Some examples of do-calculus

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aGrUM

interactive online version

In [1]:
from IPython.display import display, Math, Latex, HTML

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.causal as csl
import pyAgrum.causal.notebook as cslnb

S. Tikka and J. Karvanen, 2016 [CRAN]

In [2]:
bn = gum.fastBN("w->x->z->y;w->z")

bn.cpt("w")[:] = [0.7,0.3]

bn.cpt("x")[:] = [[0.4,0.6],[0.3,0.7]]

bn.cpt("z")[{'w':0,'x':0}]=[0.2,0.8]
bn.cpt("z")[{'w':0,'x':1}]=[0.1,0.9]
bn.cpt("z")[{'w':1,'x':0}]=[0.9,0.1]
bn.cpt("z")[{'w':1,'x':1}]=[0.5,0.5]

bn.cpt("y")[:] = [[0.1,0.9],[0.8,0.2]]


d = csl.CausalModel(bn, [("lat1", ["x","y"])])
#csl.causalImpact(d,"y",{"x":0})
cslnb.showCausalImpact(d,"y","x",values={"x":0})
cslnb.showCausalImpact(d,"y","x",values={"x":1})


G lat1 x x lat1->x y y lat1->y w w w->x z z w->z x->z z->y
Causal Model
$$\begin{equation*}P( y \mid \text{do}(x)) = \sum_{w,z}{\left(\sum_{x'}{P\left(x'\mid w\right) \cdot P\left(y\mid w,x',z\right)}\right) \cdot P\left(z\mid w,x\right) \cdot P\left(w\right)}\end{equation*}$$
Explanation : Do-calculus computations
y
0
1
0.51300.4870

Impact
G lat1 x x lat1->x y y lat1->y w w w->x z z w->z x->z z->y
Causal Model
$$\begin{equation*}P( y \mid \text{do}(x)) = \sum_{w,z}{\left(\sum_{x'}{P\left(x'\mid w\right) \cdot P\left(y\mid w,x',z\right)}\right) \cdot P\left(z\mid w,x\right) \cdot P\left(w\right)}\end{equation*}$$
Explanation : Do-calculus computations
y
0
1
0.64600.3540

Impact

Since we have the formula, let us compute by hand this intervention :

In [3]:
(((bn.cpt("x") * bn.cpt("y")).sumOut(["x"]) * bn.cpt("w") * bn.cpt("z")).sumOut(["z", "w"])).putFirst("y")
Out[3]:
y
x
0
1
0
0.51300.4870
1
0.64600.3540
In [4]:
bn = gum.fastBN("Z1->X->Z2->Y")
d=csl.CausalModel(bn,[("L1",["Z1","X"]),
                      ("L2",["Z1","Z2"]),
                      ("L3",["Z1","Y"]),
                      ("L4",["Y","X"])],
                  True)
cslnb.showCausalImpact(d,"Y","X",values={"X":1})
G L1 Z1 Z1 L1->Z1 X X L1->X L2 L2->Z1 Z2 Z2 L2->Z2 L3 L3->Z1 Y Y L3->Y L4 L4->X L4->Y Z1->X X->Z2 Z2->Y
Causal Model
Hedge Error: G={'X', 'Z1', 'Z2'}, G[S]={'Z2'}
Impossible
No result
Impact

Front door

In [5]:
modele4 = gum.BayesNet()
modele4.add(gum.LabelizedVariable("Smoking"))
modele4.add(gum.LabelizedVariable("Cancer"))
modele4.add(gum.LabelizedVariable("Tar"))

modele4.addArc(0,2)
modele4.addArc(2,1)
modele4.addArc(0,1)

#Smoking
modele4.cpt(0)[:]=[0.5, 0.5]

#Tar
modele4.cpt(2)[{"Smoking":0}] = [0.4, 0.6]
modele4.cpt(2)[{"Smoking":1}] = [0.3, 0.6]

