From 9f692e3d39b7bead111c606ad77d1301ab74f104 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Sun, 22 May 2016 20:54:33 -0700
Subject: [PATCH 1/9] start work

---
 .../functional_common_information.py          | 63 +++++++++++++++++++
 .../test_functional_common_information.py     |  0
 2 files changed, 63 insertions(+)
 create mode 100644 dit/multivariate/functional_common_information.py
 create mode 100644 dit/multivariate/tests/test_functional_common_information.py

diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
new file mode 100644
index 0000000000..980b600f6f
--- /dev/null
+++ b/dit/multivariate/functional_common_information.py
@@ -0,0 +1,63 @@
+"""
+The functional common information.
+"""
+
+from ..distconst import insert_rvf
+from ..helpers import flatten, normalize_rvs
+from ..math import close
+from ..utils import partitions
+
+from .entropy import entropy
+from .binding_information import dual_total_correlation
+
+def add_partition(dist, part):
+    invert_part = {e: str(i) for i, es in enumerate(part) for e in es}
+    dist = insert_rvf(dist, lambda j: invert_part[j])
+    return dist
+
+def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
+    rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
+    outcomes = dist.outcomes
+    f = [len(dist.rvs)]
+    parts = partitions(outcomes, tuples=True)
+    dists = [ add_partition(dist, part) for part in parts ]
+
+    B = lambda d: dual_total_correlation(d, rvs, crvs+f, rv_mode)
+
+    dists = [ d for d in dists if close(B(d), 0) ]
+    return min(dists, key=lambda d: entropy(d, rvs=f, rv_mode=rv_mode))
+
+def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):
+    """
+    Compute the functional common information, F, of `dist`. It is the entropy
+    of the smallest random variable W such that all the variables in `rvs` are
+    rendered independent conditioned on W, and W is a function of `rvs`.
+
+    Parameters
+    ----------
+    dist : Distribution
+        The distribution from which the functional common information is
+        computed.
+    rvs : list, None
+        A list of lists. Each inner list specifies the indexes of the random
+        variables used to calculate the total correlation. If None, then the
+        total correlation is calculated over all random variables, which is
+        equivalent to passing `rvs=dist.rvs`.
+    crvs : list, None
+        A single list of indexes specifying the random variables to condition
+        on. If None, then no variables are conditioned on.
+    rv_mode : str, None
+        Specifies how to interpret `rvs` and `crvs`. Valid options are:
+        {'indices', 'names'}. If equal to 'indices', then the elements of
+        `crvs` and `rvs` are interpreted as random variable indices. If equal
+        to 'names', the the elements are interpreted as random variable names.
+        If `None`, then the value of `dist._rv_mode` is consulted, which
+        defaults to 'indices'.
+
+    Returns
+    -------
+    F : float
+        The functional common information.
+    """
+    d = functional_markov_chain(dist, rvs, crvs, rv_mode)
+    return H(d.marginalize(list(flatten(dist.rvs))))
diff --git a/dit/multivariate/tests/test_functional_common_information.py b/dit/multivariate/tests/test_functional_common_information.py
new file mode 100644
index 0000000000..e69de29bb2

From bbe5642b0edd3cb7742b02451491c94eb98fdfca Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Tue, 24 May 2016 22:09:55 -0700
Subject: [PATCH 2/9] more work

---
 dit/abc.py                                    |  2 +
 dit/multivariate/__init__.py                  |  1 +
 .../functional_common_information.py          | 59 +++++++++++++++++--
 .../test_functional_common_information.py     | 44 ++++++++++++++
 4 files changed, 102 insertions(+), 4 deletions(-)

diff --git a/dit/abc.py b/dit/abc.py
index 8a99009049..ec22fbc728 100644
--- a/dit/abc.py
+++ b/dit/abc.py
@@ -21,6 +21,7 @@
                               caekl_mutual_information as J,
                               coinformation as I,
                               entropy as H,
+                              functional_common_information as F,
                               gk_common_information as K,
                               interaction_information as II,
                               joint_mss_entropy as M,
@@ -50,6 +51,7 @@
     'J',    # the CAEKL common information
     'K',    # the Gacs-Korner common information [meet entropy]
     'II',   # the interaction information
+    'F',    # the functional common information
     'M',    # the joint minimal sufficient statistic entropy
 ]
 
diff --git a/dit/multivariate/__init__.py b/dit/multivariate/__init__.py
index fb8e37bb73..69fe8abb0e 100644
--- a/dit/multivariate/__init__.py
+++ b/dit/multivariate/__init__.py
@@ -7,6 +7,7 @@
 from .caekl_mutual_information import caekl_mutual_information
 from .coinformation import coinformation
 from .entropy import entropy
+from .functional_common_information import functional_common_information
 from .gk_common_information import gk_common_information
 from .interaction_information import interaction_information
 from .joint_mss import joint_mss_entropy
diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
index 980b600f6f..65e6321e38 100644
--- a/dit/multivariate/functional_common_information.py
+++ b/dit/multivariate/functional_common_information.py
@@ -10,21 +10,72 @@
 from .entropy import entropy
 from .binding_information import dual_total_correlation
 
+__all__ = ['functional_common_information']
+
 def add_partition(dist, part):
+    """
+    Add a function of the joint distribution.
+
+    Parameters
+    ----------
+    dist : Distribution
+        The distribution to add a function to.
+    part : list of lists
+        A partition of the outcomes. Each outcome will be mapped to the id of
+        its partition element.
+
+    Returns
+    -------
+    dist : Distribution
+        The original `dist` with the function defined by `part` added.
+    """
     invert_part = {e: str(i) for i, es in enumerate(part) for e in es}
     dist = insert_rvf(dist, lambda j: invert_part[j])
     return dist
 
 def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
+    """
+    Add the smallest function of `dist` which renders `rvs` independent.
+
+    Parameters
+    ----------
+    dist : Distribution
+        The distribution for which the smallest function will be constructed.
+    rvs : list, None
+        A list of lists. Each inner list specifies the indexes of the random
+        variables used to calculate the total correlation. If None, then the
+        total correlation is calculated over all random variables, which is
+        equivalent to passing `rvs=dist.rvs`.
+    crvs : list, None
+        A single list of indexes specifying the random variables to condition
+        on. If None, then no variables are conditioned on.
+    rv_mode : str, None
+        Specifies how to interpret `rvs` and `crvs`. Valid options are:
+        {'indices', 'names'}. If equal to 'indices', then the elements of
+        `crvs` and `rvs` are interpreted as random variable indices. If equal
+        to 'names', the the elements are interpreted as random variable names.
+        If `None`, then the value of `dist._rv_mode` is consulted, which
+        defaults to 'indices'.
+
+    Returns
+    -------
+    d : Distribution
+        The distribution `dist` with the additional variable added to the end.
+    """
     rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
     outcomes = dist.outcomes
     f = [len(dist.rvs)]
-    parts = partitions(outcomes, tuples=True)
-    dists = [ add_partition(dist, part) for part in parts ]
+    parts = partitions(outcomes)
+    dists = [ (add_partition(dist, part), part) for part in parts ]
 
