Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market
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Transcript of Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market
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Plan
Introduction
Bayesian Networks
Application
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Probabilistic Structural Equations
Application to the Analysis of a Perfume Market
Dr. Lionel JOUFFE
August 20091
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Plan
Introduction
Bayesian Networks
Application
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written permission
BayesiaLab’s Probabilistic Structural Equations for Perfume Market Analysis
2
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Plan
Introduction
Bayesian Networks
Application
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INTRODUCTION
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Plan
Introduction
Bayesian Networks
Application
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Bayesian Networks
A Computational Tool to Model Uncertainty
Based both on graph theory and on probability theory
Manual modeling through brainstorming:
probabilistic expert systems
Induction by automatic learning:
data analysis, data mining
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Plan
Introduction
Bayesian Networks
Application
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Bayesian Networks
1763: Bayes’ Theorem
P(A|B) = P(B|A)P(A)/P(B)
1988: Judea Pearl“Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference”
1996:“Microsoft's competitive advantage is its expertise in Bayesian networks”, Bill Gates
2004: Bayesian Machine Learning at the 4th rank among the 10 Emerging Technologies That Will Change Your World
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Plan
Introduction
Bayesian Networks
Application
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Example of Probabilistic Reasoning
Letter from the analysis laboratory
“You recently went to our laboratory for a screening test. The targeted rare disease has a prevalence of one person out of ten thousand. We regret to inform you that this test, which has a symmetric efficiency of 99%, is positive.”
What is your feeling after reading this letter? Do you think that the probability that you are affected is
1%, 50% or 99%
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Plan
Introduction
Bayesian Networks
Application
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Example of Probabilistic Reasoning
Letter from the analysis laboratory
Among the 9 999 other persons, “99.99 persons” will receive a letter with a
positive test result
One person out of 10 000 is affected.He will receive “0.99 letter” with a
positive test result
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Plan
Introduction
Bayesian Networks
Application
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Example of Probabilistic Reasoning
- There is then a total of 0.99 + 99.99 letters with a positive test result
- Probability to be affected when one receives such letter:
0.99/(0.99+99.99) = 0.98%
Letter from the analysis laboratory
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Plan
Introduction
Bayesian Networks
Application
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Example of Probabilistic Reasoning
Letter from the analysis laboratory
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Plan
Introduction
Bayesian Networks
Application
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BAYESIAN BELIEF NETWORKS
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Plan
Introduction
Bayesian Networks
Application
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... are made of Two Distinct Parts
Structure
Directed Acyclic Graph (DAG), i.e. no directed loop
Nodes represent the domain’s variables
Arcs represent the direct probabilistic influences between the variables (possibly causal)
Parameters
Probability distributions are associated to each node, usually by using tables
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Plan
Introduction
Bayesian Networks
Application
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... are Powerful Inference Engines
21
We get some evidence on the states of a subset of variables
Hard positive evidence
Hard negative evidence
Likelihoods
Probability distributions (fixed or not)
Mean values (fixed or not)
We then want to take these findings into account in a rigorous way to update our belief on the states of the other variables
Probability distributions on their values
Multi-Directional Inference (Simulation and/or Diagnosis)
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Plan
Introduction
Bayesian Networks
Application
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How to Build a Bayesian Network?
Modeling by Brainstorming
Automatic Modeling by Data Mining
Productive exchange between experts that can ease the consensus
An Expert System with powerful computational and analytical abilities
Modeling of rare or never occurred cases
Probability estimation/updating of a network
Structural learning and probability estimation
Missing values Filtered/censored states Initial network proposed by experts Discovering of all the direct probabilistic relations Target node characterization - Supervised learning Data clustering Variable clustering Probabilistic Structural Equations
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Plan
Introduction
Bayesian Networks
Applications
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PROBABILISTIC STRUCTURAL EQUATIONS*
-Perfume Market Analysis
* see “Probabilistic Structural Equations and Path Analysis - Part I” (http://www.bayesia.com/en/products/bayesialab/resources/tutorials/probabilistic-structural-equations-I.php) for a detailed BayesiaLab’s tutorial describing the complete workflow to get Probabilistic Structural Equations
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Plan
Introduction
Bayesian Networks
Applications
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Perfume Market Analysis
To get an insight of the market (11 products), 1.300 monadic tests have been carried out (each woman has only evaluated one perfume).
