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☰  Causality_Counterfactual 
pyAgrum 0.17.3   
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generation: 2020-04-27 18:59  

Creative Commons License
This pyAgrum's notebook is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
Author: Aymen Merrouche and Pierre-Henri Wuillemin.

**The Effect of Education and Experience on Salary**

This notebook follows the example from "The Book Of Why" (Pearl, 2018) chapter 8 page 251.

Counterfactuals

In [1]:
%load_ext autoreload
%autoreload 2
In [2]:
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
import os
import math
import numpy as np
import scipy.stats

In this example we are interested in the effect of experience and education on the salary of an employee, we are in possession of the following data:

Employé EX(u) ED(u) $S_{0}(u)$ $S_{1}(u)$ $S_{2}(u)$
Alice 8 0 86,000 ? ?
Bert 9 1 ? 92,500 ?
Caroline 9 2 ? ? 97,000
David 8 1 ? 91,000 ?
Ernest 12 1 ? 100,000 ?
Frances 13 0 97,000 ? ?
etc
  • $EX(u)$ : years of experience of employee $u$. [0,20]
  • $ED(u)$ : Level of education of employee $u$ (0:high school degree (low), 1:college degree (medium), 2:graduate degree (high)) [0,2]
  • $S_{i}(u)$ [65k,150k] :
    • salary (observable) of employee $u$ if $i = ED(u)$,
    • Potential outcome (unobservable) if $i \not = ED(u)$, salary of employee $u$ if he had a level of education of $i$.

We are left with the previous data and we want to answer the counterfactual question What would Alice's salary be if she attended college ? (i.e. $S_{1}(Alice)$)

We create the causal diagram

In this model it is assumed that an employee's salary is determined by his level of education and his experience. Years of experience are also affected by the level of education. Having a higher level of education means spending more time studying hence less experience.

In [3]:
edex = gum.fastBN("Ux[-2,10]->experience[0,20]<-education{low|medium|high}->salary[65,150]<-Us[0,25];experience->salary")
edex
Out[3]:
G Ux Ux experience experience Ux->experience salary salary experience->salary education education education->experience education->salary Us Us Us->salary

However counterfactual queries are specific to one datapoint (in our case Alice), we need to add additional variables to our model to allow for individual variations:

  • Us : unobserved variables that affect salary.[0,25k]
  • Ux : unobserved variables that affect experience.[-2,10]
In [4]:
# no prior information about the individual (datapoint)
edex.cpt("Us").fillWith(1).normalize()
edex.cpt("Ux").fillWith(1).normalize()
# education level(supposed)
edex.cpt("education")[:] = [0.4, 0.4, 0.2]
In [5]:
# To have probabilistic results, we add a perturbation. (Gaussian around the exact values)
# we calculate a gaussian distribution 
x_min = 0.0
x_max = 4.0

mean = 2.0
std = 0.65

x = np.linspace(x_min, x_max, 5)

y = scipy.stats.norm.pdf(x,mean,std)
print("We'll use the following distribution \n",y)
We'll use the following distribution 
 [0.00539715 0.18794845 0.61375735 0.18794845 0.00539715]

Experience listens to Education and Ux : $$Ex = 10 -4 \times Ed + Ux$$

In [6]:
edex.cpt("experience").fillWithFunction("10-4*education+Ux",noise=list(y))
edex.cpt("experience")
Out[6]:
experience
education
Ux
0
1
2
3
4
5
6
7
8
9
10
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16
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20
low
-2
0.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
-1
0.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.0000
0
0.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.0000
1
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.0000
2
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.0000
3
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.0000
4
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.0000
5
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.0000
6
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.0000
7
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.0000
8
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.0054
9
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18890.61680.1889
10
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00670.23290.7604
medium
-2
0.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
-1
0.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0
0.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
1
0.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
2
0.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
3
0.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.0000
4
0.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.0000
5
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.0000
6
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.0000
7
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.0000
8
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.0000
9
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.0000
10
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.0000
high
-2
0.76040.23290.00670.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
-1
0.18890.61680.18890.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0
0.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
1
0.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
2
0.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
3
0.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
4
0.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
5
0.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
6
0.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
7
0.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.00000.0000
8
0.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.00000.0000
9
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.00000.0000
10
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00540.18790.61350.18790.00540.00000.00000.00000.00000.00000.0000