#Cancer
modele4.cpt(1)[{'Smoking':0,'Tar':0}]=[0.1,0.9]   #No Drug, Male -> healed in 0.8 of cases
modele4.cpt(1)[{'Smoking':0,'Tar':1}]=[0.15,0.85] #No Drug, Female -> healed in 0.4 of cases
modele4.cpt(1)[{'Smoking':1,'Tar':0}]=[0.2,0.8]   #Drug, Male -> healed 0.7 of cases
modele4.cpt(1)[{'Smoking':1,'Tar':1}]=[0.25,0.75]

d4 = csl.CausalModel(modele4, [("Genotype", ["Smoking","Cancer"])],False)
cslnb.showCausalModel(d4)
../_images/notebooks_64-Causality_DoCalculusExamples_9_0.svg
In [6]:
try:
    a = csl.doCalculusWithObservation (d4,"Cancer", {"Smoking"})
except csl.HedgeException as h:
    print (h.message)
In [7]:
display(Math(a.toLatex()))
$\displaystyle P( Cancer \mid \text{do}(Smoking)) = \sum_{Tar}{P\left(Tar\mid Smoking\right) \cdot \left(\sum_{Smoking'}{P\left(Smoking'\right) \cdot P\left(Cancer\mid Smoking',Tar\right)}\right)}$
In [8]:
try:
    adjj = a.eval()
except csl.UnidentifiableException as u:
    print (u.message)

print (adjj)


      ||  Cancer           |
Smokin||0        |1        |
------||---------|---------|
0     || 0.1774  | 0.8226  |
1     || 0.1626  | 0.7374  |

In [9]:
formula, adj, exp = csl.causalImpact(d4, "Cancer", "Smoking",values={"Smoking":0})
In [10]:
display(Math(formula.toLatex()))
adj
$\displaystyle P( Cancer \mid \text{do}(Smoking)) = \sum_{Tar}{P\left(Tar\mid Smoking\right) \cdot \left(\sum_{Smoking'}{P\left(Cancer\mid Smoking',Tar\right) \cdot P\left(Smoking'\right)}\right)}$
Out[10]:
Cancer
0
1
0.17740.8226

## Last example from R

In [11]:
m = gum.fastBN("z2->x->z1->y;z2->z1;z2->z3->y")

m.cpt("z2") [:] = [0.5, 0.5]
m.cpt("x") [:] = [[0.4,0.6], #z2=0
                  [0.4,0.6]] #z2=1
m.cpt("z3") [:] = [[0.3,0.7], #z2=0
                   [0.3,0.7]] #z2=1
m.cpt("z1")[{"z2":0, "x":0}]= [0.2, 0.8]
m.cpt("z1")[{"z2":0, "x":1}]= [0.25, 0.75]
m.cpt("z1")[{"z2":1, "x":0}]= [0.1, 0.9]
m.cpt("z1")[{"z2":1, "x":1}]= [0.15, 0.85]

m.cpt("y")[{"z1":0,"z3":0}]= [0.5,0.5]
m.cpt("y")[{"z1":0,"z3":1}]= [0.45,0.55]
m.cpt("y")[{"z1":1,"z3":0}]= [0.4,0.6]
m.cpt("y")[{"z1":1,"z3":1}]= [0.35,0.65]


d = csl.CausalModel(m, [("X-Z2",["x","z2"]),
                        ("X-Z3",["x","z3"]),
                        ("X-Y",["x","y"]),
                        ("Y-Z2",["y","z2"])],
                    True)

cslnb.showCausalModel(d)
../_images/notebooks_64-Causality_DoCalculusExamples_16_0.svg
In [12]:
try:
    formula,result,msg = csl.causalImpact(d,on={"y","z2","z1","z3"},doing={"x"})
except csl.HedgeException as h:
    print (h.message)

