Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market

Plan

Introduction

Bayesian Networks

Application

©2009 Bayesia SAAll rights reserved. Forbidden reproduction in whole or part without the Bayesia’s express

written permission

Probabilistic Structural Equations

Application to the Analysis of a Perfume Market

Dr. Lionel JOUFFE

August 20091

Plan

Introduction

Bayesian Networks

Application


written permission

BayesiaLab’s Probabilistic Structural Equations for Perfume Market Analysis

2

Plan

Introduction

Bayesian Networks

Application


written permission3

INTRODUCTION

Plan

Introduction

Bayesian Networks

Application


written permission4

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

Plan

Introduction

Bayesian Networks

Application


written permission5

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

Plan

Introduction

Bayesian Networks

Application


written permission6

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%

Plan

Introduction

Bayesian Networks

Application


written permission7



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

Plan

Introduction

Bayesian Networks

Application


written permission8


- 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%


Plan

Introduction

Bayesian Networks

Application


written permission9



Plan

Introduction

Bayesian Networks

Application


written permission10

BAYESIAN BELIEF NETWORKS

Plan

Introduction

Bayesian Networks

Application



... 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

Plan

Introduction

Bayesian Networks

Application



... 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)

Plan

Introduction

Bayesian Networks

Application



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

Plan

Introduction

Bayesian Networks

Applications



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

http://www.bayesia.com/en/products/bayesialab/resources/tutorials/probabilistic-structural-equations-I.php






Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



Step 1: Unsupervised learning on the Manifest variables only

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



Step 2: Variables’ Clustering

Plan

Introduction

Bayesian Networks

Applications



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

.....

Plan

Introduction

Bayesian Networks

Applications



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.

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications




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)

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications




Factors’ Hierarchization by using the Standardized Total Effects (STE)

Graphical representation of each Factor’s influence on the Purchase Intent

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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

Plan

Introduction

Bayesian Networks

Applications



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)

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications




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)

Plan

Introduction

Bayesian Networks

Applications




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.

Plan

Introduction

Bayesian Networks

Applications




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.!"#$

!"#%$

!"&$

!"&%$

!"'$

!"'%$

!"($

!"(%$

)$*+,-+,$ ./-01+2$ .13,4,41$ 5+,6,47/$ 81479,-:;$ .+:,<2$

Plan

Introduction

Bayesian Networks

Applications




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

Plan

Introduction

Bayesian Networks

Applications




Plan

Introduction

Bayesian Networks

Applications




Let’s compute the same Driver Analysis for Product 5

Plan

Introduction

Bayesian Networks

Applications




Plan

Introduction

Bayesian Networks

Application


written permission

Contact

Address

BAYESIA SA6 rue Léonard de Vinci BP011953001 LAVAL CedexFrance

Contact

Dr. Lionel JOUFFEManaging Director / Cofounder

Tel.: +33(0)243 49 75 58Mobile: +33(0)607 25 70 05Fax: +33(0)243 49 75 83

44

Date post:	22-Apr-2015
Category:	Technology
Upload:	jouffe
View:	6,213 times
Download:	2 times

Probabilistic Structural Equations - Bayesian Networks for the Analysis of a Perfume Market

Technology