1 target variable, the Purchase Intent: 6 numerical states
27 questions relative to the perfume : 10 numerical levels considered as continuous values and discretized into 5 numerical states (equal distances)
19 questions relative to the woman wearing the perfume: 10 numerical levels considered as continuous values and discretized into 5 numerical states (equal distances)
1 Just About Right (JAR) question for the fragrance Intensity: 5 numerical states
Questionnaire’s characteristics
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Plan
Introduction
Bayesian Networks
Applications
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Step 1: Unsupervised learning on the Manifest variables only
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the arcs’ strength
Here is the Kullback-Leibler Divergence associated to the arc, and its relative weight in the factorized representation of the Joint Probability
distribution
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Plan
Introduction
Bayesian Networks
Applications
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Step 2: Variables’ Clusteringto find the concepts
Based on those Kullback-Liebler measures, 15 clusters are automatically proposed by the BayesiaLab’s
variable clustering algorithm
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Plan
Introduction
Bayesian Networks
Applications
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Step 2: Variables’ Clustering
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Plan
Introduction
Bayesian Networks
Applications
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Step 3: Multiple Data Clustering
By using the BayesiaLab’s Multiple-Clustering algorithm, we carry out data clustering on the implied
subset of variables, for each cluster of variables.
Factor 0 is a new random variable summarizing these 5
manifest variables
Factor 2 is a new random variable that summarizes these 4 manifest variables
Factor 1 is a new random variable that summarizes these 5
manifest variables
.....
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors:Factor 0
Based on the associated variables, we name this Factor “IS SELF-CONFIDENT”
5 states have been automatically created by the BayesiaLab’s Data
Clustering algorithm. Here is the Marginal Distribution
over those 5 states.
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors:Quality measurement of Factor 0
The state’s Purity is the mean of its posterior probabilities (given the
manifest variables), over all the points that have been associated to that state with the
maximum likelihood rule
When the purity is not 100%, the remaining probabilities
are used to define the probabilistic neighborhood
The 2-dimensional representation of Factor 0. The bubble size is proportional to the prior probability, the darkness
of the blue represents the state purity, and the bubble proximity is based on the probabilistic vicinity
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors:Quality measurement of Factor 0
The 5 states of Factor 0 summarize the Joint Probability Distribution over its 5 associated manifest variables. This Joint is a 5 dimensional
hypercube, with 5 states per dimension, i.e. 5^5 cells = 3,125 probabilities
This probability density function is based on the database’s log-
Likelihood returned by Factor 0’s network
The Contingency Table Fit measures the representation quality of the Joint Probability Distribution.
100% corresponds to the perfect representation with the fully connected network (no independence hypothesis), 0% corresponds to the
representation with the fully unconnected network (no dependence hypothesis)
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors:Quality measurement of Factor 0
In the specific case of a Factor’s analysis, the dimension represented by that factor is not taken into account in the Joint. The Contingency Table Fit measures then the
quality of the Joint’s summary realized by the Factor’s states
Contingency Table Fit: 78.39% Contingency Table Fit: 85.04%
The representation of the Joint (defined over the 5 manifest variables) with the 5 states latent variable Factor 0 is more precise than the one obtained with an
unsupervised learning representing the direct probabilistic relations between the manifest variables
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors:Semantic analysis of Factor 0
The numerical value associated to each state corresponds to the mean value over the manifest variables
when this latent state is observed (weighted by the relative significance of the manifest variables wrt that state). These values
allow to have a quick insight on the meaning of the state. For example, C3 corresponds to the lowest evaluations ...
... whereas C5 corresponds to the highest ones
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Plan
Introduction
Bayesian Networks
Applications
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Analysis of the Induced Factors
Here is a table describing the Multiple Clustering key measures obtained during the data clustering of the 15 manifest variables’ clusters
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Plan
Introduction
Bayesian Networks
Applications
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Final Step: Unsupervised Learning on Manifest, Latent, and Target variables
The “Probabilistic Structural Equation” has been obtained under some constraints:no arc from Manifests toward Factorsno direct relation between Manifestsno direct relation between the Target and Manifests
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Plan
Introduction
Bayesian Networks
Applications
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The Path can be highlighted just by hiding the Manifest variables
As we can see, the Purchase Intent in only directly
connected to one Latent variable, the “ADEQUACY”
Path Analysis:Focussing on Factor variables only
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Plan
Introduction
Bayesian Networks
Applications
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Path Analysis:Focussing on Factor variables only
Factors’ Hierarchization by using the Standardized Total Effects (STE)
Graphical representation of each Factor’s influence on the Purchase Intent
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Plan
Introduction
Bayesian Networks
Applications
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Path Analysis:Focussing on Factor variables only
Our Quadrant Analysis allows to get a concise view of the Factors’ hierarchy wrt the Purchase Intent. Whereas the Y-axis is based on the Standardized Total Effect
(STE), the X-axis corresponds to the Factors’ mean value
Mean of the Mean Values
Mean of the STEs
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis: Focussing on Manifest variables only
The Bayesian network representing the Probabilistic Structural Equation (PSE) has been learnt by using the Perfume Total Market (11 products)
useful for understanding the Total Market
inappropriate for finding the levers that can be used to improve a given product
To be able to analyze the products’ drivers, we define the Product variable as a BayesiaLab’s Breakout variable
the PSE’s structure remains the same for all the products
the PSE’s parameters (conditional probability tables) are estimated, for each perfume, on its corresponding subset of lines
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis: Focussing on Manifest variables only
Only a subset of Manifest variables can be used as Drivers. The PSE below masks the non-actionable variables
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
Due to non-linearity, the Standardized Total Effect (STE) does not reflect the importance of
Intensity
This graph highlights the non linear influence of Intensity on Purchase
Intent (JAR variable)
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
Note that STE is only proposed in BayesiaLab for some analysis tools. This is not a measure used for learning Bayesian networks (BN). As the states are discrete, the
learning algorithms are not sensitive to linearity.
