Potentials : named tensors

In pyAgrum, Potentials represent multi-dimensionnal arrays with (discrete) random variables attached to each dimension. This mathematical object have tensorial operators w.r.t. to the variables attached.

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In [1]:

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb

a,b,c=[gum.LabelizedVariable(s,s,2) for s in "abc"]

potential algebra

In [2]:

p1=gum.Potential().add(a).add(b).fillWith([1,2,3,4]).normalize()
p2=gum.Potential().add(b).add(c).fillWith([4,5,2,3]).normalize()

In [3]:

gnb.flow.row(p1,p2,p1+p2,
              captions=['p1','p2','p1+p2'])
a
b
0
1
0
0.10000.2000
1
0.30000.4000

p1
b
c
0
1
0
0.28570.3571
1
0.14290.2143

p2
b
a
c
0
1
0
0
0.38570.6571
1
0.24290.5143
1
0
0.48570.7571
1
0.34290.6143

p1+p2
In [4]:

p3=p1+p2
gnb.showPotential(p3/p3.sumOut(["b"]))
c
b
a
0
1
0
0
0.36990.3208
1
0.39080.3582
1
0
0.63010.6792
1
0.60920.6418
In [5]:

p4=gum.Potential()+p3
gnb.flow.row(p3,p4,
              captions=['p3','p4'])
b
a
c
0
1
0
0
0.38570.6571
1
0.24290.5143
1
0
0.48570.7571
1
0.34290.6143

p3
b
a
c
0
1
0
0
1.38571.6571
1
1.24291.5143
1
0
1.48571.7571
1
1.34291.6143

p4

Bayes’ theorem

In [6]:

bn=gum.fastBN("a->c;b->c",3)
bn

Out[6]:
G a a c c a->c b b b->c

In such a small bayes net, we can directly manipulate \(P(a,b,c)\). For instance :

\[P(b|c)=\frac{\sum_{a} P(a,b,c)}{\sum_{a,b} P(a,b,c)}\]
In [7]:

pABC=bn.cpt("a")*bn.cpt("b")*bn.cpt("c")
pBgivenC=(pABC.sumOut(["a"])/pABC.sumOut(["a","b"]))

pBgivenC.putFirst("b") # in order to have b horizontally in the table

Out[7]:
b
c
0
1
2
0
0.00220.97860.0192
1
0.00580.97330.0209
2
0.00380.97520.0210

Joint, marginal probability, likelihood

Let’s compute the joint probability \(P(A,B)\) from \(P(A,B,C)\)

In [8]:

pAC=pABC.sumOut(["b"])
print("pAC really is a probability : it sums to {}".format(pAC.sum()))
pAC

pAC really is a probability : it sums to 0.9999999999999998

Out[8]:
a
c
0
1
2
0
0.14220.12400.0582
1
0.12990.07040.1512
2
0.06660.11150.1459

Computing \(p(A)\)

In [9]:

pAC.sumOut(["c"])

Out[9]:
a
0
1
2
0.33870.30600.3553

Computing \(p(A |C=1)\)

It is easy to compute \(p(A, C=1)\)

In [10]:

pAC.extract({"c":1})

Out[10]:
a
0
1
2
0.12990.07040.1512

Moreover, we know that \(P(C=1)=\sum_A P(A,C=1)\)

In [11]:

pAC.extract({"c":1}).sum()

Out[11]:

0.35154423108981625

Now we can compute \(p(A|C=1)=\frac{P(A,C=1)}{p(C=1)}\)

In [12]:

pAC.extract({"c":1}).normalize()

Out[12]:
a
0
1
2
0.36940.20040.4302

Computing \(P(A|C)\)

\(P(A|C)\) is represented by a matrix that verifies \(p(A|C)=\frac{P(A,C)}{P(C}\)

In [13]:

pAgivenC=(pAC/pAC.sumIn("c")).putFirst("a")
# putFirst("a") : to correctly show a cpt, the first variable have to bethe conditionned one
gnb.flow.row(pAgivenC,pAgivenC.extract({'c':1}),
               captions=["$P(A|C)$","$P(A|C=1)$"])
a
c
0
1
2
0
0.43840.38230.1793
1
0.36940.20040.4302
2
0.20560.34420.4502

$P(A|C)$
a
0
1
2
0.36940.20040.4302

$P(A|C=1)$

Likelihood \(P(A=2|C)\)

A likelihood can also be found in this matrix.

In [14]:

pAgivenC.extract({'a':2})

Out[14]:
c
0
1
2
0.17930.43020.4502

A likelihood does not have to sum to 1. It is not relevant to normalize it.

In [15]:

pAgivenC.sumIn(["a"])

Out[15]:
a
0
1
2
1.01330.92691.0598

entropy of potential

In [16]:

%matplotlib inline
from pylab import *
import matplotlib.pyplot as plt
import numpy as np

In [17]:

p1=gum.Potential().add(a)
x = np.linspace(0, 1, 100)
plt.plot(x,[p1.fillWith([p,1-p]).entropy() for p in x])
plt.show()
../_images/notebooks_93-Tools_potentials_31_0.svg 
In [18]:

t=gum.LabelizedVariable('t','t',3)
p1=gum.Potential().add(t)

def entrop(bc):
    """
    bc is a list [a,b,c] close to a distribution
    (normalized just to be sure)
    """
    return p1.fillWith(bc).normalize().entropy()

import matplotlib.tri as tri

corners = np.array([[0, 0], [1, 0], [0.5, 0.75**0.5]])
triangle = tri.Triangulation(corners[:, 0], corners[:, 1])

# Mid-points of triangle sides opposite of each corner
midpoints = [(corners[(i + 1) % 3] + corners[(i + 2) % 3]) / 2.0 \
             for i in range(3)]
def xy2bc(xy, tol=1.e-3):
    """
    From 2D Cartesian coordinates to barycentric.
    """
    s = [(corners[i] - midpoints[i]).dot(xy - midpoints[i]) / 0.75 \
         for i in range(3)]
    return np.clip(s, tol, 1.0 - tol)

def draw_entropy(nlevels=200, subdiv=6, **kwargs):
    import math

    refiner = tri.UniformTriRefiner(triangle)
    trimesh = refiner.refine_triangulation(subdiv=subdiv)
    pvals = [entrop(xy2bc(xy)) for xy in zip(trimesh.x, trimesh.y)]

    plt.tricontourf(trimesh, pvals, nlevels, **kwargs)
    plt.axis('equal')
    plt.xlim(0, 1)
    plt.ylim(0, 0.75**0.5)
    plt.axis('off')

draw_entropy()
plt.show()
../_images/notebooks_93-Tools_potentials_32_0.svg