# Basis of the theory of belief functions

This page is directly inspired from [Thierry Denoeux's talks] and [Philippe Smets' articles].

## Contents

## Introduction

Introduced by Dempster (1968) and Shafer (1976), belief functions constitute one of the main frameworks for reasoning with imperfect information.

Belief function is a mathematical tool used in different models with different semantics:

- Lower-upper probabilities (Dempster's model, Hint model); - Random sets; - Degrees of belief (Transferable Belief Model - TBM).

It generalizes both Set Theory and Probability Theory: a belief function may be viewed both as a generalized set and as a non additive measure.

The theory includes extensions of probabilistic notions (conditioning, marginalization) and set-theoretic notions (intersection, union, inclusion, etc.)

## Representing Information

Let , called the *frame of discernment*, be a finite set comprising all possible answers to a given question of interest.

The beliefs held by a rational agent *Ag* regarding the answer to question can be quantified by a *basic belief assignment (BBA)* or a *mass function* , defined as a function from to verifying:

The quantity represents the part of the unit mass allocated to the hypothesis that the answer to question is in the subset of . When there is no ambiguity on the agent or the frame of discernment, the notation will be simplified to or .

### Some definitions

- A subset of such that is called afocal set of. - A BBA with only one focal set is called acategorical BBAand is denoted ; we then have . - Total ignorance is represented by the BBA such that , called thevacuous mass function. - Anormal BBAsatisfies the condition . - A BBA whose focal sets are nested is said to beconsonant.

### Other representations

The belief and plausibility functions associated with a BBA are defined, respectively, as:

and

for all .

Functions , and are in one-to-one correspondence, and thus constitute different forms of the same information.

## Combining evidence

The basic operation for combining BBAs induced by distinct sources of information is the *conjunctive rule of combination*, also referred to as the *unnormalized Dempster's rule of combination*, defined as

Other rules exist: disjunctive rule, cautious rule, ...

## Making decisions

In Bayesian decision theory, modeling a decision process implies defining:

- A set of decisions that can be made; - A set of considered states of nature; - A cost function , such that represents the cost of making decision when is the true state of nature.

Rationality principles justify the choice of the decision corresponding to the minimum expected cost or *risk* according to some probability measure on :

with

In the TBM, this decision-theoretic framework is accepted (see Smets 2005 for instance).

The set , called the **betting frame**, is often taken equal to . However, it may be any refinement or coarsening of ,or it may be obtained from by a succession of refinings and coarsenings.

The probability measure is obtained by the *pignistic transformation*, which consists in firstly expressing the piece of evidence , initially defined on the frame of discernment, on the betting frame , and then computing the pignistic probability as:

where denotes the cardinality of .