     B = lambda d: dual_total_correlation(d, rvs, crvs+f, rv_mode)
 
-    dists = [ d for d in dists if close(B(d), 0) ]
+    dists = [ (d, p) for d, p in dists if close(B(d), 0) ]
+    return dists
+    dists = [(entropy(d, rvs=f, rv_mode=rv_mode), d) for d in dists ]
+    return dists
+    dists = [(h, d) for h, d in dists if h == min(h for h, d in dists)]
+    return dists
     return min(dists, key=lambda d: entropy(d, rvs=f, rv_mode=rv_mode))
 
 def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):
@@ -60,4 +111,4 @@ def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):
         The functional common information.
     """
     d = functional_markov_chain(dist, rvs, crvs, rv_mode)
-    return H(d.marginalize(list(flatten(dist.rvs))))
+    return entropy(d.marginalize(list(flatten(dist.rvs))))
diff --git a/dit/multivariate/tests/test_functional_common_information.py b/dit/multivariate/tests/test_functional_common_information.py
index e69de29bb2..fae04a5d72 100644
--- a/dit/multivariate/tests/test_functional_common_information.py
+++ b/dit/multivariate/tests/test_functional_common_information.py
@@ -0,0 +1,44 @@
+"""
+Tests for dit.multivariate.functional_common_information.
+"""
+
+from __future__ import division
+
+from nose.tools import assert_almost_equal, assert_less_equal
+
+from dit import Distribution, random_distribution
+from dit.multivariate import (functional_common_information as F,
+                              dual_total_correlation as B,
+                              joint_mss_entropy as M
+                             )
+
+def test_fci1():
+    """
+    Test known values.
+    """
+    d = Distribution(['000', '011', '101', '110'], [1/4]*4)
+    assert_almost_equal(F(d), 2.0)
+    assert_almost_equal(F(d, [[0], [1]]), 0.0)
+    assert_almost_equal(F(d, [[0], [1]], [2]), 1.0)
+
+def test_fci2():
+    """
+    Test known values w/ rv names.
+    """
+    d = Distribution(['000', '011', '101', '110'], [1/4]*4)
+    d.set_rv_names('XYZ')
+    assert_almost_equal(F(d), 2.0)
+    assert_almost_equal(F(d, [[0], [1]]), 0.0)
+    assert_almost_equal(F(d, [[0], [1]], [2]), 1.0)
+
+def test_fci3():
+    """
+    Test that B <= F <= M.
+    """
+    dists = [ random_distribution(2, 2) for _ in range(10) ]
+    for d in dists:
+        b = B(d)
+        f = F(d)
+        m = M(d)
+        yield assert_less_equal, b, f
+        yield assert_less_equal, f, m

From fcea626a508bdeec8d6ed08daa41526a9a941618 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Tue, 24 May 2016 22:12:42 -0700
Subject: [PATCH 3/9] remove some testing code

---
 dit/multivariate/functional_common_information.py | 11 ++---------
 1 file changed, 2 insertions(+), 9 deletions(-)

diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
index 65e6321e38..96b25fa10e 100644
--- a/dit/multivariate/functional_common_information.py
+++ b/dit/multivariate/functional_common_information.py
@@ -66,16 +66,9 @@ def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
     outcomes = dist.outcomes
     f = [len(dist.rvs)]
     parts = partitions(outcomes)
-    dists = [ (add_partition(dist, part), part) for part in parts ]
-
+    dists = [ add_partition(dist, part) for part in parts ]
     B = lambda d: dual_total_correlation(d, rvs, crvs+f, rv_mode)
-
-    dists = [ (d, p) for d, p in dists if close(B(d), 0) ]
-    return dists
-    dists = [(entropy(d, rvs=f, rv_mode=rv_mode), d) for d in dists ]
-    return dists
-    dists = [(h, d) for h, d in dists if h == min(h for h, d in dists)]
-    return dists
+    dists = [ d for d in dists if close(B(d), 0) ]
     return min(dists, key=lambda d: entropy(d, rvs=f, rv_mode=rv_mode))
 
 def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):

From d6ec36775c907fc91bbbc13d5f56a7a8c62e9748 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Thu, 26 May 2016 17:37:05 -0700
Subject: [PATCH 4/9] do some renaming, cleaning things up, starting docs

---
 dit/abc.py                                    |  4 +-
 dit/multivariate/__init__.py                  |  7 +-
 ...formation.py => dual_total_correlation.py} | 22 +++--
 .../functional_common_information.py          | 95 ++++++++++++++++++-
 ...joint_mss.py => mss_common_information.py} | 10 +-
 ...tion.py => test_dual_total_correlation.py} |  6 +-
 .../test_functional_common_information.py     | 19 +++-
 ..._mss.py => test_mss_common_information.py} |  2 +-
 docs/conf.py                                  |  6 ++
 .../multivariate/dual_total_correlation.rst   |  4 +-
 .../functional_common_information.rst         |  6 ++
 .../multivariate/mss_common_information.rst   |  6 ++
 docs/measures/multivariate/multivariate.rst   |  4 +-
 docs/references.bib                           | 12 ++-
 site/src/MathJax/local/dit.js                 |  9 ++
 15 files changed, 181 insertions(+), 31 deletions(-)
 rename dit/multivariate/{binding_information.py => dual_total_correlation.py} (82%)
 rename dit/multivariate/{joint_mss.py => mss_common_information.py} (73%)
 rename dit/multivariate/tests/{test_binding_information.py => test_dual_total_correlation.py} (93%)
 rename dit/multivariate/tests/{test_joint_mss.py => test_mss_common_information.py} (95%)
 create mode 100644 docs/measures/multivariate/functional_common_information.rst
 create mode 100644 docs/measures/multivariate/mss_common_information.rst

