pyAgrum 1.8.1 on Jupyter

Tutorials

• ▶ Tutorial
  • ▷ Tutorial
  • ▷ Tutorial2
• ▶ Examples
  • ▷ Asthma
  • ▷ Kaggle Titanic
  • ▷ Naive Credit Default Modeling
  • ▷ Causality And Learning
  • ▷ Sensitivity Analysis Using Credal Networks
  • ▷ Quasi Continuous
  • ▷ Parameters Learning With Pandas
  • ▷ Bayesian Beta Coin
• ▶ Models
  • ▷ Influence Diagram
  • ▷ Dynamic Bn
  • ▷ Markov Random Field
  • ▷ Credal Networks
  • ▷ O3PRM
• ▶ Learning
  • ▷ Structural Learning
  • ▷ Learning Classifier
  • ▷ Learning And Essential Graphs
  • ▷ Dirichlet Prior And Weigthed Database
  • ▷ Parametric Em
  • ▷ Chi2 And Scores From Bn Learner
• ▶ Inference
  • ▷ Graphical Inference
  • ▷ Relevance Reasoning
  • ▷ Lazy Propagation Advanced Features
  • ▷ Approximate Inference
  • ▷ Sampling Inference
• ▶ Classifier
  • ▷ Learning
  • ▷ Discretizer
  • ▷ Compare Classifiers With Sklearn
  • ▷ Cross Validation
  • ▷ Binary And Nary Classifier From Bn
• ▶ Causality
  • ▷ Tobacco
  • ▷ Simpson Paradox
  • ▷ Multinomial Simpson Paradox
  • ▷ Do Calculus Examples
  • ▷ Counterfactual
• ▶ Applications
  • ▷ Ipywidgets
• ▶ Tools
  • ▷ Potentials
  • ▷ Aggregators
  • ▷ Explain
  • ▷ Kl For BNs
  • ▷ Comparing Bn
  • ▷ Colouring And Exporting BNs
  • ▷ Config For PyAgrum

pyAgrum

Counterfactual : 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

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.

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

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]

# 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]

# 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 : $$Ex = 10 -4 \times Ed + Ux$$

edex.cpt("experience").fillWithFunction("10-4*education+Ux",noise=list(y))
edex.cpt("experience")

		experience
education	Ux	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
low	-2	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	-1	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	1	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	2	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	3	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000
	4	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000
	5	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000
	6	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000
	7	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000
	8	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054
	9	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1889	0.6168	0.1889
	10	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0067	0.2329	0.7604
medium	-2	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	-1	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	1	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	2	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	3	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	4	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	5	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	6	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	7	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000
	8	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000
	9	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000
	10	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000
high	-2	0.7604	0.2329	0.0067	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	-1	0.1889	0.6168	0.1889	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	0	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	1	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	2	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	3	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	4	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	5	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	6	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	7	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	8	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	9	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	10	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0054	0.1879	0.6135	0.1879	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

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

edex.cpt("salary").fillWithFunction("round(65+2.51*experience+5*education+Us)",noise=list(y))
gnb.showInference(edex)

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.

ie=gum.LazyPropagation(edex)
ie.setEvidence({'experience':8, 'education': 'low', 'salary' : "86"})
ie.makeInference()
newUs = ie.posterior("Us")
newUs

Us
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
0.1889	0.6168	0.1889	0.0054	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

ie=gum.LazyPropagation(edex)
ie.setEvidence({'experience':8, 'education': 'low', 'salary' : "86"})
ie.makeInference()
newUx = ie.posterior("Ux")
newUx

Ux
-2	-1	0	1	2	3	4	5	6	7	8	9	10
0.7604	0.2329	0.0067	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

gnb.showInference(edex,evs={'experience':8, 'education': "low", 'salary' : "86"},targets={'Ux','Us'})

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:

# the counterfactual world
edexCounterfactual = gum.BayesNet(edex)

# 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")

counterfactual world created

# We operate the intervention
edexModele = csl.CausalModel(edexCounterfactual)
cslnb.showCausalImpact(edexModele,"salary",doing="education",values={"education":"medium"})

Causal Model

$$\begin{equation*}P( salary \mid \hookrightarrow\mkern-6.5mueducation) = \sum_{Us,Ux,experience}{P\left(Us\right) \cdot P\left(salary\mid Us,education,experience\right) \cdot P\left(experience\mid Ux,education\right) \cdot P\left(Ux\right)}\end{equation*}$$
Explanation : Do-calculus computations

salary
65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88	89	90	91	92	93	94	95	96	97	98	99	100	101	102	103	104	105	106	107	108	109	110	111	112	113	114	115	116	117	118	119	120	121	122	123	124	125	126	127	128	129	130	131	132	133	134	135	136	137	138	139	140	141	142	143	144	145	146	147	148	149	150
0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0002	0.0010	0.0020	0.0066	0.0342	0.0846	0.1525	0.2357