Junction tree and belief propagation algorithms
Project description
junctiontree
Implementation of discrete factor graph inference utilizing the Junction Tree algorithm
Requirements
 Python3 (>= 3.5.5), NumPy (>= 1.13.3), SciPy (>= 1.1), attrs (>=17.4)
Factor graphs:
A factor graph is given as a list of variables that indicate which variables are in the factor.
[vars1, ..., varsN] # a list of N factors
The index in the list can be used as an ID for the factor, that is, the first factor in the list has ID 0 and the last factor has ID N1.
A companion list (of numpy arrays) of the same length as the factor list is provided as a representation for the factor values
[values1, ..., valuesN]
Also, the size of each of the M variables is given as a dictionary:
{
var1: size1,
...
varM: sizeM
}
Here, size is an integer representing the size of the variable. It is the same as the length of the corresponding axis in the numeric array.
Generic trees (recursive definition)
[index, vars, child_tree1, ..., child_treeN]
Junction trees
tree structure (composed of node indices found in node list):
[
index,
(
separator1_index,
child_tree1
),
...,
(
separatorN_index,
child_treeN
)
]
node list (elements are list of variables which define node):
[node0_vars, node1_vars,...,nodeN_vars]
maxcliques and separators are both types of nodes
Potentials in (junction) trees
A list of arrays. The node IDs in the tree graphs map to the arrays in this data structure in order to get the numeric arrays in the execution phase. The numeric arrays are not needed in the compilation phase.
Usage
Junction Tree construction
Starting with the definition of a factor graph
(Example taken from http://mensxmachina.org/files/software/demos/jtreedemo.html)
import junctiontree.beliefpropagation as bp
import junctiontree.junctiontree as jt
import numpy as np
var_sizes = {
"cloudy": 2,
"sprinkler": 2,
"rain": 2,
"wet_grass": 2
}
factors = [
["cloudy"],
["cloudy", "sprinkler"],
["cloudy", "rain"],
["rain", "sprinkler", "wet_grass"]
]
values = [
np.array([0.5,0.5]),
np.array(
[
[0.5,0.5],
[0.9,0.1]
]
),
np.array(
[
[0.8,0.2],
[0.2,0.8]
]
),
np.array(
[
[
[1,0],
[0.1,0.9]
],
[
[0.1,0.9],
[0.01,0.99]
]
]
)
]
tree = jt.create_junction_tree(factors, var_sizes)
Global Propagation
The initial clique potentials are inconsistent. The potentials are made consistent through global propagation on the junction tree
prop_values = tree.propagate(values)
Observing Data
Alternatively, clique potentials can be made consistent after observing data for the variables in the junction tree
# Update the size of observed variable
cond_sizes = var_sizes.copy()
cond_sizes["wet_grass"] = 1
cond_tree = jt.create_junction_tree(factors, cond_sizes)
# Then, also similarly the values:
cond_values = values.copy()
# remove axis corresponding to "wet_grass" == 0
cond_values[3] = cond_values[3][:,:,1:2]
# Perform global propagation using conditioned values
prop_values = tree.propagate(cond_values)
Marginalization
From a collection of consistent clique potentials, the marginal value of variables of interest can be calculated
# Pr(sprinklerwet_grass = 1)
marginal = np.sum(prop_values[1], axis=0)
# The probabilities are unnormalized but we can calculate the normalized values:
norm_marginal = marginal/np.sum(marginal)
References

S. M. Aji and R. J. McEliece, "The generalized distributive law," in IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 325343, Mar 2000. doi: 10.1109/18.825794

Cecil Huang, Adnan Darwiche, Inference in belief networks: A procedural guide, International Journal of Approximate Reasoning, Volume 15, Issue 3, 1996, Pages 225263, ISSN 0888613X, http://dx.doi.org/10.1016/S0888613X(96)000692.

F. R. Kschischang, B. J. Frey and H. A. Loeliger, "Factor graphs and the sumproduct algorithm," in IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498519, Feb 2001. doi: 10.1109/18.910572

Kjærulff, Uffe. 1997. Nested junction trees. In Proceedings of the Thirteenth conference on Uncertainty in artificial intelligence (UAI’97). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 294–301.
Other Python Junction Tree Implementations

symfer  a tool suite, written mostly in Python, for performing probabilistic inference

bayesianbeliefnetworks  a library for the creation of and exact inference on Bayesian Belief Networks specified as pure Python functions
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