support gradients in TensorFlow for Python, PyMC3, and Stan models

[dustinvtran]

This pull request is a work in progress. Comments are welcome from interested parties, and also feedback on the implementation.

• fixes Support TensorFlow gradients for PyMC3Model, PythonModel, StanModel #163; related to discussion in Feature list: Graphical Models #154

Gradients of the log density are now supported for all model wrappers for use within a TensorFlow session.

• PythonModel uses autograd's automatic differentiation.
• PyMC3Model uses PyMC3's automatic differentiation.
• StanModel uses Stan's automatic differentiation.

In all wrappers, the gradient can be overwritten by manually specifying it. The manual gradient is written in native Python. This is useful for speeding up computation: manual gradients are often faster than automatic gradients (especially reverse-mode autodiff). It is also useful for the PythonModel in cases where its gradient is not supported by autograd.

Examples showing proof of concept are added. Each example is a normal posterior; inference is done via variational inference with the reparameterization trick, which uses the model gradient. The model is written in 4 ways:

1. PythonModel (with autograd).
2. PythonModel with a manually specified gradient.
3. PyMC3Model.
4. StanModel.

References.

• TensorFlow discussion on how to define gradients of a tf.py_func op.
• API for tf.RegisterGradient.
• API for tf.Graph.gradient_override_map.
• GPU support for Python gradients, and in autograd.

[twiecki]

Neat. Want to merge this first and then I could add pymc3 gradients?

[dustinvtran]

Sure sounds good! Once one of them is written and working properly, e.g., PythonModel, it should be easy to do the same for the others. The only nuance may be changing the various input types so they are compatible to be passed into PyMC3 or Stan or autograd's gradient function.

[dustinvtran]

model wrappers are now deprecated