diff --git a/dit/abc.py b/dit/abc.py
index ec22fbc728..559c3ad81e 100644
--- a/dit/abc.py
+++ b/dit/abc.py
@@ -24,7 +24,7 @@
                               functional_common_information as F,
                               gk_common_information as K,
                               interaction_information as II,
-                              joint_mss_entropy as M,
+                              mss_common_information as M,
                               residual_entropy as R,
                               total_correlation as T,
                               tse_complexity as TSE,
@@ -41,7 +41,7 @@
     'H',    # the joint conditional entropy
     'I',    # the multivariate conditional mututal information
     'T',    # the conditional total correlation [multi-information/integration]
-    'B',    # the conditional binding information [dual total correlation]
+    'B',    # the conditional dual total correlation [binding information]
     'R',    # the conditional residual entropy [erasure entropy]
     'TSE',  # the TSE complexity
 ]
diff --git a/dit/multivariate/__init__.py b/dit/multivariate/__init__.py
index 69fe8abb0e..b88c3f6ddf 100644
--- a/dit/multivariate/__init__.py
+++ b/dit/multivariate/__init__.py
@@ -3,13 +3,16 @@
 measures and others are more distantly related.
 """
 
-from .binding_information import binding_information, dual_total_correlation, residual_entropy, variation_of_information
 from .caekl_mutual_information import caekl_mutual_information
 from .coinformation import coinformation
+from .dual_total_correlation import (binding_information,
+                                     dual_total_correlation,
+                                     independent_information, residual_entropy,
+                                     variation_of_information)
 from .entropy import entropy
 from .functional_common_information import functional_common_information
 from .gk_common_information import gk_common_information
 from .interaction_information import interaction_information
-from .joint_mss import joint_mss_entropy
+from .mss_common_information import mss_common_information
 from .total_correlation import total_correlation
 from .tse_complexity import tse_complexity
diff --git a/dit/multivariate/binding_information.py b/dit/multivariate/dual_total_correlation.py
similarity index 82%
rename from dit/multivariate/binding_information.py
rename to dit/multivariate/dual_total_correlation.py
index 1db8bcda7b..e1ba33b0d4 100644
--- a/dit/multivariate/binding_information.py
+++ b/dit/multivariate/dual_total_correlation.py
@@ -1,5 +1,5 @@
 """
-The binding information and residual entropy.
+The dual total correlation and variation of information.
 """
 
 from ..shannon import conditional_entropy as H
@@ -7,21 +7,23 @@
 
 __all__ = ('binding_information',
            'dual_total_correlation',
+           'independent_information',
            'residual_entropy',
+           'variation_of_information',
           )
 
-def binding_information(dist, rvs=None, crvs=None, rv_mode=None):
+def dual_total_correlation(dist, rvs=None, crvs=None, rv_mode=None):
     """
-    Calculates the binding information, also known as the dual total
-    correlation.
+    Calculates the dual total correlation, also known as the binding
+    information.
 
     Parameters
     ----------
     dist : Distribution
-        The distribution from which the binding information is calculated.
+        The distribution from which the dual total correlation is calculated.
     rvs : list, None
         The indexes of the random variable used to calculate the binding
-        information. If None, then the binding information is calculated
+        information. If None, then the dual total correlation is calculated
         over all random variables.
     crvs : list, None
         The indexes of the random variables to condition on. If None, then no
@@ -37,7 +39,7 @@ def binding_information(dist, rvs=None, crvs=None, rv_mode=None):
     Returns
     -------
     B : float
-        The binding information.
+        The dual total correlation.
 
     Raises
     ------
@@ -59,6 +61,8 @@ def binding_information(dist, rvs=None, crvs=None, rv_mode=None):
 
 def residual_entropy(dist, rvs=None, crvs=None, rv_mode=None):
     """
+    Compute the residual entropy.
+
     Parameters
     ----------
     dist : Distribution
@@ -99,6 +103,6 @@ def residual_entropy(dist, rvs=None, crvs=None, rv_mode=None):
     return R
 
 
-dual_total_correlation = binding_information
+binding_information = dual_total_correlation
 
-variation_of_information = residual_entropy
+independent_information = variation_of_information = residual_entropy
diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
index 96b25fa10e..05b685d0e8 100644
--- a/dit/multivariate/functional_common_information.py
+++ b/dit/multivariate/functional_common_information.py
@@ -2,13 +2,16 @@
 The functional common information.
 """
 
-from ..distconst import insert_rvf
+from collections import deque
+from itertools import combinations
+
+from ..distconst import insert_rvf, modify_outcomes
 from ..helpers import flatten, normalize_rvs
 from ..math import close
 from ..utils import partitions
 
 from .entropy import entropy
-from .binding_information import dual_total_correlation
+from .dual_total_correlation import dual_total_correlation
 
 __all__ = ['functional_common_information']
 
@@ -29,11 +32,11 @@ def add_partition(dist, part):
     dist : Distribution
         The original `dist` with the function defined by `part` added.
     """
-    invert_part = {e: str(i) for i, es in enumerate(part) for e in es}
+    invert_part = {e: (i,) for i, es in enumerate(part) for e in es}
     dist = insert_rvf(dist, lambda j: invert_part[j])
     return dist
 
-def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
+def functional_markov_chain_naive(dist, rvs=None, crvs=None, rv_mode=None):
     """
     Add the smallest function of `dist` which renders `rvs` independent.
 
@@ -71,6 +74,88 @@ def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
     dists = [ d for d in dists if close(B(d), 0) ]
     return min(dists, key=lambda d: entropy(d, rvs=f, rv_mode=rv_mode))
 