Salary listens to Education, Experience and Us : $$S = 65 + 2.5 \times Ex + 5 \times Ed + Us$$

In [83]:
edex.cpt("salary").fillWithFunction("round(65+2.51*experience+5*education+Us)",noise=list(y))
gnb.showInference(edex)
structs Inference in   1.33ms Ux experience Ux->experience salary experience->salary education education->experience education->salary Us Us->salary

To answer this counterfactual question we will follow the three steps algorithm from "The Book Of Why" (Pearl 2018) chapter 8 page 253 :

Step 1 : Abduction

Use the data to retrieve all the information that characterizes Alice

From the data we can retrieve Alice's profile :

  • $Ed(Alice)$ : 0
  • $Ex(Alice)$ : 8
  • $S_{0}(Alice)$ : 86k

We will use Alice's profile to get $U_s$ and $U_x$, which tell Alice apart from the rest of the data.

In [84]:
ie=gum.LazyPropagation(edex)
ie.setEvidence({'experience':8, 'education': 'low', 'salary' : "86"})
ie.makeInference()
newUs = ie.posterior("Us")
newUs
Out[84]:
Us
0
1
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3
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5
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7
8
9
10
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12
13
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25
0.18890.61680.18890.00540.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
In [85]:
ie=gum.LazyPropagation(edex)
ie.setEvidence({'experience':8, 'education': 'low', 'salary' : "86"})
ie.makeInference()
newUx = ie.posterior("Ux")
newUx
Out[85]:
Ux
-2
-1
0
1
2
3
4
5
6
7
8
9
10
0.76040.23290.00670.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
In [86]:
gnb.showInference(edex,evs={'experience':8, 'education': "low", 'salary' : "86"},targets={'Ux','Us'})
structs Inference in   1.66ms Ux experience experience Ux->experience salary salary experience->salary education education education->experience education->salary Us Us->salary

Step 2 & 3 : Action And Prediction

Change the model to match the hypothesis implied by the query (if she had attended university) and then use the data that characterizes Alice to calculate her salary.

We create a counterfactual world with Alice's idiosyncratic factors, and we operate the intervention:

In [87]:
# the counterfactual world
edexCounterfactual = gum.BayesNet(edex)
In [88]:
# we replace the prior probabilities of idiosynatric factors with potentials calculated earlier
edexCounterfactual.cpt("Ux").fillWith(newUx)
edexCounterfactual.cpt("Us").fillWith(newUs)
gnb.showInference(edexCounterfactual,size="10")
print("counterfactual world created")
structs Inference in   1.11ms Ux experience Ux->experience salary experience->salary education education->experience education->salary Us Us->salary
counterfactual world created
In [89]:
# We operate the intervention
edexModele = csl.CausalModel(edexCounterfactual)
cslnb.showCausalImpact(edexModele,"salary",doing="education",values={"education":"medium"})
G Ux Ux experience experience Ux->experience salary salary experience->salary education education education->experience education->salary Us Us Us->salary
$$\begin{equation}P( salary \mid \hookrightarrow\mkern-6.5mueducation) = \sum_{Us,Ux,experience}{P\left(Us\right) \cdot P\left(salary\mid Us,education,experience\right) \cdot P\left(experience\mid Ux,education\right) \cdot P\left(Ux\right)}\end{equation}$$
salary
65
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0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00020.00100.00200.00660.03420.08460.15250.23570.13070.08840.13200.07920.03540.01280.00280.00120.00060.00010.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
Causal Model
Explanation : Do-calculus computations
Impact : $P( salary \mid \hookrightarrow\mkern-6.5mueducation=medium)$

Since education has no parents in our model (no graph surgery, no causes to emancipate it from), an intervention is equivalent to an observation, the only thing we need to do is to set the value of education:

In [90]:
gnb.showInference(edexCounterfactual,targets={"salary",'experience'},evs={'education':"medium"},size="10")
structs Inference in   3.27ms Ux Ux experience Ux->experience salary experience->salary education education education->experience education->salary Us Us Us->salary

The result (salary if she had attended college) is given by the formaula: $$\sum_{salary} salary \times P(salary^* \mid RealSalary = 86k, education = 0, experience = 8, education^*=1)$$ Where variables marked with an asterisk are inobservable.

