Basic Formal Properties of A Relational Model of The Mathematical Theory of Evidence

TL;DR

This paper offers a new relational and rough-set based interpretation of Dempster-Shafer belief functions, linking belief combination to database join operations and emphasizing diversity over frequency.

Contribution

It introduces a novel qualitative interpretation of belief functions using relational data and rough sets, connecting belief operators with database operations.

Findings

Dempster rule corresponds to relational join

Belief measures can be derived through domain operations

Interpretation emphasizes diversity over frequency

Abstract

The paper presents a novel view of the Dempster-Shafer belief function as a measure of diversity in relational data bases. It is demonstrated that under the interpretation The Dempster rule of evidence combination corresponds to the join operator of the relational database theory. This rough-set based interpretation is qualitative in nature and can represent a number of belief function operators. The interpretation has the property that Given a definition of the belief measure of objects in the interpretation domain we can perform operations in this domain and the measure of the resulting object is derivable from measures of component objects via belief operator. We demonstrated this property for Dempster rule of combination, marginalization, Shafer's conditioning, independent variables, Shenoy's notion of conditional independence of variables. The interpretation is based on rough sets (in connection with decision tables), but differs from previous interpretations of this type in that it counts the diversity rather than frequencies in a decision table.

Tables

Table 1. Table 1 : Example: Public offering: erection of buildings of a school, a restaurant and a shopping center. Decision table: BUILDing. ”Information part” - the firm, ”Decision part” - the object to be erected

	I	D
1.	ABD A.G.	center
2.	LQR Inc.	school
3.	PTS Ltd.	center
	PTS Ltd.	restaurant
4.	XYZ Inc.	center
5.	ZZZ Ltd.	restaurant
	ZZZ Ltd.	school

Table 2. Table 2: Shafer’s Conditioning. Relational Interpretation. Calculating B e l ( . | | { s c h o o l , r e s t a u r a n t } ) Bel(.||\{school,restaurant\}) as a B ​ e ​ l 𝐵 𝑒 𝑙 Bel from the table obtained by select I,D from BUILD where D=school or D= restaurant

	I	D
1.	LQR Inc.	school
2.	PTS Ltd.	restaurant
5.	ZZZ Ltd.	restaurant
	ZZZ Ltd.	school

Table 3. Table 3: Public offering: equipment for buildings of a school, a restaurant and a shopping center. Decision table: EQUIPment ”Information part” - the firm, ”Decision part” - the object to be equipped

	I2	D
1.	AAA GmbH	school
2.	BBB Ltd.	center
	BBB Ltd.	restaurant
3.	CCC Inc.	center
	CCC Inc.	restaurant

Table 4. Table 4: Combination of Independent Evidence. Decision table FINISH obtained as select I,I2,D from BUILD, EQUIP where BUILD.D=EQUIP.D

	I	I2	D
1.	ABD A.G.	BBB Ltd.	center
2.	ABD A.G.	CCC Inc.	center
3.	LQR Inc.	AAA GmbH	school
4.	PTS Ltd.	BBB Ltd.	center
	PTS Ltd.	BBB Ltd.	restaurant
5.	PTS Ltd.	CCC Inc.	center
	PTS Ltd.	CCC Inc.	restaurant
6.	XYZ Inc.	BBB Ltd.	center
7.	XYZ Inc.	CCC Inc.	center
8.	ZZZ Ltd.	BBB Ltd.	restaurant
9.	ZZZ Ltd.	CCC Inc.	restaurant
10.	ZZZ Ltd.	AAA GmbH	school

Table 5. Table 5: Multivariate MTE. The table MADEOF and its projection select distinct I,D from MADEOF

Decision table MADEOF
	I	D	D2
1.	ABD A.G.	center	wooden
2.	LQR Inc.	school	stone
	LQR Inc.	school	wooden
3.	PTS Ltd.	center	stone
	PTS Ltd.	center	wooden
	PTS Ltd.	restaurant	stone
4.	XYZ Inc.	center	stone
5.	ZZZ Ltd.	restaurant	stone
	ZZZ Ltd.	restaurant	wooden
	ZZZ Ltd.	school	wooden
m({(school,wooden)})=1/5
m({(school,stone)})=2/5