+def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
+    """
+    Add the smallest function of `dist` which renders `rvs` independent.
+
+    Parameters
+    ----------
+    dist : Distribution
+        The distribution for which the smallest function will be constructed.
+    rvs : list, None
+        A list of lists. Each inner list specifies the indexes of the random
+        variables used to calculate the total correlation. If None, then the
+        total correlation is calculated over all random variables, which is
+        equivalent to passing `rvs=dist.rvs`.
+    crvs : list, None
+        A single list of indexes specifying the random variables to condition
+        on. If None, then no variables are conditioned on.
+    rv_mode : str, None
+        Specifies how to interpret `rvs` and `crvs`. Valid options are:
+        {'indices', 'names'}. If equal to 'indices', then the elements of
+        `crvs` and `rvs` are interpreted as random variable indices. If equal
+        to 'names', the the elements are interpreted as random variable names.
+        If `None`, then the value of `dist._rv_mode` is consulted, which
+        defaults to 'indices'.
+
+    Returns
+    -------
+    d : Distribution
+        The distribution `dist` with the additional variable added to the end.
+
+    Notes
+    -----
+    The implimentation of this function is quite slow. It is approximately
+    doubly exponential in the size of the sample space. This method is several
+    times faster than the naive method however. It remains an open question as
+    to whether a method to directly construct this variable exists (as it does
+    with the GK common variable, minimal sufficient statistic, etc).
+    """
+    rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
+
+    dist = modify_outcomes(dist, lambda x: tuple(x))
+
+    part = frozenset([ frozenset([o]) for o in dist.outcomes ]) # make copy
+
+    W = [dist.outcome_length()]
+
+    H = lambda d: entropy(d, W, rv_mode=rv_mode)
+    B = lambda d: dual_total_correlation(d, rvs, crvs+W, rv_mode)
+
+    optimal_b = dual_total_correlation(dist, rvs, crvs, rv_mode)
+
+    initial = add_partition(dist, part)
+    optimal = (H(initial), initial)
+
+    queue = deque([part])
+
+    checked = set()
+
+    while queue:
+        part = queue.popleft()
+
+        checked.add(part)
+
+        d = add_partition(dist, part)
+
+        if close(B(d), 0):
+
+            h = H(d)
+
+            if h <= optimal[0]:
+                optimal = (h, d)
+
+            if close(h, optimal_b):
+                break
+
+            new_parts = [frozenset([ p for p in part if p not in pair ] +
+                                   [pair[0]|pair[1]])
+                         for pair in combinations(part, 2) ]
+            new_parts = sorted([ part for part in new_parts if part not in checked ], key=lambda p: sorted(map(len, p)))
+            queue.extendleft(new_parts)
+
+    return optimal[1]
+
 def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):
     """
     Compute the functional common information, F, of `dist`. It is the entropy
@@ -104,4 +189,4 @@ def functional_common_information(dist, rvs=None, crvs=None, rv_mode=None):
         The functional common information.
     """
     d = functional_markov_chain(dist, rvs, crvs, rv_mode)
-    return entropy(d.marginalize(list(flatten(dist.rvs))))
+    return entropy(d, [dist.outcome_length()])
diff --git a/dit/multivariate/joint_mss.py b/dit/multivariate/mss_common_information.py
similarity index 73%
rename from dit/multivariate/joint_mss.py
rename to dit/multivariate/mss_common_information.py
index 8ad2028761..e7997e1c10 100644
--- a/dit/multivariate/joint_mss.py
+++ b/dit/multivariate/mss_common_information.py
@@ -1,13 +1,16 @@
 """
-Compute the entropy of the joint minimal sufficient statistic.
+Compute the minimal sufficient statistic common information.
 """
 
 from ..algorithms.minimal_sufficient_statistic import insert_joint_mss
+from ..helpers import normalize_rvs
 from .entropy import entropy
 
-def joint_mss_entropy(dist, rvs=None, crvs=None, rv_mode=None):
+def mss_common_information(dist, rvs=None, crvs=None, rv_mode=None):
     """
-    Compute the entropy of the join of the minimal sufficent statistic of each variable about the others.
+    Compute the minimal sufficient statistic common information, which is the
+    entropy of the join of the minimal sufficent statistic of each variable
+    about the others.
 
     Parameters
     ----------
@@ -26,6 +29,7 @@ def joint_mss_entropy(dist, rvs=None, crvs=None, rv_mode=None):
         defaults to 'indices'.
 
     """
+    rvs, crvs, rv_mode = normalize_rvs(dist, rvs, crvs, rv_mode)
     d = insert_joint_mss(dist, -1, rvs, rv_mode)
 
     M = entropy(d, [d.outcome_length() - 1], crvs, rv_mode)
diff --git a/dit/multivariate/tests/test_binding_information.py b/dit/multivariate/tests/test_dual_total_correlation.py
similarity index 93%
rename from dit/multivariate/tests/test_binding_information.py
rename to dit/multivariate/tests/test_dual_total_correlation.py
index 5029d5e195..11f4308589 100644
--- a/dit/multivariate/tests/test_binding_information.py
+++ b/dit/multivariate/tests/test_dual_total_correlation.py
@@ -1,5 +1,5 @@
 """
-Tests for dit.multivariate.binding_information.
+Tests for dit.multivariate.dual_total_correlation.
 """
 
 from __future__ import division
@@ -7,8 +7,8 @@
 from nose.tools import assert_almost_equal, assert_raises
 
 from dit import Distribution as D, ScalarDistribution as SD
-from dit.multivariate import (binding_information as B,
-                            residual_entropy as R)
+from dit.multivariate import (dual_total_correlation as B,
+                              residual_entropy as R)
 from dit.shannon import (entropy as H,
                          mutual_information as I)
 from dit.exceptions import ditException
diff --git a/dit/multivariate/tests/test_functional_common_information.py b/dit/multivariate/tests/test_functional_common_information.py
index fae04a5d72..7c8ce77cbf 100644
--- a/dit/multivariate/tests/test_functional_common_information.py
+++ b/dit/multivariate/tests/test_functional_common_information.py
@@ -4,12 +4,13 @@
 
 from __future__ import division
 
+from nose.plugins.attrib import attr
 from nose.tools import assert_almost_equal, assert_less_equal
 
 from dit import Distribution, random_distribution
 from dit.multivariate import (functional_common_information as F,
                               dual_total_correlation as B,
-                              joint_mss_entropy as M
+                              mss_common_information as M
                              )
 
 def test_fci1():
@@ -31,11 +32,25 @@ def test_fci2():
     assert_almost_equal(F(d, [[0], [1]]), 0.0)
     assert_almost_equal(F(d, [[0], [1]], [2]), 1.0)
 
