Utility function to assess conditional independence (#791)

* Adds utility functions to assess conditional independence.

* Do not return random variables that are part of the query

* Added unit tests for edward.util.get_irrelevant

* Fixed a typo in the docstring for get_irrelevant

* Added unit tests for ed.is_independent

* Removed helper function get_irrelevant

* Update comments to respect PEP8

* More PEP-8 related changes.

* Updated docstrings and added bibtex entry for bayes-ball.

Author: bkayalibay

docs/tex/bib.bib

@@ -265,6 +265,14 @@ @article{lecun1998gradient
   publisher={IEEE}
 }
 
+@inproceedings{schachter1998bayes,
+  title={Bayes-Ball: The Rational Pastime (for Determining Irrelevance and Requisite Information in Belief Networks and Influence Diagrams)},
+  author={Schachter, Ross D},
+  booktitle={Proceedings of the Fourteenth Conference in Uncertainty in Artificial Intelligence},
+  pages={480--487},
+  year={1998}
+}
+
 @article{watts1998collective,
   title={Collective dynamics of ‘small-world’networks},
   author={Watts, Duncan J and Strogatz, Steven H},

edward/__init__.py

@@ -20,7 +20,8 @@
 from edward.util import check_data, check_latent_vars, copy, dot, \
     get_ancestors, get_blanket, get_children, get_control_variate_coef, \
     get_descendants, get_parents, get_session, get_siblings, get_variables, \
-    Progbar, random_variables, rbf, set_seed, to_simplex, transform
+    is_independent, Progbar, random_variables, rbf, set_seed, \
+    to_simplex, transform
 from edward.version import __version__, VERSION
 
 from tensorflow.python.util.all_util import remove_undocumented
@@ -74,6 +75,7 @@
     'get_session',
     'get_siblings',
     'get_variables',
+    'is_independent',
     'Progbar',
     'random_variables',
     'rbf',

edward/util/__init__.py

@@ -27,6 +27,7 @@
     'get_session',
     'get_siblings',
     'get_variables',
+    'is_independent',
     'Progbar',
     'random_variables',
     'rbf',

edward/util/random_variables.py

@@ -9,6 +9,7 @@
 from copy import deepcopy
 from edward.models.random_variable import RandomVariable
 from edward.models.random_variables import TransformedDistribution
+from edward.models import PointMass
 from edward.util.graphs import random_variables
 from tensorflow.contrib.distributions import bijectors
 from tensorflow.core.framework import attr_value_pb2
@@ -714,6 +715,85 @@ def get_variables(x, collection=None):
   return list(output)
 
 
+def is_independent(a, b, condition=None):
+  """Assess whether a is independent of b given the random variables in
+  condition.
+
+  Implemented using the Bayes-Ball algorithm [@schachter1998bayes].
+
+  Args:
+    a: RandomVariable or list of RandomVariable.
+       Query node(s).
+    b: RandomVariable or list of RandomVariable.
+       Query node(s).
+    condition: RandomVariable or list of RandomVariable, optional.
+       Random variable(s) to condition on.
+
+  Returns:
+    bool.
+    True if a is independent of b given the random variables in condition.
+
+  #### Examples
+
+  ```python
+  a = Normal(0.0, 1.0)
+  b = Normal(a, 1.0)
+  c = Normal(a, 1.0)
+  assert ed.is_independent(b, c, condition=a)
+  ```
+  """
+  if condition is None:
+    condition = []
+  if not isinstance(a, list):
+    a = [a]
+  if not isinstance(b, list):
+    b = [b]
+  if not isinstance(condition, list):
+    condition = [condition]
+  A = set(a)
+  B = set(b)
+  condition = set(condition)
+
+  top_marked = set()
+  # The Bayes-Ball algorithm will traverse the belief network
+  # and add each node that is relevant to B given condition
+  # to the set bottom_marked. A and B are conditionally
+  # independent if no node in A is in bottom_marked.
+  bottom_marked = set()
+
+  schedule = [(node, "child") for node in B]
+  while schedule:
+    node, came_from = schedule.pop()
+
+    if node not in condition and came_from == "child":
+      if node not in top_marked:
+        top_marked.add(node)
+        for parent in get_parents(node):
+          schedule.append((parent, "child"))
+
+      if not isinstance(node, PointMass) and node not in bottom_marked:
+        bottom_marked.add(node)
+        if node in A:
+          return False  # node in A is relevant to B
+        for child in get_children(node):
+          schedule.append((child, "parent"))
+
+    elif came_from == "parent":
+      if node in condition and node not in top_marked:
+        top_marked.add(node)
+        for parent in get_parents(node):
+          schedule.append((parent, "child"))
+
+      elif node not in condition and node not in bottom_marked:
+        bottom_marked.add(node)
+        if node in A:
+          return False  # node in A is relevant to B
+        for child in get_children(node):
+          schedule.append((child, "parent"))
+
+  return True
+
+
 def transform(x, *args, **kwargs):
   """Transform a continuous random variable to the unconstrained space.
 

tests/util/test_is_independent.py

@@ -0,0 +1,64 @@
+from __future__ import absolute_import
+from __future__ import division
+from __future__ import print_function
+
+import tensorflow as tf
+
+from edward.models import Normal
+from edward.util import is_independent
+
+
+class test_is_independent_class(tf.test.TestCase):
+
+  def test_chain_structure(self):
+    """a -> b -> c -> d -> e"""
+    a = Normal(0.0, 1.0)
+    b = Normal(a, 1.0)
+    c = Normal(b, 1.0)
+    d = Normal(c, 1.0)
+    e = Normal(d, 1.0)
+    self.assertTrue(is_independent(c, e, d))
+    self.assertTrue(is_independent([a, b, c], e, d))
+    self.assertTrue(is_independent([a, b], [d, e], c))
+    self.assertFalse(is_independent([a, b, e], d, c))
+
+  def test_binary_structure(self):
+    """f <- c <- a -> b -> d
+            |         |
+            v         v
+            g         e
+    """
+    a = Normal(0.0, 1.0)
+    b = Normal(a, 1.0)
+    c = Normal(a, 1.0)
+    d = Normal(b, 1.0)
+    e = Normal(b, 1.0)
+    f = Normal(c, 1.0)
+    g = Normal(c, 1.0)
+    self.assertFalse(is_independent(b, c))
+    self.assertTrue(is_independent(b, c, a))
+    self.assertTrue(is_independent(d, [a, c, e, f, g], b))
+    self.assertFalse(is_independent(b, [e, d], a))
+    self.assertFalse(is_independent(a, [b, c, d, e, f, g]))
+
+  def test_grid_structure(self):
+    """a -> b -> c
+       |    |    |
+       v    v    v
+       d -> e -> f
+    """
+    a = Normal(0.0, 1.0)
+    b = Normal(a, 1.0)
+    c = Normal(b, 1.0)
+    d = Normal(a, 1.0)
+    e = Normal(b + d, 1.0)
+    f = Normal(e + c, 1.0)
+    self.assertFalse(is_independent(f, [a, b, d]))
+    self.assertTrue(is_independent(f, [a, b, d], [e, c]))
+    self.assertTrue(is_independent(e, [a, c], [b, d]))
+    self.assertFalse(is_independent(e, f, [b, d]))
+    self.assertFalse(is_independent(e, f, [a, b, c, d]))
+
+
+if __name__ == '__main__':
+  tf.test.main()