Table 6. Table 6: Variable Independence

	I2	I3	D	D3
1.	AAA GmbH	EC	school	electric
2.	AAA GmbH	GC	school	gas
3.	BBB Ltd.	EC	center	electric
	BBB Ltd.	EC	restaurant	electric
4.	BBB Ltd.	GC	center	gas
	BBB Ltd.	GC	restaurant	gas
5.	CCC Inc.	EC	center	electric
	CCC Inc.	EC	restaurant	electric
6.	CCC Inc.	GC	center	gas
	CCC Inc.	GC	restaurant	gas

Table 7. Table 7: Conditional Variable Independence

	I2	I4	D	D5	D4
1.	AAA GmbH	Messer	school	wood	green
	AAA GmbH	Messer	school	wood	red
	AAA GmbH	Messer	school	plastic	green
	AAA GmbH	Messer	school	plastic	red
2.	BBB Ltd.	Messer	center	metallic	white
	BBB Ltd.	Messer	center	metallic	yellow
	BBB Ltd.	Messer	center	marble	white
	BBB Ltd.	Messer	center	marble	yellow
3.	BBB Ltd.	Gabel	restaurant	wood	red
4.	CCC Inc.	Messer	center	metallic	white
	CCC Inc.	Messer	center	metallic	yellow
	CCC Inc.	Messer	center	laminated	white
	CCC Inc.	Messer	center	laminated	yellow
5.	CCC Inc.	Gabel	restaurant	plastic	red

Equations

$$\sum _{A\in 2^{\Xi }}m(A)=1m(\varnothing )=0,{\forall }_{A\in 2^{\Xi }}0\le m(A)$$

$$Bel=Be{l}^{\downarrow D}\oplus Be{l}^{\downarrow D2}$$

Full text

111**Keywords: ** soft computing, knowledge representation and integration, Dempster-Shafer theory, rough set theory, relational databases, qualitative interpretation of Dempster rule. I.2.3, I.2.4.

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Basic Formal Properties of A Relational Model of The Mathematical Theory of Evidence

1 Introduction

Belief functions and their mathematical properties were investigated by A.P. Dempster in a series of papers in the late sixties [4]. They were intended as a generalisation of Bayesian inference in the sampling situation. In the mid-seventies G. Shafer [17] (who coined the term belief functions) developed a set of formal tools for the representation and combination of evidence. His aim was to construct a rather general theory, termed Mathematical Theory of Evidence, or MTE for short, for coping with uncertain but non-probabilistic information. In spite of its numerous interesting formal properties, the theory caused great discussion centred around the validity of the axioms and manageable interpretation of a belief function - see e.g. [4, 5, 7, 8, 10, 11, 13, 14, 18, 24].

Dempster [4] initiated interval interpretation of MTE, which in fact is compatible with the random set theory [28]. H. Kyburg [10] showed that the belief function may be represented by an envelope of a family of probability functions and claimed that the behaviour of combining evidence via belief functions may be properly explained in statistics under proper independence assumptions. Hummel and Landy [8] considered MTE as a ”statistics of expert opinions” so that it ”contains nothing more than Bayes’ formula applied to Boolean assertions, …(and) tracks multiple opinion as opposed to a single probabilistic assessment”. Pearl [13] and Provan [14] considered belief functions as ”probabilities of provability”. Still another view has been developed in connection with rough set theory [6, 22, 23, 24]. Belief function is considered there as the lower approximation of the set of possible decisions in a (partial) decision table. Fagin and Halpern [5] postulated probabilistic interpretation of MTE around lower and upper probability measures defined over a probability structure (rather than space). The list of other attempts is quite long.

Though a tendency to consider belief functions as subjective uncertainty measures is visible [20], the need for case-based interpretation as a pre-condition for practical applicability has been explicitly stressed [26]. Still this interpretation should be qualitative rather than quantitative in nature [25]. However, these requirements seem not to be met so far.