+@attr('slow')
 def test_fci3():
     """
     Test that B <= F <= M.
     """
-    dists = [ random_distribution(2, 2) for _ in range(10) ]
+    dists = [ random_distribution(2, 3) for _ in range(3) ]
+    for d in dists:
+        b = B(d)
+        f = F(d)
+        m = M(d)
+        yield assert_less_equal, b, f
+        yield assert_less_equal, f, m
+
+@attr('slow')
+def test_fci4():
+    """
+    Test that B <= F <= M.
+    """
+    dists = [ random_distribution(3, 2) for _ in range(3) ]
     for d in dists:
         b = B(d)
         f = F(d)
diff --git a/dit/multivariate/tests/test_joint_mss.py b/dit/multivariate/tests/test_mss_common_information.py
similarity index 95%
rename from dit/multivariate/tests/test_joint_mss.py
rename to dit/multivariate/tests/test_mss_common_information.py
index b014dd3d25..0a9e6fdf6d 100644
--- a/dit/multivariate/tests/test_joint_mss.py
+++ b/dit/multivariate/tests/test_mss_common_information.py
@@ -7,7 +7,7 @@
 from nose.tools import assert_almost_equal
 
 from dit import Distribution
-from dit.multivariate import joint_mss_entropy as M
+from dit.multivariate import mss_common_information as M
 
 def test_M1():
     """ Test M """
diff --git a/docs/conf.py b/docs/conf.py
index 30a300dea4..513816852e 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -212,15 +212,21 @@
 \newcommand{\op}[1]{\ensuremath{\operatorname{#1}}}
 
 \renewcommand{\H}  {\op{H}}   % No more Erd\H{o}s
+
 \newcommand{\I}    {\op{I}}
 \newcommand{\T}    {\op{T}}
 \newcommand{\B}    {\op{B}}
 \newcommand{\J}    {\op{J}}
+
 \newcommand{\R}    {\op{R}}
 \newcommand{\II}   {\op{II}}
 \newcommand{\TSE}  {\op{TSE}}
 
 \newcommand{\K}    {\op{K}}
+\newcommand{\C}    {\op{C}}
+\newcommand{\G}    {\op{G}}
+\newcommand{\F}    {\op{F}}
+\newcommand{\M}    {\op{M}}
 
 \renewcommand{\P}  {\op{P}}
 \newcommand{\X}    {\op{X}}
diff --git a/docs/measures/multivariate/dual_total_correlation.rst b/docs/measures/multivariate/dual_total_correlation.rst
index f988fd0389..c0a238d1f8 100644
--- a/docs/measures/multivariate/dual_total_correlation.rst
+++ b/docs/measures/multivariate/dual_total_correlation.rst
@@ -1,11 +1,11 @@
 .. dual_total_correlation.rst
-.. py:module:: dit.multivariate.binding_information
+.. py:module:: dit.multivariate.dual_total_correlation
 
 **********************
 Dual Total Correaltion
 **********************
 
-The dual total correlation :cite:, or binding information :cite:`Abdallah2012`, is yet another generalization of the mutual information. It is the amount of information that is shared among the variables. It is defined as:
+The dual total correlation :cite:`Han1975linear`, or binding information :cite:`Abdallah2012`, is yet another generalization of the mutual information. It is the amount of information that is shared among the variables. It is defined as:
 
 .. math::
 
diff --git a/docs/measures/multivariate/functional_common_information.rst b/docs/measures/multivariate/functional_common_information.rst
new file mode 100644
index 0000000000..d20aecbf4e
--- /dev/null
+++ b/docs/measures/multivariate/functional_common_information.rst
@@ -0,0 +1,6 @@
+.. functional_common_information.rst
+.. py:module:: dit.multivariate.functional_common_information
+
+*****************************
+Functional Common Information
+*****************************
diff --git a/docs/measures/multivariate/mss_common_information.rst b/docs/measures/multivariate/mss_common_information.rst
new file mode 100644
index 0000000000..108ba8e006
--- /dev/null
+++ b/docs/measures/multivariate/mss_common_information.rst
@@ -0,0 +1,6 @@
+.. mss_common_information.rst
+.. py:module:: dit.multivariate.mss_common_information
+
+**********************
+MSS Common Information
+**********************
diff --git a/docs/measures/multivariate/multivariate.rst b/docs/measures/multivariate/multivariate.rst
index 05cd584ec0..cb4d194eb2 100644
--- a/docs/measures/multivariate/multivariate.rst
+++ b/docs/measures/multivariate/multivariate.rst
@@ -37,7 +37,8 @@ These measures all somehow measure shared information, but do not equal the mutu
    :maxdepth: 1
 
    gk_common_information
-   tse_complexity
+   functional_common_information
+   mss_common_information
 
 Others
 ======
@@ -47,3 +48,4 @@ These measures quantify other aspects of a joint distribution.
    :maxdepth: 1
 
    residual_entropy
+   tse_complexity
diff --git a/docs/references.bib b/docs/references.bib
index 60d2d05af5..3aaf13892b 100644
--- a/docs/references.bib
+++ b/docs/references.bib
@@ -68,7 +68,7 @@ @article{Gacs1973
     year={1973}
 }
 @article{Han1980,
-    author = {Han, Te Sun},
+    author = {Han, T. S.},
     doi = {10.1016/S0019-9958(80)90478-7},
     issn = {00199958},
     journal = {Information and Control},
@@ -79,6 +79,16 @@ @article{Han1980
     volume = {46},
     year = {1980}
 }
+@article{Han1975linear,
+  title={Linear dependence structure of the entropy space},
+  author={Han, T. S.},
+  journal={Information and Control},
+  volume={29},
+  pages={337--368},
+  year={1975},
+  publisher={Elsevier}
+}
+
 @article{Verdu2008,
   title={The information lost in erasures},
   author={Verdu, Sergio and Weissman, Tsachy},
diff --git a/site/src/MathJax/local/dit.js b/site/src/MathJax/local/dit.js
index 9277a2d880..099f085caf 100644
--- a/site/src/MathJax/local/dit.js
+++ b/site/src/MathJax/local/dit.js
@@ -32,16 +32,25 @@ MathJax.Hub.Register.StartupHook("TeX Jax Ready",function () {
   var TEX = MathJax.InputJax.TeX;
 
   TEX.Macro("op", "\\operatorname{#1}", 1);
+
   TEX.Macro("H", "\\op{H}");
+
+  // mutual informations
   TEX.Macro("I", "\\op{I}");
   TEX.Macro("T", "\\op{T}");
   TEX.Macro("B", "\\op{B}");
   TEX.Macro("J", "\\op{J}");
+
   TEX.Macro("R", "\\op{R}");
   TEX.Macro("II", "\\op{II}");
   TEX.Macro("TSE", "\\op{TSE}");
 
