`*.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.

Content retrieved from: https://github.com/google-research/google-research/tree/master/simple_probabilistic_programming/.