Switch to EliminateDiscrete for max-product

[dellaert]

• added tiny example in c++
• made SumFrontals a method
• added test for SumFrontals
• added test for elimination in python, and it works (now we need to move to the next simplest example):
• below shows estimate of marginals, which match the eliminated result! And rations of posterior resulting from elimination also match! Awesome job :-)

..True mode: 0
P(mode=0; z0, z1) = 0.776760546738582
P(mode=1; z0, z1) = 0.22323945326141803
HybridBayesNet
 
size: 2
conditional 0: Hybrid  P( x0 | m0)
 Discrete Keys = (m0, 2), 
 Choice(m0) 
 0 Leaf  p(x0)
  R = [ 2.83549 ]
  d = [ 33.4989 ]
  mean: 1 elements
  x0: 11.8142
  No noise model

 1 Leaf  p(x0)
  R = [ 0.512076 ]
  d = [ 5.53226 ]
  mean: 1 elements
  x0: 10.8036
  No noise model

conditional 1:  P( m0 ):
 Choice(m0) 
 0 Leaf 0.76887015
 1 Leaf 0.23112985

expected_ratio: 9.889012746635817

Ratio: 9.889012746635933
Ratio: 9.889012746636144
Ratio: 9.889012746635778
Ratio: 9.889012746633508
Ratio: 9.889012746635919
Ratio: 9.889012746635949
Ratio: 9.889012746633531
Ratio: 9.889012746633517
Ratio: 9.889012746633536
Ratio: 9.889012746635643

[dellaert]

Not sure why it works yet, we should talk.
Switch to EliminateDiscrete in this PR was key I think. And maybe your constant, although was that used already?

[dellaert]

@varunagrawal So, I thought the code would now be correct, but it's not working out. The ratios also stopped working. Maybe we can investigate together. In a nutshell, in the C++ code, because we only use measurement, the discretefactor should in the final posterior be exactly equal to the prior. That means that the factor computed in the first elimination step should be zero information: DiscreteFactor with two equal leaves. But the constants do not add up to the same value (which is what is required).

[varunagrawal]

@varunagrawal So, I thought the code would now be correct, but it's not working out. The ratios also stopped working. Maybe we can investigate together. In a nutshell, in the C++ code, because we only use measurement, the discretefactor should in the final posterior be exactly equal to the prior. That means that the factor computed in the first elimination step should be zero information: DiscreteFactor with two equal leaves. But the constants do not add up to the same value (which is what is required).

I guess the issue with this was the assumption that one measurement wasn't informative enough to distinguish the modes? If yes, then glad we figured it out today. Making the note here for posterity.

[varunagrawal]

I'm making a PR to address some of these things and fix the tests.

[varunagrawal]

Are we keeping these for the future? My intuition says yes, but this would probably move to a base class.

[varunagrawal]

We need this. Or will normalization be handled later?

[varunagrawal]

@dellaert I am merging this.