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README.md

Simple, Distributed, and Accelerated Probabilistic Programming

This directory contains reference code for the NIPS 2018 paper, "Simple, Distributed, and Accelerated Probabilistic Programming". It's organized as follows:

  • *.py: Edward2, an implementation of the idea.
  • Companion.ipynb: Jupyter notebook which expands on the paper's code snippets and examples.
  • no_u_turn_sampler/: Example implementation of the No-U-Turn Sampler.

The implementation, Edward2, is a probabilistic programming language in TensorFlow and Python. It extends the TensorFlow ecosystem so that one can declare models as probabilistic programs and manipulate a model's computation for flexible training, latent variable inference, and predictions.

Are you upgrading from Edward? Check out the guide Upgrading_from_Edward_to_Edward2.md.

1. Models as Probabilistic Programs

Random Variables

In Edward2, we use RandomVariables to specify a probabilistic model's structure. A random variable rv carries a probability distribution (rv.distribution), which is a TensorFlow Distribution instance governing the random variable's methods such as log_prob and sample.

Random variables are formed like TensorFlow Distributions.

import simple_probabilistic_programming as ed

normal_rv = ed.Normal(loc=0., scale=1.)
## <ed.RandomVariable 'Normal/' shape=() dtype=float32>
normal_rv.distribution.log_prob(1.231)
## <tf.Tensor 'Normal/log_prob/sub:0' shape=() dtype=float32>

dirichlet_rv = ed.Dirichlet(concentration=tf.ones([2, 10]))
## <ed.RandomVariable 'Dirichlet/' shape=(2, 10) dtype=float32>

By default, instantiating a random variable rv creates a sampling op to form the tensor rv.value ~ rv.distribution.sample(). The default number of samples (controllable via the sample_shape argument to rv) is one, and if the optional value argument is provided, no sampling op is created. Random variables can interoperate with TensorFlow ops: the TF ops operate on the sample.

x = ed.Normal(loc=tf.zeros(10), scale=tf.ones(10))
y = 5.
x + y, x / y
## (<tf.Tensor 'add:0' shape=(10,) dtype=float32>,
##  <tf.Tensor 'div:0' shape=(10,) dtype=float32>)
tf.tanh(x * y)
## <tf.Tensor 'Tanh:0' shape=(10,) dtype=float32>
x[2]  # 3rd normal rv
## <tf.Tensor 'strided_slice:0' shape=() dtype=float32>

Probabilistic Models

Probabilistic models in Edward2 are expressed as Python functions that instantiate one or more RandomVariables. Typically, the function ("program") executes the generative process and returns samples. Inputs to the function can be thought of as values the model conditions on.

Below we write Bayesian logistic regression, where binary outcomes are generated given features, coefficients, and an intercept. There is a prior over the coefficients and intercept. Executing the function adds operations to the TensorFlow graph, and asking for the result node in a TensorFlow session will sample coefficients and intercept from the prior, and use these samples to compute the outcomes.

def logistic_regression(features):
  """Bayesian logistic regression p(y | x) = int p(y | x, w, b) p(w, b) dwdb."""
  coeffs = ed.Normal(loc=tf.zeros(features.shape[1]), scale=1., name="coeffs")
  intercept = ed.Normal(loc=0., scale=1., name="intercept")
  outcomes = ed.Bernoulli(
      logits=tf.tensordot(features, coeffs, [[1], [0]]) + intercept,
      name="outcomes")
  return outcomes

num_features = 10
features = tf.random_normal([100, num_features])
outcomes = logistic_regression(features)

# Execute the model program, returning a sample np.ndarray of shape (100,).
with tf.Session() as sess:
  outcomes_ = sess.run(outcomes)

Edward2 programs can also represent distributions beyond those which directly model data. For example, below we write a learnable distribution with the intention to approximate it to the logistic regression posterior.

def logistic_regression_posterior(num_features):
  """Posterior of Bayesian logistic regression p(w, b | {x, y})."""
  posterior_coeffs = ed.MultivariateNormalTriL(
      loc=tf.get_variable("coeffs_loc", [num_features]),
      scale_tril=tfp.trainable_distributions.tril_with_diag_softplus_and_shift(
          tf.get_variable("coeffs_scale", [num_features*(num_features+1) / 2])),
      name="coeffs_posterior")
  posterior_intercept = ed.Normal(
      loc=tf.get_variable("intercept_loc", []),
      scale=tf.nn.softplus(tf.get_variable("intercept_scale", [])) + 1e-5,
      name="intercept_posterior")
  return coeffs, intercept

coeffs, intercept = logistic_regression_posterior(num_features)

# Execute the program, returning a sample
# (np.ndarray of shape (55,), np.ndarray of shape ()).
with tf.Session() as sess:
  sess.run(tf.global_variables_initializer())
  posterior_coeffs_, posterior_ntercept_ = sess.run(
      [posterior_coeffs, posterior_intercept])

2. Manipulating Model Computation

Tracing

Training and testing probabilistic models typically require more than just samples from the generative process. To enable flexible training and testing, we manipulate the model's computation using tracing.