print(msg)
display(Math(formula.toLatex()))
Do-calculus computations
$\displaystyle P( z3,z2,y,z1 \mid \text{do}(x)) = P\left(z3\mid z2\right) \cdot P\left(z1\mid x,z2\right) \cdot P\left(z2\right) \cdot \frac {\sum_{x'}{P\left(x'\mid z2\right) \cdot P\left(z2\right) \cdot P\left(y\mid x',z1,z2,z3\right) \cdot P\left(z3\mid x',z2\right)}}{\sum_{x',y'}{P\left(x'\mid z2\right) \cdot P\left(z2\right) \cdot P\left(y'\mid x',z1,z2,z3\right) \cdot P\left(z3\mid x',z2\right)}}$
In [13]:
# computation for this formula directly in pyAgrum
f1=m.cpt("x")*m.cpt("z2")*m.cpt("z3")*m.cpt("y")
f2=f1.sumOut(["x"])
f3=f1.sumOut(["x","y"])
f4=f2/f3
pyResult=m.cpt("z3")*m.cpt("z1")*m.cpt("z2")*f4
In [14]:
# computation for this formula directly by creating the causal AST
a = csl.ASTposteriorProba(m,{"z1"},{"x","z2"})
b= csl.ASTposteriorProba(m,{"y","z3"},{"x","z1","z2"})
c = csl.ASTjointProba(["x","z2"])
correct = csl.ASTmult(a,csl.ASTsum(["x"],csl.ASTmult(b,c)))

print("According to [ref], the result should be :")
display(Math(correct.toLatex()))
According to [ref], the result should be :
$\displaystyle P\left(z1\mid x,z2\right) \cdot \left(\sum_{x}{P\left(y,z3\mid z1,z2\right) \cdot P\left(x,z2\right)}\right)$
In [15]:
# computation for that formula
ie=gum.LazyPropagation(m)
refResult=(ie.evidenceJointImpact(["y","z3"],["x","z1","z2"])*
     ie.evidenceJointImpact(["x","z2"],[])
    ).sumOut(["x"])* m.cpt("z1")

In [16]:
print("Maximum error between these 3 versions : {}".format(max((refResult-pyResult).abs().max(),
                                                               (refResult-result).abs().max(),
                                                               (pyResult-result).new_abs().max())))
Maximum error between these 3 versions : 5.551115123125783e-17

Unidentifiabilty

In [17]:
m1 = gum.fastBN("z1->x->z2->y")

cdg = csl.CausalModel(m1, [("Z1−X",["z1","x"]),
                           ("Z1-Y",["z1","y"]),
                           ("Z1-Z1",["z1","z2"]),
                           ("X−Y",["x","y"])
                          ], True )
cslnb.showCausalModel(cdg)
../_images/notebooks_64-Causality_DoCalculusExamples_23_0.svg
In [18]:
err = cslnb.showCausalImpact(cdg,"y","x",values={"x":0})

G Z1−X z1 z1 Z1−X->z1 x x Z1−X->x Z1-Y Z1-Y->z1 y y Z1-Y->y Z1-Z1 Z1-Z1->z1 z2 z2 Z1-Z1->z2 X−Y X−Y->x X−Y->y z1->x x->z2 z2->y
Causal Model
Hedge Error: G={'x', 'z2', 'z1'}, G[S]={'z2'}
Impossible
No result
Impact

another one

In [19]:
# EXEMPLE PAGE 17 : http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf


m1 = gum.BayesNet()
m1.add(gum.LabelizedVariable("x"))
m1.add(gum.LabelizedVariable("y"))
m1.add(gum.LabelizedVariable("z1"))
m1.add(gum.LabelizedVariable("z2"))
m1.add(gum.LabelizedVariable("z3"))

m1.addArc(2,4)
m1.addArc(2,0)
m1.addArc(3,4)
m1.addArc(3,1)
m1.addArc(4,1)
m1.addArc(4,0)
m1.addArc(0,1)

