☰  CausalityAndLearning
In :
import pyAgrum as gum
import pyAgrum.lib.notebook as gnb


# Model¶

Let's assume a process $X_1\rightarrow Y_1$ with a control on $X_1$ by $C_a$ and a parameter $P_b$ on $Y_1$.

In :
bn=gum.fastBN("Ca->X1->Y1<-Pb")
bn.cpt("Ca").fillWith([0.8,0.2])
bn.cpt("Pb").fillWith([0.3,0.7])

bn.cpt("X1")[:]=[[0.9,0.1],[0.1,0.9]]

bn.cpt("Y1")[{"X1":0,"Pb":0}]=[0.8,0.2]
bn.cpt("Y1")[{"X1":1,"Pb":0}]=[0.2,0.8]
bn.cpt("Y1")[{"X1":0,"Pb":1}]=[0.6,0.4]
bn.cpt("Y1")[{"X1":1,"Pb":1}]=[0.4,0.6]

gnb.sideBySide(bn,bn.cpt("Ca"),bn.cpt("Pb"),bn.cpt("X1"),bn.cpt("Y1"))

Ca
0
1
0.80000.2000
Pb
0
1
0.30000.7000
X1
Ca
0
1
0
0.90000.1000
1
0.10000.9000
Y1
Pb
X1
0
1
0
0
0.80000.2000
1
0.20000.8000
1
0
0.60000.4000
1
0.40000.6000

Actually the process is duplicated in the system but the control $C_a$ and the parameter $P_b$ are shared.

In :
bn.add("X2",2)

bn.cpt("X2").fillWith(bn.cpt("X1"),["X1","Ca"]) # copy cpt(X1) with the translation X2<-X1,Ca<-Ca
bn.cpt("Y2").fillWith(bn.cpt("Y1"),["Y1","X1","Pb"]) # copy cpt(Y1) with translation Y2<-Y1,X2<-X1,Pb<-Pb

gnb.sideBySide(bn,bn.cpt("X2"),bn.cpt("Y2"))

X2
Ca
0
1
0
0.90000.1000
1
0.10000.9000
Y2
Pb
X2
0
1
0
0
0.80000.2000
1
0.20000.8000
1
0
0.60000.4000
1
0.40000.6000

# Simulation of the data¶

The process is partially observed : the control has been taken into account. However the parameter has not been identified and therefore is not collected.

In :
#the base will be saved in completeData="out/complete_data.csv", observedData="out/observed_data.csv"
import os
completeData=os.path.join("out","complete_data.csv")
observedData=os.path.join("out","observed_data.csv")
fixedObsData=os.path.join("res","fixed_observed_data.csv")

# generating complete date with pyAgrum
size=35000
#gum.generateCSV(bn,"data.csv",5000,random_order=True)
generator=gum.BNDatabaseGenerator(bn)
generator.setRandomVarOrder()
generator.drawSamples(size)
generator.toCSV(completeData)

In :
# selecting some variables using pandas
import pandas as pd
keep_col = ["X1","Y1","X2","Y2","Ca"] # Pb is removed
new_f = f[keep_col]
new_f.to_csv(observedData, index=False)


In order to have fixed results, we will use now a database fixed_observed_data.csv generated with the process above.

# statistical learning¶

Using a classical statistical learning method, one can approximate a model from the observed data.

In :
learner=gum.BNLearner(fixedObsData)
learner.useGreedyHillClimbing()
bn2=learner.learnBN()

In :
gnb.sideBySide(bn,bn2,
captions=['Original model','Learned model (with no $P_b$ in the base)'])

 G Ca Ca X1 X1 Ca->X1 X2 X2 Ca->X2 Y1 Y1 X1->Y1 Pb Pb Pb->Y1 Y2 Y2 Pb->Y2 X2->Y2 G X1 X1 Y1 Y1 X1->Y1 X2 X2 Y2 Y2 X2->Y2 Y2->Y1 Ca Ca Ca->X1 Ca->X2 Original model Learned model (with no $P_b$ in the base)

# Evaluating the impact of $X2$ on $Y1$¶

Using the database, a question for the user is to evaluate the impact of the value of $X2$ on $Y1$.

In :
target="Y1"
evs="X2"
ie=gum.LazyPropagation(bn)
ie2=gum.LazyPropagation(bn2)
p1=ie.evidenceImpact(target,[evs])
p2=gum.Potential(p1).fillWith(ie2.evidenceImpact(target,[evs]),[target,evs])
errs=((p1-p2)/p1).scale(100)
captions=['in original model','in learned model','relative errors','quadratic error'])

Y1
X2
0
1
0
0.62110.3789
1
0.45080.5492
Y1
X2
0
1
0
0.61830.3817
1
0.44150.5585
Y1
X2
0
1
0
0.4422-0.7249
1
2.0453-1.6787
7.722258840106108
in original model
in learned model
relative errors

# Evaluating the causal impact of $X2$ on $Y1$ with the learned model¶

The statistician notes that the change wanted by the user to apply on $X_2$ is not an observation but rather an intervention.