A tracer is a function that acts on another function f and its arguments *args, **kwargs. It performs various computations before returning an output (typically f(*args, **kwargs): the result of applying the function itself). The ed.trace context manager pushes tracers onto a stack, and any traceable function is intercepted by the stack. All random variable constructors are traceable.

Below we trace the logistic regression model's generative process. In particular, we make predictions with its learned posterior means rather than with its priors.

def set_prior_to_posterior_mean(f, *args, **kwargs):
  """Forms posterior predictions, setting each prior to its posterior mean."""
  name = kwargs.get("name")
  if name == "coeffs":
    return posterior_coeffs.distribution.mean()
  elif name == "intercept":
    return posterior_intercept.distribution.mean()
  return f(*args, **kwargs)

with ed.trace(set_prior_to_posterior_mean):
  predictions = logistic_regression(features)

training_accuracy = (
    tf.reduce_sum(tf.cast(tf.equal(predictions, outcomes), tf.float32)) /
    tf.cast(tf.shape(outcomes), tf.float32))

Program Transformations

Using tracing, one can also apply program transformations, which map from one representation of a model to another. This provides convenient access to different model properties depending on the downstream use case.

For example, Markov chain Monte Carlo algorithms often require a model's log-joint probability function as input. Below we take the Bayesian logistic regression program which specifies a generative process, and apply the built-in ed.make_log_joint transformation to obtain its log-joint probability function. The log-joint function takes as input the generative program's original inputs as well as random variables in the program. It returns a scalar Tensor summing over all random variable log-probabilities.

In our example, features and outcomes are fixed, and we want to use Hamiltonian Monte Carlo to draw samples from the posterior distribution of coeffs and intercept. To this use, we create target_log_prob_fn, which takes just coeffs and intercept as arguments and pins the input features and output rv outcomes to its known values.

from simple_probabilistic_programming import no_u_turn_sampler

tf.enable_eager_execution()

# Set up training data.
features = tf.random_normal([100, 55])
outcomes = tf.random_uniform([100], minval=0, maxval=2, dtype=tf.int32)

# Pass target log-probability function to MCMC transition kernel.
log_joint = ed.make_log_joint_fn(logistic_regression)

def target_log_prob_fn(coeffs, intercept):
  """Target log-probability as a function of states."""
  return log_joint(features,
                   coeffs=coeffs,
                   intercept=intercept,
                   outcomes=outcomes)

coeffs_samples = []
intercept_samples = []
coeffs = tf.random_normal([55])
intercept = tf.random_normal([])
target_log_prob = None
grads_target_log_prob = None
for _ in range(1000):
  [
      [coeffs, intercepts],
      target_log_prob,
      grads_target_log_prob,
  ] = no_u_turn_sampler.kernel(
          target_log_prob_fn=target_log_prob_fn,
          current_state=[coeffs, intercept],
          step_size=[0.1, 0.1],
          current_target_log_prob=target_log_prob,
          current_grads_target_log_prob=grads_target_log_prob)
  coeffs_samples.append(coeffs)
  intercept_samples.append(coeffs)

The returned coeffs_samples and intercept_samples contain 1,000 posterior samples for coeffs and intercept respectively. They may be used, for example, to evaluate the model's posterior predictive on new data.

References

Tran, D., Hoffman, M. D., Moore, D., Suter, C., Vasudevan S., Radul A., Johnson M., and Saurous R. A. (2018). Simple, Distributed, and Accelerated Probabilistic Programming. In Neural Information Processing Systems.

@inproceedings{tran2018simple,
  author = {Dustin Tran and Matthew D. Hoffman and Dave Moore and Christopher Suter and Srinivas Vasudevan and Alexey Radul and Matthew Johnson and Rif A. Saurous},
  title = {Simple, Distributed, and Accelerated Probabilistic Programming},
  booktitle = {Neural Information Processing Systems},
  year = {2018},
}