refactor variational module, add histogram approximation

[ferrine]

Here I implement Histogram Approx with the same interface as other approximations

[ferrine]

Histogram is needed to put trace into theano graph for general purposes

[twiecki]

What's Histogram? Is there a reference?

[ferrine]

Not yet

[ferrine]

@twiecki, do you mean a paper on arxiv under reference?

[twiecki]

@ferrine Anything that makes me understand what this is useful for.

[ferrine]

@twiecki I'll use posterior widely for minimizing Bayesian Risk.

1. Consider you have a model that for given inputs yields posterior predictive distribution.
   X -> p(y|theta)
2. You know the desired y you want, name it y_target, and a loss function L(y_target, y_true)
3. Your decision is to choose the best X that minimizes L

So the objective is E_theta[L(y_target, y_true)] -> min X

Here I do not need to know anything about p(theta|D) I want to sample from it. I can do it with variational approximation only now. After implementing Histogram I can use SVGD/NUTS to make a trace and sample from it.

[twiecki]

OK, what's an example application? Also, do you assume theta is already estimated?

[ferrine]

Ye posterior is already estimated. You will be able to use any PyMC3 posterior for cost-benefit optimization. At my work I do exactly what I described. I want to minimize costs of inputs with archiving my target and be confident I can do it. I can't give more details because of NDA:(
I'm planning to implement all that stuff in few weeks to write a post including toy examples of

• Maximizing stochastic output of model E_{p(theta|D)}[f(X)] -> max X
• Minimizing costs of inputs to archive targetE_{p(y|X, theta, D)}[L(y_target, X, y)] -> min X
• Minimizing Bayesian risk: E_{p(y|X, theta, D)}[L(y_p, y)] -> min y_p
  I'm going to use ADAM for optimization

[ferrine]

@twiecki I want to merge Histogram tomorrow when I'm sure tests pass as I have ~95% coverage. I also think that sample_vp method should be renamed to sample as here I use traces from step methods. For other approximations I want to do the same as I want consistent interface for sampling. Your thoughts?

[ferrine]

I've forgot Docs for the class:( I'll add them in a separate PR

[fonnesbeck]

Can we get a working example of this, either as a notebook if you have time to do a fully-documented case study, or a model in the pymc3/examples folder.