+  // common information
   TEX.Macro("K", "\\op{K}");
+  TEX.Macro("C", "\\op{C}");
+  TEX.Macro("G", "\\op{G}");
+  TEX.Macro("F", "\\op{F}");
+  TEX.Macro("M", "\\op{M}");
 
   TEX.Macro("P", "\\op{P}");
   TEX.Macro("X", "\\op{X}");

From f5b56af320bb5ae475d869bc690934154ff2d327 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Thu, 26 May 2016 22:33:28 -0700
Subject: [PATCH 5/9] some docs

---
 docs/measures/multivariate/functional_common_information.rst | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/docs/measures/multivariate/functional_common_information.rst b/docs/measures/multivariate/functional_common_information.rst
index d20aecbf4e..ba94a2b9b0 100644
--- a/docs/measures/multivariate/functional_common_information.rst
+++ b/docs/measures/multivariate/functional_common_information.rst
@@ -4,3 +4,8 @@
 *****************************
 Functional Common Information
 *****************************
+
+The functional common information captures the minimum amount of information neccessary to capture all of a distribution's share information using a function of that information. In other words:
+
+.. math::
+   \F[X_{0:n}] = \min_{\substack{W = f(X_{0:n} \\ \B[X_{0:n}|W] = 0} \H[W]

From 8a11c7a6b3227907a1484a8704aacedf0b0c9c29 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Fri, 27 May 2016 12:57:18 -0700
Subject: [PATCH 6/9] add documentation

---
 docs/conf.py                                  |  5 ++
 .../functional_common_information.rst         | 17 ++++++-
 .../multivariate/mss_common_information.rst   | 11 +++++
 docs/measures/multivariate/multivariate.rst   |  9 ++++
 docs/notation.rst                             | 48 +++++--------------
 site/src/MathJax/local/dit.js                 |  6 +++
 6 files changed, 58 insertions(+), 38 deletions(-)

diff --git a/docs/conf.py b/docs/conf.py
index 513816852e..ac13198d58 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -245,6 +245,11 @@
 \newcommand{\imore}{\succeq}
 \newcommand{\ieq}  {\cong}
 \newcommand{\mss}  {\searrow}
+
+\DeclareMathOperator*{\meetop}{\scalerel*{\meet}{\textstyle\sum}}
+\DeclareMathOperator*{\joinop}{\scalerel*{\join}{\textstyle\sum}}
+
+\newcommand{\ind}{\mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}}}
 '''
 
 latex_elements = {
diff --git a/docs/measures/multivariate/functional_common_information.rst b/docs/measures/multivariate/functional_common_information.rst
index ba94a2b9b0..6e58871759 100644
--- a/docs/measures/multivariate/functional_common_information.rst
+++ b/docs/measures/multivariate/functional_common_information.rst
@@ -8,4 +8,19 @@ Functional Common Information
 The functional common information captures the minimum amount of information neccessary to capture all of a distribution's share information using a function of that information. In other words:
 
 .. math::
-   \F[X_{0:n}] = \min_{\substack{W = f(X_{0:n} \\ \B[X_{0:n}|W] = 0} \H[W]
+
+   \F[X_{0:n}] = \min_{\substack{\ind X_{0:n} \mid W \\ W = f(X_{0:n})}} \H[W]
+
+Relationship To Other Measures of Common Information
+====================================================
+
+Since this is an additional constraint on the Exact common information, it is generally larger than it, and since its constraint is weaker than that of the :doc:`mss_common_information`, it is generally less than it:
+
+.. math::
+
+   \G[X_{0:n}] \leq \F[X_{0:n}] \leq \M[X_{0:n}]
+
+API
+===
+
+.. autofunction:: functional_common_information
diff --git a/docs/measures/multivariate/mss_common_information.rst b/docs/measures/multivariate/mss_common_information.rst
index 108ba8e006..51138c370a 100644
--- a/docs/measures/multivariate/mss_common_information.rst
+++ b/docs/measures/multivariate/mss_common_information.rst
@@ -4,3 +4,14 @@
 **********************
 MSS Common Information
 **********************
+
+The Minimal Sufficient Statistic Common Information is the entropy of the join of the minimal sufficient statistic of each variable about the others:
+
+.. math::
+
+   \M[X_{0:n}] = \H\left[ \joinop_i X_i \mss X_\overline{\{i\}} \right]
+
+API
+===
+
+.. autofunction:: mss_common_information
diff --git a/docs/measures/multivariate/multivariate.rst b/docs/measures/multivariate/multivariate.rst
index cb4d194eb2..9a7d59866a 100644
--- a/docs/measures/multivariate/multivariate.rst
+++ b/docs/measures/multivariate/multivariate.rst
@@ -40,6 +40,15 @@ These measures all somehow measure shared information, but do not equal the mutu
    functional_common_information
    mss_common_information
 
+Ordering
+--------
+
+The common information measures (togehter with the :doc:`dual_total_correlation`) form an ordering:
+
+.. math::
+
+   \K[X_{0:n}] \B[X_{0:n}] \leq \F[X_{0:n}] \leq \M[X_{0:n}]
+
 Others
 ======
 These measures quantify other aspects of a joint distribution.
diff --git a/docs/notation.rst b/docs/notation.rst
index 3c5ab34fb5..e574927cef 100644
--- a/docs/notation.rst
+++ b/docs/notation.rst
@@ -4,50 +4,24 @@
 Notation
 ********
 
-``dit`` is a scientific tool, and so, much of this documentation will contain
-mathematical expressions. Here we will describe this notation.
+``dit`` is a scientific tool, and so, much of this documentation will contain mathematical expressions. Here we will describe this notation.
 
