Those slides describe how the probabilistic computations are handle in the application http://worldcup.bayesialab.com. This application computes the Stage 2 qualification probability of any team of any Group of the next FIFA World Cup. Based on the user input (with probability distributions on the Group matches' result: win, draw, and loss), a Bayesian Network is used to rigorously compute the qualification probabilities.

1. How it Works?
A Probabilistic Expert System based on a
Bayesian Belief Network

2. 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
CONDITIONAL
PROBABILITY DISTRIBUTION Here, for a France’s defeat
MARGINAL PROBABILITY The result of the first match has an impact on the against Uruguay, we set a 45%
DISTRIBUTION team’s spirit and then on the probability chance that France wins the
We here consider that Uruguay has a 15% chance distribution of the second match second match vs Mexico, 40%
to win the match against France, 60% that it will for a draw, and 15% for a
be a draw, and 25% that France will win it 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

3. 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?”

4. The Bayesian Network used for the Application

5. The structure: 3 layers
The 6 matches of
Group A
The points for each team
The qualification for
each team

6. The Parameters
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

7. Probabilistic Computation: Simulation
“What are the consequences on the Qualification probability for Stage 2 when the team loses its first match?”
Initially, without modifying the
equiprobable distribution on the
matches’ results, the Qualification
probability is 50%
If Uruguay loses the first match,
the Qualification probability falls from
50% to 23.59% (without any
information on the other matches’
results)

8. 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

9. 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?”
Given that France made a draw
during the first match ....
... and France is qualified for
Stage 2 .....
... there is then a 58.49%
chance that France won the second
match

10. 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

11. We wish you pleasant
simulations ...
and a great World Cup
http://worldcup.bayesialab.com

12. Contact
6 rue Léonard de Vinci BP0119
53001 LAVAL Cedex
FRANCE
Dr. Lionel JOUFFE
President / CEO
Tel.: +33(0)243 49 75 58
Skype: +33(0)970 46 42 68
Mobile: +33(0)607 25 70 05
Fax: +33(0)243 49 75 83