Inference
Inference
We describe how to perform inference in probabilistic models. For background, see the Inference tutorial.
Suppose we have a model $(p(\mathbf{x}, \mathbf{z}, \beta))$ of data $(\mathbf{x}_{\text{train}})$ with latent variables $((\mathbf{z}, \beta))$. Consider the posterior inference problem, \[q(\mathbf{z}, \beta)\approx p(\mathbf{z}, \beta\mid \mathbf{x}_{\text{train}}),\] in which the task is to approximate the posterior $(p(\mathbf{z}, \beta\mid \mathbf{x}_{\text{train}}))$ using a family of distributions, $(q(\mathbf{z},\beta; \lambda))$, indexed by parameters $(\lambda)$.
In Edward, let z and beta be latent variables in the model, where we observe the random variable x with data x_train. Let qz and qbeta be random variables defined to approximate the posterior. We write this problem as follows:
inference = ed.Inference({z: qz, beta: qbeta}, {x: x_train})Inference is an abstract class which takes two inputs. The first is a collection of latent random variables beta and z, along with “posterior variables” qbeta and qz, which are associated to their respective latent variables. The second is a collection of observed random variables x, which is associated to the data x_train.
Inference adjusts parameters of the distribution of qbeta and qz to be close to the posterior $(p(\mathbf{z}, \beta\,|\,\mathbf{x}_{\text{train}}))$.
Running inference is as simple as running one method.
inference = ed.Inference({z: qz, beta: qbeta}, {x: x_train})
inference.run()Inference also supports fine control of the training procedure.
inference = ed.Inference({z: qz, beta: qbeta}, {x: x_train})
inference.initialize()
tf.global_variables_initializer().run()
for _ in range(inference.n_iter):
info_dict = inference.update()
inference.print_progress(info_dict)
inference.finalize()initialize() builds the algorithm’s update rules (computational graph) for $(\lambda)$; tf.global_variables_initializer().run() initializes $(\lambda)$ (TensorFlow variables in the graph); update() runs the graph once to update $(\lambda)$, which is called in a loop until convergence; finalize() runs any computation as the algorithm terminates.
The run() method is a simple wrapper for this procedure.
Other Settings
We highlight other settings during inference.
Model parameters. Model parameters are parameters in a model that we will always compute point estimates for and not be uncertain about. They are defined with tf.Variables, where the inference problem is \[\hat{\theta} \leftarrow^{\text{optimize}} p(\mathbf{x}_{\text{train}}; \theta)\]
from edward.models import Normal
theta = tf.Variable(0.0)
x = Normal(loc=tf.ones(10) * theta, scale=1.0)
inference = ed.Inference({}, {x: x_train})Only a subset of inference algorithms support estimation of model parameters. (Note also that this inference example does not have any latent variables. It is only about estimating theta given that we observe $(\mathbf{x} = \mathbf{x}_{\text{train}})$. We can add them so that inference is both posterior inference and parameter estimation.)
For example, model parameters are useful when applying neural networks from high-level libraries such as Keras and TensorFlow Slim. See the model compositionality page for more details.
Conditional inference. In conditional inference, only a subset of the posterior is inferred while the rest are fixed using other inferences. The inference problem is \[q(\mathbf{z}\mid\beta)q(\beta)\approx p(\mathbf{z}, \beta\mid\mathbf{x}_{\text{train}})\] where parameters in $(q(\mathbf{z}\mid\beta))$ are estimated and $(q(\beta))$ is fixed. In Edward, we enable conditioning by binding random variables to other random variables in data.
inference = ed.Inference({z: qz}, {x: x_train, beta: qbeta})In the compositionality page, we describe how to construct inference by composing many conditional inference algorithms.
Implicit prior samples. Latent variables can be defined in the model without any posterior inference over them. They are implicitly marginalized out with a single sample. The inference problem is \[q(\beta)\approx p(\beta\mid\mathbf{x}_{\text{train}}, \mathbf{z}^*)\] where $(\mathbf{z}^*\sim p(\mathbf{z}\mid\beta))$ is a prior sample.
inference = ed.Inference({beta: qbeta}, {x: x_train})For example, implicit prior samples are useful for generative adversarial networks. Their inference problem does not require any inference over the latent variables; it uses samples from the prior.