 Basic Notation
 ==============
 
-A random variable :math:`X` consists of *outcomes* :math:`x` from an *alphabet*
-:math:`\mathcal{X}`. As such, we write the entropy of a distribution as
-:math:`\H[X] = \sum_{x \in \mathcal{X}} p(x) \log_2 p(x)`, where :math:`p(x)`
-denote the probability of the outcome :math:`x` occuring.
-
-Many distributions are *joint* distribution. In the absence of variable names,
-we index each random variable with a subscript. For example, a distribution over
-three variables is written :math:`X_0X_1X_2`. As a shorthand, we also denote
-those random variables as :math:`X_{0:3}`, meaning start with :math:`X_0` and go
-through, but not including :math:`X_3` — just like python slice notation.
-
-If we ever need to describe an infinitely long chain of variables we drop the
-index from the side that is infinite. So :math:`X_{:0} = \ldots
-X_{-3}X_{-2}X_{-1}` and :math:`X_{0:} = X_0X_1X_2\ldots`. For an arbitrary set
-of indices :math:`A`, the corresponding collection of random variables is
-denoted :math:`X_A`. For example, if :math:`A = \{0,2,4\}`, then :math:`X_A =
-X_0 X_2 X_4`. The complement of :math:`A` (with respect to some universal set)
-is denoted :math:`\bar{A}`.
+A random variable :math:`X` consists of *outcomes* :math:`x` from an *alphabet* :math:`\mathcal{X}`. As such, we write the entropy of a distribution as :math:`\H[X] = \sum_{x \in \mathcal{X}} p(x) \log_2 p(x)`, where :math:`p(x)` denote the probability of the outcome :math:`x` occuring.
+
+Many distributions are *joint* distribution. In the absence of variable names, we index each random variable with a subscript. For example, a distribution over three variables is written :math:`X_0X_1X_2`. As a shorthand, we also denote those random variables as :math:`X_{0:3}`, meaning start with :math:`X_0` and go through, but not including :math:`X_3` — just like python slice notation.
+
+If a set of variables :math:`X_{0:n}` are independent, we will write :math:`\ind X_{0:n}`. If a set of variables :math:`X_{0:n}` are independent conditioned on :math:`V`, we write :math:`\ind X_{0:n} \mid V`.
+
+If we ever need to describe an infinitely long chain of variables we drop the index from the side that is infinite. So :math:`X_{:0} = \ldots X_{-3}X_{-2}X_{-1}` and :math:`X_{0:} = X_0X_1X_2\ldots`. For an arbitrary set of indices :math:`A`, the corresponding collection of random variables is denoted :math:`X_A`. For example, if :math:`A = \{0,2,4\}`, then :math:`X_A = X_0 X_2 X_4`. The complement of :math:`A` (with respect to some universal set) is denoted :math:`\overline{A}`.
 
 Furthermore, we define :math:`0 \log_2 0 = 0`.
 
 Advanced Notation
 =================
 
-When there exists a function :math:`Y = f(X)` we write :math:`X \imore Y`
-meaning that :math:`X` is *informationally richer* than :math:`Y`. Similarly, if
-:math:`f(Y) = X` then we write :math:`X \iless Y` and say that :math:`X` is
-*informationally poorer* than :math:`Y`. If :math:`X \iless Y` and :math:`X
-\imore Y` then we write :math:`X \ieq Y` and say that :math:`X` is
-*informationally equivalent* to :math:`Y`. Of all the variables that are poorer
-than both :math:`X` and :math:`Y`, there is a richest one. This variable is
-known as the *meet* of :math:`X` and :math:`Y` and is denoted :math:`X \meet Y`.
-By definition, :math:`\forall Z s.t. Z \iless X` and :math:`Z \iless Y, Z \iless
-X \meet Y`. Similarly of all variables richer than both :math:`X` and :math:`Y`,
-there is a poorest. This variable is known as the *join* of :math:`X` and
-:math:`Y` and is denoted :math:`X \join Y`. The joint random variable
-:math:`(X,Y)` and the join are informationally equivalent: :math:`(X,Y) \ieq X
-\join Y`.
-
-Lastly, we use :math:`X \mss Y` to denote the minimal sufficient statistic of
-:math:`X` about the random variable :math:`Y`.
+When there exists a function :math:`Y = f(X)` we write :math:`X \imore Y` meaning that :math:`X` is *informationally richer* than :math:`Y`. Similarly, if :math:`f(Y) = X` then we write :math:`X \iless Y` and say that :math:`X` is *informationally poorer* than :math:`Y`. If :math:`X \iless Y` and :math:`X \imore Y` then we write :math:`X \ieq Y` and say that :math:`X` is *informationally equivalent* to :math:`Y`. Of all the variables that are poorer than both :math:`X` and :math:`Y`, there is a richest one. This variable is known as the *meet* of :math:`X` and :math:`Y` and is denoted :math:`X \meet Y`. By definition, :math:`\forall Z s.t. Z \iless X` and :math:`Z \iless Y, Z \iless X \meet Y`. Similarly of all variables richer than both :math:`X` and :math:`Y`, there is a poorest. This variable is known as the *join* of :math:`X` and :math:`Y` and is denoted :math:`X \join Y`. The joint random variable :math:`(X,Y)` and the join are informationally equivalent: :math:`(X,Y) \ieq X \join Y`.
+
+Lastly, we use :math:`X \mss Y` to denote the minimal sufficient statistic of :math:`X` about the random variable :math:`Y`.
diff --git a/site/src/MathJax/local/dit.js b/site/src/MathJax/local/dit.js
index 099f085caf..a090f42c98 100644
--- a/site/src/MathJax/local/dit.js
+++ b/site/src/MathJax/local/dit.js
@@ -71,6 +71,12 @@ MathJax.Hub.Register.StartupHook("TeX Jax Ready",function () {
   TEX.Macro("ieq", "\\cong");
   TEX.Macro("mss", "\\searrow");
 
+  TEX.Macro("meetop", "\\DeclareMathOperator*{\\meetop}{\\scalerel*{\\meet}{\\textstyle\\sum}}")
+  TEX.Macro("joinop", "\\DeclareMathOperator*{\\joinop}{\]scalerel*{\]join}{\]textstyle\]sum}}")
+
+  TEX.Macro("ind", "\\mathrel{\\text{\\scalebox{1.07}{$\\perp\\mkern-10mu\\perp$}}}")
+
+
   // don't use stix, it's pretty ugly
   MathJax.Hub.Config({
     "HTML-CSS": { availableFonts: ["TeX"] }

From f0978b726ca477152cb1ece0199f3d703e48f8dc Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Fri, 27 May 2016 13:13:58 -0700
Subject: [PATCH 7/9] missed a \leq

---
 docs/measures/multivariate/multivariate.rst | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/measures/multivariate/multivariate.rst b/docs/measures/multivariate/multivariate.rst
index 9a7d59866a..4df01de4fd 100644
--- a/docs/measures/multivariate/multivariate.rst
+++ b/docs/measures/multivariate/multivariate.rst
@@ -47,7 +47,7 @@ The common information measures (togehter with the :doc:`dual_total_correlation`
 
 .. math::
 