gnb.showBN(m1)
d = csl.CausalModel(m1)
../_images/notebooks_64-Causality_DoCalculusExamples_26_0.svg
In [20]:
display(Math(csl.identifyingIntervention (d,{"z1","z2","z3","y"}, {"x"}).toLatex()))
$\displaystyle P\left(z3\mid z1,z2\right) \cdot P\left(z2\right) \cdot P\left(z1\right) \cdot P\left(y\mid x,z2,z3\right)$
In [21]:
display(Math(csl.identifyingIntervention(d,{"y"}, {"x"}).toLatex()))
$\displaystyle \sum_{z1,z2,z3}{P\left(z3\mid z1,z2\right) \cdot P\left(z2\right) \cdot P\left(z1\right) \cdot P\left(y\mid x,z2,z3\right)}$
In [22]:
display(Math(csl.identifyingIntervention(d,{"z1","z3","y"}, {"x","z2"}).toLatex()))
$\displaystyle P\left(z3\mid z1,z2\right) \cdot P\left(z1\right) \cdot P\left(y\mid x,z2,z3\right)$
In [23]:
display(Math(csl.identifyingIntervention(d,{"y"}, {"x","z2"}).toLatex()))
$\displaystyle \sum_{z1,z3}{P\left(z3\mid z1,z2\right) \cdot P\left(z1\right) \cdot P\left(y\mid x,z2,z3\right)}$

Other example

In [24]:
#http://www.stats.ox.ac.uk/~lienart/gml15_causalinference.html

m1 = gum.BayesNet()
m1.add(gum.LabelizedVariable("a"))
m1.add(gum.LabelizedVariable("p"))
m1.add(gum.LabelizedVariable("b"))
m1.add(gum.LabelizedVariable("y"))

m1.addArc(0,1)
m1.addArc(1,2)
m1.addArc(0,3)
m1.addArc(1,3)
m1.addArc(2,3)
gnb.showBN(m1)
d = csl.CausalModel(m1)
../_images/notebooks_64-Causality_DoCalculusExamples_32_0.svg
In [25]:
display(Math(csl.identifyingIntervention(d,{"y"}, {"a","b"}).toLatex()))
$\displaystyle \sum_{p}{P\left(y\mid a,b,p\right) \cdot P\left(p\mid a\right)}$

example f

In [26]:
#https://cse.sc.edu/~mgv/talks/AIM2010.ppt , example (f)

m1 = gum.BayesNet()
m1.add(gum.LabelizedVariable("X"))
m1.add(gum.LabelizedVariable("Y"))
m1.add(gum.LabelizedVariable("Z1"))
m1.add(gum.LabelizedVariable("Z2"))

m1.addArc(0,1)
m1.addArc(0,2)
m1.addArc(2,3)
m1.addArc(3,1)
m1.addArc(2,1)
d = csl.CausalModel(m1, [("l1",["X","Z1"]) ,("l2",["Y","Z1"])],True)
cslnb.showCausalModel(d)
../_images/notebooks_64-Causality_DoCalculusExamples_35_0.svg
In [27]:
try:
    display(Math(csl.identifyingIntervention (d,{"Y"}, {"X"}).toLatex()))
except csl.HedgeException as e:
    print("Hedge exception : {}".format(e))
Hedge exception : Hedge Error: G={'Y', 'X', 'Z1'}, G[S]={'Y', 'Z1'}

Example [Pearl,2009] Causality, p66

In [28]:
bn = gum.fastBN("Z1->Z2->Z3->Y<-X->Z2;Z2->Y;Z1->X->Z3<-Z1")
gnb.showBN(bn)
../_images/notebooks_64-Causality_DoCalculusExamples_38_0.svg
In [29]:
c = csl.CausalModel(bn, [("Z0", ("X", "Z1", "Z3"))], False)
cslnb.showCausalModel(c)
../_images/notebooks_64-Causality_DoCalculusExamples_39_0.svg
In [30]:
formula, impact, explanation = csl.causalImpact(c, "Y", "X")
cslnb.showCausalImpact(c,"Y","X")
G Z0 Z1 Z1 Z0->Z1 Z3 Z3 Z0->Z3 X X Z0->X Z2 Z2 Z1->Z2 Z2->Z3 Y Y Z2->Y Z3->Y X->Z2 X->Y
Causal Model
$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{Z1,Z2,Z3}{P\left(Y\mid X,Z2,Z3\right) \cdot P\left(Z2\mid X,Z1\right) \cdot \left(\sum_{X'}{P\left(Z3\mid X',Z1,Z2\right) \cdot P\left(Z1\right) \cdot P\left(X'\mid Z1\right)}\right)}\end{equation*}$$
Explanation : Do-calculus computations
Y
X
0
1
0
0.54820.4518
1
0.61700.3830

Impact