The analysis below ranks the Drivers wrt the Mutual Information criterion.
As we can see, Intensity is now in the 4th position
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
To be able to use STE properly, we can use BayesiaLab to linearize Intensity. It will then associate numerical values to the states in order to get a positive linear
relation (sorting of the states wrt to their relation to Purchase Intent).
Intensity is now in the 4th position with STE and with the Slopes in
the Graphical representation
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
Quadrant based on the potential Drivers
Usually this kind of quadrant can be used to quickly see what
the Drivers to prioritize are1: Concentrate here
2: Keep on the good work3: Possible overkill
4: Low priority
1 2
34
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
However, this kind of interpretation is not appropriate here. Indeed, quadrants are defined with the means (STEs and Mean Values) of the studied product. Even if a variable is located in Quadrants 1 or 4,
its value can be the highest of the Total Market. Conversely, variables belonging to Quadrants 2 and 3 can also have low values compared with the other products.
Thanks to the scales associated to each
variable, this new BayesiaLab’s Quadrant allows to quickly have an insight on how the
variables are ranked wrt the other products. Product 10 has the best Intensity value, but a
poor Flowery value (lower than the mean value over the products)
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
By hovering over the point, it is possible to have a specific view of the
variable values for all the products. The best ranked product on Flowery is then Product 11, the
worse one being Product 1
This Multiple-Quadrant tool allows to export the variation percentage needed to reach the best market value, for each product and each variable.
For Product 10, we need to apply a 10.02% increase on the Flowery mean to reach Product 11’s level.
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
We use our Target Dynamic Profile tool to estimate the most realistic action policy. Here are the optimization parameters:
maximize the Purchase Intent Mean valuetake into account the Joint Probability of the actionstake the costs into account (1 per action consisting in reaching the max authorized value)“Soft Increase” of the drivers’ mean by taking into account the exported variation values
The induced policy is then to work on Flowery, then Feminine, ....,
and Fruity, to increase the Purchase Intent Value from 3.65 to 3.92. The Joint is 50.35%, which means that
half of those product evaluations corresponds to this setting. The column “Value/Mean at T” indicates the
impact of each action on the other drivers. As we see, those impacts reduce the cost for
the actions.!"#$
!"#%$
!"&$
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!"'$
!"'%$
!"($
!"(%$
)$*+,-+,$ ./-01+2$ .13,4,41$ 5+,6,47/$ 81479,-:;$ .+:,<2$
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
Here is the complete policy over all the
drivers. The BayesiaLab’s Soft Increase allows to get a targeted mean value by using
the closest probability distribution to the initial one. It then means that the corresponding action should be the easiest one, as it is
close to the current state
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Plan
Introduction
Bayesian Networks
Applications
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Driver Analysis for Product 10
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Plan
Introduction
Bayesian Networks
Applications
©2009 Bayesia SAAll rights reserved. Forbidden reproduction in whole or part without the Bayesia’s express
written permission42
Driver Analysis for Product 5
Let’s compute the same Driver Analysis for Product 5
![Page 43: Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market](https://reader034.fdocuments.us/reader034/viewer/2022051314/553866fb550346722e8b47ee/html5/thumbnails/43.jpg)
Plan
Introduction
Bayesian Networks
Applications
©2009 Bayesia SAAll rights reserved. Forbidden reproduction in whole or part without the Bayesia’s express
written permission43
Driver Analysis for Product 5
![Page 44: Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market](https://reader034.fdocuments.us/reader034/viewer/2022051314/553866fb550346722e8b47ee/html5/thumbnails/44.jpg)
Plan
Introduction
Bayesian Networks
Application
©2009 Bayesia SAAll rights reserved. Forbidden reproduction in whole or part without the Bayesia’s express
written permission
Contact
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Dr. Lionel JOUFFEManaging Director / Cofounder
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