How it Works?

A Probabilistic Expert System based on a Bayesian Belief Network

Bayesian Belief Networks are made of Two Distinct parts

Structure

Directed Acyclic Graph

Nodes represent the variables of the studied domain (e.g.: URU-FRA to model the Match Uruguay versus France)

Each node has exclusive states (e.g.: FRA, Draw, URU)

Arcs represent the direct probabilistic influences between the variables (possibly causal), e.g.: the results of the matches implying France have a direct impact on the final number of points of France

Parameters

Probability distributions are associated to each node, usually by using tables

MARGINAL PROBABILITY DISTRIBUTION

We here consider that Uruguay has a 15% chance to win the match against France, 60% that it will

be a draw, and 25% that France will win it

CONDITIONAL PROBABILITY DISTRIBUTION

The result of the first match has an impact on the team’s spirit and then on the probability

distribution of the second match

Here, for a France’s defeat against Uruguay, we set a 45%

chance that France wins the second match vs Mexico, 40%

for a draw, and 15% for a defeat.

On the other hand, if France wins, we set a 85% chance for a win in the second match, 10% for a draw, and 5% for a defeat

Bayesian Belief Networks are Powerful Inference Engines

We exploit all the information available on a subset of variables for updating, in a rigorous way, the probability distribution of the other variables

All kinds of inference are allowed:

Simulation: from “causes” toward “effects”

“What are the consequences on the Qualification probability for Stage 2 when the team loses its first match?”

Diagnosis: from “effects” toward “causes”

“When a team is qualified for Stage 2, what is the probability that this team has lost its first match?”

All the combinations of those two kinds of inference:

“When a team is qualified for Stage 2, with a draw during its first match, what is the probability that this team has won its second match?”

The Bayesian Network used for the Application

The structure: 3 layers

The 6 matches of Group A

The points for each team

The qualification for each team

Marginal probability distribution defined as Equiprobable.The user will define his/her own distribution by using the

web application’s sliders, for describing his/her own knowledge/belief

Deterministic relation between the 3

matches and the total number of points for

Stage 1

A probabilistic equation describes the different

qualification scenarios

The Parameters

Initially, without modifying the equiprobable distribution on the

matches’ results, the Qualification probability is 50%

Probabilistic Computation: Simulation

“What are the consequences on the Qualification probability for Stage 2 when the team loses its first match?”

If Uruguay loses the first match, the Qualification probability falls from

50% to 23.59% (without any information on the other matches’

results)

Probabilistic Computation: Diagnosis

“When a team is qualified for Stage 2, what is the probability that this team has lost its first match?”

Given that France is qualified for Stage 2 .....

... there is a 15.73% chance that France has lost the first match

Probabilistic Computation

“When a team is qualified for Stage 2, with a draw during its first match, what is the probability that this team has won its second match?”

... and France is qualified for Stage 2 .....

Given that France made a draw during the first match ....

... there is then a 58.49% chance that France won the second

match

Probabilistic Computation

“Is it possible to be qualified for Stage 2 with 2 points only?”

Given that South Africa gets 2 points only ....

... there is still a 1.23% chance that South Africa is qualified

We wish you pleasant simulations ...

and a great World Cuphttp://worldcup.bayesialab.com

Contact

6 rue Léonard de Vinci BP011953001 LAVAL CedexFRANCE

Dr. Lionel JOUFFEPresident / CEO

Tel.: +33(0)243 49 75 58Skype: +33(0)970 46 42 68Mobile: +33(0)607 25 70 05Fax: +33(0)243 49 75 83