In [93]:
formula, adj, exp = csl.causalImpact(edexModele,"salary",doing="education",values={"education":"medium"})
gnb.showProba(adj)
In [94]:
i = gum.Instantiation(adj)
i.setFirst()
mean = 0
while (not i.end()):
    v = i.val(0)
    mean = mean + (v+65)*adj.get(i)
    i.inc()
print(mean)
81.84325639929719

$$S_1(Alice) = 81k $$ Alice's salary would be $\$81.843$ if she had attended college !

We can now use a function that answers counterfactual queries using the previous algorithm (in pyAgrum.causal.counterfactual)

In [95]:
help(csl.counterfactual)
Help on function counterfactual in module pyAgrum.causal._causalImpact:

counterfactual(cm: pyAgrum.causal._CausalModel.CausalModel, profile: Union[Dict[str, int], NoneType], on: Union[str, Set[str]], whatif: Union[str, Set[str]], values: Union[Dict[str, int], NoneType] = None) -> pyAgrum.pyAgrum.Potential
    Determines the estimation of a counterfactual query following the the three steps algorithm from "The Book Of Why" (Pearl 2018) chapter 8 page 253.
    
    Determines the estimation of the counterfactual query: Given the "profile" (dictionary <variable name>:<value>),what would variables in "on" (single or list of variables) be if variables in "whatif" (single or list of variables) had been as specified in "values" (dictionary <variable name>:<value>)(optional).
    
    This is done according to the following algorithm:
        -Step 1-2: compute the twin causal model
        -Step 3 : determine the causal impact of the interventions specified in  "whatif" on the single or list of variables "on" in the causal model.
    
    This function returns the potential calculated in step 3, representing the probability distribution of  "on" given the interventions  "whatif", if it had been as specified in "values" (if "values" is omitted, every possible value of "whatif")
    
    :param cm: CausalModel
    :param profile: Dictionary
    :param on: variable name or variable names set
    :param whatif: variable name or variable names set
    :param values: Dictionary
    :type cm: pyAgrum.causal.CausalModel
    :type profile: Union[Dict[str, int], type(None)]
    :type on: Union[str, Set[str]]
    :type whatif: Union[str, Set[str]]
    :type values: Union[Dict[str, int], type(None)]
    :return: the computation
    :rtype: gum.Potential

Let's try with the previous query :

In [96]:
cm_edex= csl.CausalModel(edex)
pot=csl.counterfactual(cm =cm_edex, 
                       profile = {'experience':8, 'education': "low", 'salary' : "86"},
                       whatif={"education"},
                       on={"salary"}, 
                       values = {"education" : "medium"})
In [97]:
gnb.showProba(pot)

We get the same result !

If we omit values:

We get every potential outcome :

In [98]:
pot=csl.counterfactual(cm =cm_edex, 
                       profile = {'experience':8, 'education': 'low', 'salary' : '86'},
                       whatif={"education"},
                       on={"salary"})
In [99]:
gnb.showPotential(pot)
education
salary
low
medium
high
65
0.00000.00000.0000
66
0.00000.00000.0000
67
0.00000.00000.0000
68
0.00000.00000.0000
69
0.00000.00000.0000
70
0.00000.00000.0000
71
0.00000.00000.0000
72
0.00000.00000.0000
73
0.00000.00000.0006
74
0.00000.00020.0242
75
0.00000.00100.1449
76
0.00000.00200.2800
77
0.00000.00660.1580
78
0.00000.03420.1012
79
0.00020.08460.1480
80
0.00100.15250.0877
81
0.00200.23570.0375
82
0.00660.13070.0132
83
0.03420.08840.0028
84
0.08460.13200.0012
85
0.15250.07920.0006
86
0.23570.03540.0001
87
0.13070.01280.0000
88
0.08840.00280.0000
89
0.13200.00120.0000
90
0.07920.00060.0000
91
0.03540.00010.0000
92
0.01280.00000.0000
93
0.00280.00000.0000
94
0.00120.00000.0000
95
0.00060.00000.0000
96
0.00010.00000.0000
97
0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
100
0.00000.00000.0000
101
0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
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0.00000.00000.0000
108
0.00000.00000.0000
109
0.00000.00000.0000
110
0.00000.00000.0000
111
0.00000.00000.0000
112
0.00000.00000.0000