Probabilistic Graphical Models: On Reasoning, Learning,

and Revision (Extended Abstract)

Rudolf Kruse

Faculty of Computer Science, Otto von Guericke University, Magdeburg, Germany

Keywords: Bayesian Network, Markov Network, Item Set Planning.

Abstract: Probabilistic Graphical Models are of high relevance for complex industrial applications. The Bayesian

network and the Markov network approach are the most prominent representatives and an important tool to

structure uncertain knowledge about high dimensional domains. This extended abstract serves to highlight

that the decomposition of the underlying high dimensional spaces turns out to be useful to make reasoning,

learning and revision in such domains feasible. The methods are explained by using a real-world industrial

application from automotive industry.

1 INTRODUCTION

In the automotive industry, customers prefer to opt for

individual vehicle specifications. For this reason,

some manufacturers prefer a marketing policy that

offers maximum freedom in choosing individual

vehicle specifications. This means that a customer

can select from a large number of options, according

to his personal preferences. One can choose the body

variant, engine, circuit, door layout, seat cover, radio

and navigation system, and this only reflect a small

part of the entire product family. In the case of a

typical popular car, there are about 200 such

variables, each typically having 4-8 values and a total

range of cardinalities from 2 to 150. Of course, not all

possible instantiations of these so-called item

variables result in valid vehicle configurations, as

technical rules, manufacturing restrictions and selling

requirements induce a common rule system that

restricts the acceptable ways of item combination.

However, with more than 10,000 technical rules in a

class and many more rules supplied by the sales

programs for the specific needs of different countries,

there remains a huge number of correct vehicle

specifications.

2 ITEM SET PLANNING

The main goal of item planning is the development

and implementation of a software system that

supports item planning, parts requirements

calculation and capacity management with the aim of

short and medium-term range forecasts for future

vehicle production. In order to achieve high quality

planning results, all relevant sources of information

must be considered, namely rules for the right

combination of items to form complete vehicle

specifications, samples of produced vehicles

reflecting customer preferences, market forecasts

leading to changed specifications, item sets for

planning intervals, capacity constraints and

production programs, which determine the number of

planned vehicles.

From a logistical point of view, the most essential

result of the item planning process is the evaluation

of the rates of all item combinations that are known

to be relevant for the parts requirements calculation,

always related to a specific vehicle class in a specific

planning interval. The importance of these item

combinations derives from the fact that a vehicle can

be interpreted as a large set of fitting locations, each

characterized by a set of alternative fitting parts for

that location. Within the framework of a typical

passenger car class, a total of around 70,000 different

article combinations are required as installation

conditions for all of the installation locations. The

task of predicting the total demand for a specific part

in relation to a future planning interval is to add up

the demands across all of its installation locations.

The need for any installation location is obtained by

multiplying the rate of the combination of items that

represents its installation condition by the installed

Kruse, R.

Probabilistic Graphical Models: On Reasoning, Learning, and Revision (Extended Abstract).

DOI: 10.5220/0011598200003335

In Proceedings of the 14th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2022) - Volume 3: KMIS, pages 9-10

ISBN: 978-989-758-614-9; ISSN: 2184-3228

one in quantity and the total number of vehicles

intended to be produced in the respective planning

interval.

3 PROBABILISTIC NETWORKS

The domain and expert knowledge in this application

about installation rates can be formally represented by

a probability distribution

over the set of relevant item

families or attributes. Conditional independences are

used to decompose this distribution into lower

dimensional distributions. Since it is possible to

connect the concepts of conditional independence

with the separation concept in graphs, graphical

models turn out to be extremely helpful for the

problem of item set planning. Two well-know models

are Bayesian networks and Markov networks

(Borgelt 2009).

A Bayesian networks is a directed acyclic graph

(DAG’s), representing a set of random variables and

the dependencies between these random variables. A

Markov network is an undirected conditional

independence graph G = (V,E) of a probability

distribution together with a family of conditional

probabilities of the factorization induced by the

graph.

Probabilistic graphical models allow for an

efficient knowledge representation as well as an

integration of new evidence via conditioning. The

basic idea is to distribute (propagate) the evidence

through the network to reach all attributes.

Probabilistic graphical models can also be learned

from given data. Classical statistical techniques for

parameters learning as well as other methods for

learning the network structure are useful (Drton

2017). Approaches for learning graphical models

typically fall into one of two categories: score-based

approaches and constraint-based approaches. Score-

based approaches consist of two elements: a score

function to evaluate how well graph candidates fit the

database, and some search heuristic (possibly guided

by the scores) to traverse the set of graphs. The goal

of constraint-based approaches is to use conditional

(in)dependence tests to construct a graphical model

which is a perfect map (or independence map) of the

data-generating distribution. Several efficient and

user-friendly commercial tools such as Hugin

(HuginExpert 2022) are available for this task.

In practice, however, there is also a need to revise

the probability distribution represented by a graphical

model in such a way that it satisfies the given

framework conditions, for example given marginal

distributions. Pure evidence propagation methods

such as join tree propagation and bucket elimination

are unsuitable for this task. We present a knowledge-

based probabilistic formalization and solution of the

fundamental revision problem for Markov networks,

constrained to a set of unconstrained single-variable

boundary conditions (Gebhardt 2005). This

probabilistic approach avoids all concepts offered by

calculi with deviating semantic foundations, for

example to minimize probabilistic difference

measures that could be inherited from information

theory. From multivariate statistics, iterative

proportional fitting gives a convenient algorithm to fit

the marginal distributions of a given joint distribution

to desired values.

4 CONCLUSIONS

The probabilistic graphical network approach has

proven to be very successful for assistance systems.,

Thousands of Markov networks for different planning

scenarios and different model groups are in use every

day for item set planning.

The methodology used for item set planning can

be easily transferred to other areas. In the monograph

(Kruse 2022) a tutorial introduction to this type of

knowledge representation, updating, revision and

learning is given.

In the item planning project we have mainly

benefited from the decomposition aspect of

probabilistic graphical networks. We are convinced

that the concept of causality (Pearl 2018) will play a

central role in many future applications.

REFERENCES

Borgelt, C., Steinbrecher, M., Kruse, R. (2009) Graphical

Models, Representations for Learning, Reasoning and

Data Mining, Wiley, Chichester, 2

edition

Drton, M., Maathuis M. (2017) Structure Learning in

Graphical Modeling, Annual Review of Statistics and

its Application Vol.4, 365-393

Gebhardt, J., Kruse, R. (2005). Knowledge-Based

Operations for Graphical Models in Planning. In

ECSQARU 2005, Springer LNAI 3571, pp 3-14.

HuginExpert (2022), Bayesian network software,

http://www.hugin.com

Kruse, R., et al (2022). Computational Intelligence, A

Methodological Introduction, Springer, London, 3

edition.

Pearl, J., Mackenzie, D. (2018), The book of why: the new

science of cause and effect, Basic Books, New York.

IC3K 2022 - 14th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management