Apart from manageable interpretation, the MTE caused troubles of purely numerical nature. Unlike ordinary probabilities, which assign mass to each possible outcome, belief functions assign mass to each subset of the outcome space. As a consequence the amount of memory space required to store a belief function in a computer will grow exponentially with the size of the outcome space we consider. Searching for economical representation of a belief function, the researches made use of the already known concepts of sets factorisation analysed in the theory of relational databases. In our search for non-quantitative interpretation of MTE, our attention was attracted by the nature of the join operator of relational databases [2] or in general the multivalued dependency [3] the study of which led to invention of local computation method for uncertainty propagation of Shenoy and Shafer [21] for MTE. This in turn made the rough-set theoretic interpretation of MTE belief functions of Skowron and Busse [24] best choice for further investigation, as it was purely-case based and relational, though it is frequency based. The rough-set interpretation sheds some light onto what the concept of ”evidence” may mean in experimental terms. The ”evidence” there is the information part of a database record and it ”supports” the decision part of a record. The complexity of combination of evidence according to the Dempster rule in [24] gave an impulse for search of a simpler way to accomplish it. It turns out that with frequency approach updating of decision parts of cases is needed (Dempster’s combination is destructive). Out of this experience we decided to abandon the frequencies and concentrated on purely relational operations.

The paper is organised as follows: Section 2 briefly introduces basic MTE concepts. Section 3 recalls traditional rough set theoretic frequency interpretation of MTE from [24] and explains our insights of destructive nature of Dempster’s combination with respect to frequencies. Section 4 presents our new qualitative interpretation. The paper ends with some concluding remarks.

Throughout the paper, as relational data tables are subject of rough set theory, SQL [27] query language is used to express semantics of MTE measures and operators in terms of decision tables, both with respect to traditional and our rough sets based interpretation of MTE, as SQL has the capability of expressing purely relational and frequentistic data processing.

2 Basics of the Dempster-Shafer Theory

We understand MTE measures in a very traditional way (see [14]). Let $\Xi$ be a finite set of elements called elementary events. Any subset of $\Xi$ is a composite event, or hypothesis. $\Xi$ be called also the frame of discernment. A basic probability (or belief) assignment (bpa) function is any set function m:$2^{\Xi}\rightarrow[0,1]$ such that

[TABLE]

We say that a bpa is vacuous iff $m(\Xi)=1$ and $m(A)=0$ for every $A\neq\Xi$. A belief function is defined as Bel:$2^{\Xi}\rightarrow[0,1]$ so that $Bel(A)=\sum_{B\subseteq A}m(B)$. A plausibility function is Pl:$2^{\Xi}\rightarrow[0,1]$ with $\forall_{A\in 2^{\Xi}}Pl(A)=1-Bel(\Xi-A)$. A commonality function is Q:$2^{\Xi}-\{\emptyset\}\rightarrow[0,1]$ with $\forall_{A\in 2^{\Xi}-\{\emptyset\}}\quad Q(A)=\sum_{A\subseteq B}m(B)$.

The Rule-of-Combination of two Independent Belief Functions $Bel_{E_{1}}$, $Bel_{E_{2}}$ Over-the-Same-Frame-of-Discernment (the so-called Dempster-Rule), denoted $Bel_{E_{1},E_{2}}=Bel_{E_{1}}\oplus Bel_{E_{2}}$ is defined interms of bpa’s as follows: $m_{E_{1},E_{2}}(A)=c\cdot\sum_{B,C;A=B\cap C}m_{E_{1}}(B)\cdot m_{E_{2}}(C)$ (c - constant normalizing the sum of $m$ to 1).

Under multivariate settings $\Xi$ is a set of vectors in n-dimensional space spanned by the set of variables X={ $X_{1},X_{2},\dots X_{n}$}. If $A\subseteq\Xi$, then by projection $A^{\downarrow{\bf Y}}$ of the set $A$ onto a subspace spanned by the set of variables ${\bf Y}\subseteq{\bf X}$ we understand the set $B$ of vectors from $A$ projected onto Y. Then marginalization operator of MTE is defined as follows: $m^{\downarrow{\bf Y}}(B)=\sum_{A;B=A^{\downarrow{\tiny X}}}m(A)$.