-   \K[X_{0:n}] \B[X_{0:n}] \leq \F[X_{0:n}] \leq \M[X_{0:n}]
+   \K[X_{0:n}] \leq \B[X_{0:n}] \leq \F[X_{0:n}] \leq \M[X_{0:n}]
 
 Others
 ======

From 966e1cce6ed26a86eb13a74bbbe788b53cc9c0a0 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Fri, 27 May 2016 13:42:50 -0700
Subject: [PATCH 8/9] don't cover the naive version

---
 dit/multivariate/functional_common_information.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
index 05b685d0e8..68cbe3e024 100644
--- a/dit/multivariate/functional_common_information.py
+++ b/dit/multivariate/functional_common_information.py
@@ -36,7 +36,7 @@ def add_partition(dist, part):
     dist = insert_rvf(dist, lambda j: invert_part[j])
     return dist
 
-def functional_markov_chain_naive(dist, rvs=None, crvs=None, rv_mode=None):
+def functional_markov_chain_naive(dist, rvs=None, crvs=None, rv_mode=None): # pragma: no cover
     """
     Add the smallest function of `dist` which renders `rvs` independent.
 

From 0c4a149e045421e8dc4fabb24f3ece996e01a605 Mon Sep 17 00:00:00 2001
From: Ryan James <ryangregoryjames@gmail.com>
Date: Fri, 27 May 2016 14:04:21 -0700
Subject: [PATCH 9/9] more tests, better coverage

---
 .../functional_common_information.py          |  2 +-
 .../test_functional_common_information.py     | 26 +++++++++++++++--
 dit/profiles/entropy_triangle.py              |  2 +-
 dit/profiles/information_partitions.py        |  2 +-
 dit/profiles/tests/test_entropy_triangle.py   | 28 ++++++++++++++++++-
 5 files changed, 54 insertions(+), 6 deletions(-)

diff --git a/dit/multivariate/functional_common_information.py b/dit/multivariate/functional_common_information.py
index 68cbe3e024..d2ac0360ad 100644
--- a/dit/multivariate/functional_common_information.py
+++ b/dit/multivariate/functional_common_information.py
@@ -131,7 +131,7 @@ def functional_markov_chain(dist, rvs=None, crvs=None, rv_mode=None):
 
     checked = set()
 
-    while queue:
+    while queue: # pragma: no branch
         part = queue.popleft()
 
         checked.add(part)
diff --git a/dit/multivariate/tests/test_functional_common_information.py b/dit/multivariate/tests/test_functional_common_information.py
index 7c8ce77cbf..f5ece67a53 100644
--- a/dit/multivariate/tests/test_functional_common_information.py
+++ b/dit/multivariate/tests/test_functional_common_information.py
@@ -32,8 +32,30 @@ def test_fci2():
     assert_almost_equal(F(d, [[0], [1]]), 0.0)
     assert_almost_equal(F(d, [[0], [1]], [2]), 1.0)
 
-@attr('slow')
 def test_fci3():
+    """ Test against known values """
+    outcomes = ['000',
+                'a00',
+                '00c',
+                'a0c',
+                '011',
+                'a11',
+                '101',
+                'b01',
+                '01d',
+                'a1d',
+                '10d',
+                'b0d',
+                '110',
+                'b10',
+                '11c',
+                'b1c',]
+    pmf = [1/16]*16
+    d = Distribution(outcomes, pmf)
+    assert_almost_equal(F(d), 2.0)
+
+@attr('slow')
+def test_fci4():
     """
     Test that B <= F <= M.
     """
@@ -46,7 +68,7 @@ def test_fci3():
         yield assert_less_equal, f, m
 
 @attr('slow')
-def test_fci4():
+def test_fci5():
     """
     Test that B <= F <= M.
     """
diff --git a/dit/profiles/entropy_triangle.py b/dit/profiles/entropy_triangle.py
index 1d618bba0a..edcbc291eb 100644
--- a/dit/profiles/entropy_triangle.py
+++ b/dit/profiles/entropy_triangle.py
@@ -61,7 +61,7 @@ def __init__(self, dists):
 
     @staticmethod
     @abstractmethod
-    def _compute_point(dist):
+    def _compute_point(dist): # pragma: no cover
         """
         Compute the three normalized axis.
 
diff --git a/dit/profiles/information_partitions.py b/dit/profiles/information_partitions.py
index 9f22f257be..25059da187 100644
--- a/dit/profiles/information_partitions.py
+++ b/dit/profiles/information_partitions.py
@@ -62,7 +62,7 @@ def __init__(self, dist):
 
     @staticmethod
     @abstractmethod
-    def _symbol(rvs, crvs):
+    def _symbol(rvs, crvs): # pragma: no cover
         """
         This method should return the information symbol for an atom.
         """
diff --git a/dit/profiles/tests/test_entropy_triangle.py b/dit/profiles/tests/test_entropy_triangle.py
index fa2dd7fca5..e659f8c6b7 100644
--- a/dit/profiles/tests/test_entropy_triangle.py
+++ b/dit/profiles/tests/test_entropy_triangle.py
@@ -4,7 +4,7 @@
 
 from __future__ import division
 
-from nose.tools import assert_tuple_equal
+from nose.tools import assert_in, assert_tuple_equal
 
 from dit import Distribution
 from dit.profiles import EntropyTriangle, EntropyTriangle2
@@ -27,6 +27,19 @@ def test_et_1():
     for d, val in zip(examples, vals):
         yield assert_tuple_equal, EntropyTriangle(d).points[0], val
 
+def test_et_2():
+    """
+    Test EntropyTriangle against known values.
+    """
+    vals = [(0, 0, 1),
+            (0, 1, 0),
+            (0, 2/3, 1/3),
+            (0, 1, 0),
+           ]
+    et = EntropyTriangle(examples)
+    for val in vals:
+        yield assert_in, val, et.points
+
 
 def test_et2_1():
     """
@@ -39,3 +52,16 @@ def test_et2_1():
            ]
     for d, val in zip(examples, vals):
         yield assert_tuple_equal, EntropyTriangle2(d).points[0], val
+
+def test_et_2():
+    """
+    Test EntropyTriangle against known values.
+    """
+    vals = [(1, 0, 0),
+            (0, 2/3, 1/3),
+            (1/3, 1/3, 1/3),
+            (0, 1/3, 2/3),
+           ]
+    et = EntropyTriangle2(examples)
+    for val in vals:
+        yield assert_in, val, et.points