cov))

for ax in axes.ravel():
  rv = tfp.experimental.stats.RunningCovariance(
      num_samples=0., mean=tf.zeros(2), sum_squared_residuals=tf.zeros((2, 2)),
      event_ndims=1)

  for idx, x in enumerate(mvn.sample(tot)):
    rv = rv.update(x)
    ax.plot(*cov_to_ellipse(t, rv.covariance(), rv.mean),
            color='k', alpha=(idx + 1) / tot)
  ax.plot(*cov_to_ellipse(t, mvn.covariance(), mvn.mean()), 'r')
fig.suptitle("Twenty tries to approximate the red covariance with 10 draws");

png

Mathématiques, statistiques

Fonctions de Bessel : ive, kve, log-ive

xs = tf.linspace(0.5, 20., 100)
ys = tfp.math.bessel_ive([[0.5], [1.], [np.pi], [4.]], xs)
zs = tfp.math.bessel_kve([[0.5], [1.], [2.], [np.pi]], xs)

for i in range(4):
  plt.plot(xs, ys[i])
plt.show()

for i in range(4):
  plt.plot(xs, zs[i])
plt.show()

png

Facultatif `weights` arg à `tfp.stats.histogram`

edges = tf.linspace(-4., 4, 31)
samps = tfd.TruncatedNormal(0, 1, -4, 4).sample(100_000, seed=(123, 456))
_, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3))
ax1.bar(edges[:-1], tfp.stats.histogram(samps, edges))
ax1.set_title('samples histogram')
ax2.bar(edges[:-1], tfp.stats.histogram(samps, edges, weights=1 / tfd.Normal(0, 1).prob(samps)))
ax2.set_title('samples, weighted by inverse p(sample)');

png

`tfp.math.erfcinv`

x = tf.linspace(-3., 3., 10)
y = tf.math.erfc(x)
z = tfp.math.erfcinv(y)
print(x)
print(z)

tf.Tensor(
[-3.         -2.3333333  -1.6666666  -1.         -0.33333325  0.3333335

  1.          1.666667    2.3333335   3.        ], shape=(10,), dtype=float32)
tf.Tensor(
[-3.0002644  -2.3333426  -1.6666666  -0.9999997  -0.3333332   0.33333346
  0.9999999   1.6666667   2.3333335   3.0000002 ], shape=(10,), dtype=float32)

Bloc-notes Notes de version TFP (0.12.1)

Installe et importe

Bijecteurs

Glow

RayleighCDF

Ascending() remplace Invert(Ordered())

Ajouter low arg: Softplus(low=2.)

tfb.ScaleMatvecLinearOperatorBlock supporte par blocs LinearOperator , args plusieurs parties

Répartition

Skellam

JointDistributionCoroutine[AutoBatched] produits namedtuple -comme échantillons

VonMisesFisher supports dim > 5 , entropy()

ExpGamma , ExpInverseGamma

JointDistribution*AutoBatched supporte échantillonnage reproductible (avec tuple / graines Tenseur longueur-2)

KL(VonMisesFisher || SphericalUniform)

parameter_properties

experimental_default_event_space_bijector

JointDistribution.experimental_pin

tfd.NegativeBinomial.experimental_from_mean_dispersion

tfp.experimental.distribute