Definition 1

(See [19]) Let B be a subset of $\Xi$, called evidence, $m_{B}$ be a basic probability assignment such that $m_{B}(B)=1$ and $m_{B}(A)=0$ for any A different from B. Then the conditional belief function $Bel(.||B)$ representing the belief function $Bel$ conditioned on evidence B is defined as: $Bel(.||B)=Bel\oplus Bel_{B}$.

3 Rough Set Theory. The Traditional Interpretation of Belief

Functions

Skowron and Grzymala-Busse [24] and others studying rough sets developed more specifically the proposal of Shafer with respect to frequency interpretation of MTE.

Let us introduce the following denotation concerning decision tables. Let a tuple $\mu$ mean a function $\mu:A\rightarrow DOM(A)$, with $A$ being a set of attributes $A_{j}$, $DOM(A_{j})$ being the domain of the attribute $A_{j}$, $DOM(A)=\bigcup_{A_{j}\in A}DOM(A_{j})$. $A$ be called the scheme of $\mu$, $A=S(\mu)$. A relational table $TAB$ be any set of tuples with identical scheme. This common scheme be denoted by $S(TAB)$. Let $\mu[R]$ with $R\subseteq S(\mu)$ denote the restriction of the tuple $\mu$ to the scheme $R$: $\mu[R]=\{(A_{j},a_{jk})|A_{j}\in R\land a_{jk}=\mu(A_{j})\}$. The restriction of a relational table $TAB$ to $R$, denoted $TAB[R]$, be defined $TAB[R]=\{\mu[R]|\mu\in TAB\}$. A relational join of two relational tables $TAB_{1},TAB_{2}$ be defined as: $TAB_{1}\otimes TAB_{2}=\{\mu_{1}\cup\mu_{2}|\mu_{1}\in TAB_{1},\mu_{2}\in TAB_{2}\land\forall_{A_{j}\in S(\mu_{1})\cap S(\mu_{2})}\ \mu_{1}(A_{j})=\mu_{2}(A_{j})\}$.

A decision table is a relational table in which we split the scheme into two distinct parts: the information part **I ** and the decision part D.

Let $card(SET)$ denote the cardinality of the set SET.

Let us assume that a decision table $TAB$ of decisions $D$ (atomic values) under conditions (information) **I ** (atomic value vectors) is available. However, **I ** may not contain the complete information to make decision $D$. This gives rise in a natural way to a mapping $\Gamma$ assigning different values of $D$ to the same value of I. Under these circumstances the belief in a set $A$ (subset of the domain of $D$) $Bel(A)$ may be derived from a case database as follows: $Bel(A)=1-card(\{\mu|\mu\in TAB\land\mu(D)\not\in A\})/card(TAB)$, which may be implemented the SQL query language [27] as:

[TABLE]

Skowron and Grzymała-Busse[24] elaborated also a notion of conditioning under rough set interpretation [24, p.219 ff.] as respective measures for subtables (that is tables consisting of cases selected by a criterion). Let us condition on $D$ belonging to the set $B$ selecting tuples fitting the condition $B$ $Subtable=\{\mu|\mu\in TAB\land\mu(D)\in B$ which may be implemented as

[TABLE]

The belief distribution for the subtable can be calculated from the view SubtableTAB in the same way as for the TAB database. However, it is a matter of a calculation exercise to show that their notion of conditionality does not agree with that of Shafer from Def.1.

Therefore, to achieve consistency with Shafer’s conditioning from Def.1 we propose the following interpretation, derivable from our approach described in the paper [9]: Let $v\in B$. Then in SQL

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Let $c_{1}$,$c_{2}$ be two cases from the database TAB such that ${\bf I}(c_{1})={\bf I}(c_{2})$ but $D(c_{1})\neq D(c_{2})$ Let be $D(c_{1})\in B$ and $D(c_{2})\not\in B$ Then after the above update and delete operations both cases $c_{1},c_{2}$ are retained in the database TAB. $c_{1}$ has retained its value $D(c_{1})$. But the case $c_{2}$ was subject to a metamorphose: its $D(c_{2})$ has been changed (to v). This means that the Dempster rule of combination is ”destructive”. Preservation of frequency interpretation under conditioning enforces ignoring the intrinsic (observed) value of an attribute and replacement of it with some other value. It should be stressed at this point that the above SQL operation cannot be easily expressed in terms of sets and relations, because an update operation is engaged which may make distinct tuples identical.