In :
import pyAgrum.causal as csl
import pyAgrum.causal.notebook as cslnb

model=csl.CausalModel(bn)
model2=csl.CausalModel(bn2)
cslnb.showCausalImpact(model,target, {evs})
cslnb.showCausalImpact(model2,target, {evs})

$$\begin{equation}P( Y1 \mid \hookrightarrow\mkern-6.5muX2) = \sum_{Ca}{P\left(Y1\mid Ca\right) \cdot P\left(Ca\right)}\end{equation}$$
Y1
0
1
0.57680.4232
Causal Model
Explanation : backdoor ['Ca'] found.
Impact : $P( Y1 \mid \hookrightarrow\mkern-6.5muX2)$
$$\begin{equation}P( Y1 \mid \hookrightarrow\mkern-6.5muX2) = \sum_{X1}{P\left(Y1\mid X1,X2\right) \cdot P\left(X1\right)}\end{equation}$$
Y1
X2
0
1
0
0.57430.4257
1
0.56280.4372
Causal Model
Explanation : backdoor ['X1'] found.
Impact : $P( Y1 \mid \hookrightarrow\mkern-6.5muX2)$

Unfortunately, due to the fact that $P_a$ is not learned, the computation of the causal impact still is imprecise.

In :
_, impact1, _ = csl.causalImpact(model, on=target, doing={evs})
_, impact2orig, _ = csl.causalImpact(model2, on=target, doing={evs})

impact2=gum.Potential(p2).fillWith(impact2orig,['Y1','X2'])
errs=((impact1-impact2)/impact1).scale(100)
captions=['$P( Y_1 \mid \hookrightarrow X_2)$ in original model',
'$P( Y_1 \mid \hookrightarrow X_2)$ in learned model','relative errors','quadratic error'])

Y1
0
1
0.57680.4232
Y1
X2
0
1
0
0.57430.4257
1
0.56280.4372
Y1
X2
0
1
0
0.4414-0.6015
1
2.4251-3.3052
17.36224902446034
$P( Y_1 \mid \hookrightarrow X_2)$ in original model
$P( Y_1 \mid \hookrightarrow X_2)$ in learned model
relative errors

Just to be certain, we can verify that in the original model, $P( Y_1 \mid \hookrightarrow X_2)=P(Y_1)$

In :
gnb.sideBySide(impact1,ie.evidenceImpact(target,[]),
captions=["$P( Y_1 \mid \hookrightarrow X_2)$ in the original model","$P(Y_1)$ in the original model"])

Y1
0
1
0.57680.4232
Y1
0
1
0.57680.4232
$P( Y_1 \mid \hookrightarrow X_2)$ in the original model
$P(Y_1)$ in the original model

# Causal learning and causal impact¶

Some learning algorthims such as MIIC (Verny et al., 2017) aim to find the trace of latent variables in the data

In :
learner=gum.BNLearner(fixedObsData)
learner.useMIIC()
bn3=learner.learnBN()

In :
gnb.sideBySide(bn,bn3,[(bn3.variable(i).name(),bn3.variable(j).name()) for (i,j) in learner.latentVariables()],
captions=['original model','learned model','Latent variables found'])

 G Ca Ca X1 X1 Ca->X1 X2 X2 Ca->X2 Y1 Y1 X1->Y1 Pb Pb Pb->Y1 Y2 Y2 Pb->Y2 X2->Y2 G X1 X1 Y1 Y1 X1->Y1 X2 X2 Y2 Y2 X2->Y2 Y2->Y1 Ca Ca Ca->X1 Ca->X2 [('Y2', 'Y1')] original model learned model Latent variables found

Then we can build a causal model taking into account this latent variable found by MIIC.

In :
model3=csl.CausalModel(bn2,[("L1",("Y1","Y2"))])
cslnb.showCausalImpact(model3,target, {evs})

$$\begin{equation}P( Y1 \mid \hookrightarrow\mkern-6.5muX2) = \sum_{X1}{P\left(Y1\mid X1\right) \cdot P\left(X1\right)}\end{equation}$$
Y1
0
1
0.57250.4275
Causal Model
Explanation : backdoor ['X1'] found.
Impact : $P( Y1 \mid \hookrightarrow\mkern-6.5muX2)$

Then at least, the statistician can say that $X_2$ has no impact on $Y_1$ from the data. The error is just due to the approximation of the parameters in the database.

In :
_, impact1, _ = csl.causalImpact(model, on=target, doing={evs})
_, impact3orig, _ = csl.causalImpact(model3, on=target, doing={evs})

impact3=gum.Potential(impact1).fillWith(impact3orig,['Y1'])
errs=((impact1-impact3)/impact1).scale(100)
captions=['in original model','in learned model','relative errors','quadratic error'])

Y1
0
1
0.57680.4232
Y1
0
1
0.57250.4275
Y1
0
1
0.7454-1.0160
1.5878334363943256
in original model
in learned model
relative errors