A still more complex task is the interpretation of combination of two independent pieces of evidence. Skowron and Grzymała-Busse[24] elaborated a procedure consisting in transforming the combined decision tables into a kind of summary with multivalued decision columns (since non-relational) and then applying complex rational arithmetic to get finally a decision table (derived by so-called $\Psi$-independent combination) implementing Dempster combination of independent belief functions (consult [24] for details).

4 New Interpretation

Below we present a slight modification of the rough set interpretation of belief functions that surprisingly turns out to be both simple, elegant and straight forward and at the same time fulfils the requirement that was not matched by any known interpretations: it is qualitative and not quantitative in nature and still case-based. Furthermore we demonstrate that our new interpretation corresponds strictly to the notion of multivalued dependency, that is combination of belief functions parallels the relational join operator from database theory.

The results of any experiment with multiple outcomes can evaluated along two dimensions: the quantitative and the qualitative one. If we say e.g. that in a series of coin tossing experiments we got 57 heads and 43 tails, then this is a quantitative evaluation. But if we say that there were heads and tails (and not e.g. edges) then we say something about the qualitative aspects of the experiment. In the quantitative evaluation we say that 57 % of all the cases we got heads, in the qualitative evaluation we say that 50 % of all possibilities of diverse outcomes are heads. In most real life cases we are more interested in the quantitative aspects. Sometimes, however, the qualitative side may be of more interest. E.g. if 40 witnesses say that they saw the suspect had killed the victim, but their testimonies are suspiciously similar, and 20 say they saw the contrary, and their testimonies made impressions of individuality, then we would say that there are 1:20 chances of the guilt of the suspect rather than 40:20, because the qualitative aspects (diversity) are more important than quantitative ones (frequency).

Behaviour of frequencies under reasoning was historically the foundation of probability theory. As probabilistic (frequency based) models of MTE fail in general, we considered just the diversity as a possible alternative for a model of MTE. The diversity is well handled by relational model of databases. Though at a first glance the count of cases in a relational database may appear identical with counting frequencies of objects / events, the difference starts as soon as we make a projection on the subset of attributes. It turns out that projected frequencies differ significantly from the counts of cases in a projected relational database. Therefore we say that our approach is non-frequency, non-quantitative, that is qualitative one.

Let us define the plausibility $Pl_{TAB}(SET)$ derived from a decision table TAB with decision variable D and the set I of information variables as: $Pl_{TAB}(SET)=card(\{\mu[\mbox{\bf I}]|\mu\in TAB\land\mu(D)\in SET\})/card(TAB[\mbox{\bf I}])$, implemented as

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Example 1 explains the detailed numerical procedure for calculation of $Pl$ from the above SQL expression.

Theorem 1

The function $Pl_{TAB}(SET)$ derived from a decision table TAB with decision variable D and the set I of information variables is plausibility function $Pl(SET)$ in the sense of Dempster-Shafer theory.

**Proof: **

In MTE, the plausibility of a set SET is just the sum of basic probability assignments $m(A)$ such that $SET\cap A\neq\emptyset$. Let $r$ be a record, $I(r)$ its information part and $D(r)$ its decision part. For the set SET, let us consider a subset $R$ of all records from the decision table TAB such that: if $r\in R$ then $D(r)\in SET$ and for every $d\in SET$ there exists $r^{\prime}\in R$ such that $D(r)=D(r^{\prime})$ and there exists no record $r"\not\in R$ such that $I(r")=I(r)$. Obviously, for two distinct sets $SET^{\prime}$ and $SET"$ their respective sets $R^{\prime}$ and $R"$ will share no records. Furthermore, $\bigcup_{SET\subseteq domain(D)}R(SET)$ will be the (relationally) identical with TAB. Hence we can consider the ratio number of records with distinct information part in $R(SET)$ divided by the number of records with distinct information parts in DT as the bpa function $m(SET)$ in the sense of MTE. But the function $Pl_{TAB}(SET)$ counts (the relative share of) the records with distinct information part such that the decision part belongs to SET. Hence in practice it is just the sum of basic probability assignments $m(A)$ such that $SET\cap A\neq\emptyset$. Therefore it is a plausibility function.

$\Box$

• Example 1

Let us first look at the relational table BUILD in tab. 1). The column D is the decision column, I is information part of the table. The domain of the decision variable D is {center, restaurant, school}.

Let us calculate now the plausibility Pl({school, restaurant}) from this table. There are 7 cases (rows) in the dataset. But there are only 5 cases with distinct information part (firms) I. And there are only 3 cases with decision either $school$ or $restaurant$ with distinct information part I (LQR Inc., PTS Ltd., ZZZ Ltd).So the plausibility222Notice that under Skowron/Busse interpretation [24] we get $Pl=4/7$ which is obviously a different value. The difference stems from the fundamental difference between frequency (Skowron/Busse) and relational(ours) view of the world.is equal to Pl({school, restaurant})=$3/5$. One can check that Pl({school})=2/5 (LQR Inc., ZZZ Ltd) and Pl({restaurant})=2/5 (PTS Ltd., ZZZ Ltd). $\Diamond$

Notice that from the calculational rules for Dempster-Shafer theory we can derive also relational views calculating other measures:

Belief $Bel_{TAB}(SET)=1-card(\{\mu[\mbox{\bf I}]|\mu\in TAB\land\mu(D)\not\in SET\})/card(TAB[\mbox{\bf I}])$, implemented as

[TABLE]

Commonality $Q_{TAB}(SET)=card(\{\mu[\mbox{\bf I}]|\forall_{d\in SET}\ \mu[\mbox{\bf I}]\cup\{(D,d)\}\in TAB\})$ $/card(TAB[\mbox{\bf I}])$, implemented as

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Basic belief assignment $m_{TAB}(SET)=card(\{\mu[\mbox{\bf I}]|\forall_{d\in SET}\ \mu[\mbox{\bf I}]\cup\{(D,d)\}\in TAB\}\land\forall_{d\not\in SET}\ \mu[\mbox{\bf I}]\cup\{(D,d)\}\not\in TAB\})/card(TAB[\mbox{\bf I}])$, implemented as

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Theorem 2

The functions $Bel_{TAB}(SET)$, $Q_{TAB}(SET)$, $m_{TAB}(SET)$ derived from a decision table TAB with decision variable D and the set I of information variables are belief, commonality, basic probability/belief assignment functions resp. $Bel(SET),Q(SET),m(SET)$ in the sense of Dempster-Shafer theory.

• Example 2

From table 1 we easily calculate that:

Commonality Q({school, restaurant}=1/5 (Number of firms ready to build either the school and the restaurant: ZZZ Ltd).

Belief Q({school, restaurant}=2/5 Number of firms ready to build nothing but the school or the restaurant (LQR Inc., ZZZ Ltd)

bpa - No of firms exactly offering erecting of: m({school, restaurant}=1/5 (ZZZ Ltd), m({restaurant}=0 (none), m({school}=1/5 (LQR Inc.)

$\Diamond$

4.1 Conditioning as Selection of a Subtable

Let us define now the conditional belief function $Bel_{TAB}(.||B)$ representing the belief function $Bel$ conditioned on evidence B as the belief function $Bel_{TAB\_B}(.)$ define over the view table

[TABLE]

• Example 3

In our example, for the relational table BUILD from tab. 1, $Bel_{BUILD}(.||\{school,restaurant\})$ is just $Bel$ calculated from the respective projection *select I,D from BUILD where D=school or D= restaurant * visible in tab. 2. It is easily seen that $Pl_{BUILD}(\{restaurant\}||\{school,restaurant\})=2/3$ (PTS Ltd.,ZZZ Ltd.) and $m_{BUILD}(\{restaurant\}||\{school,restaurant\})=1/3$ (PTS Ltd.). $\Diamond$

Our conditional belief function matches perfectly the Shafer’s definition of $Bel(.||B)$ cited above ( Def.1).Notice that under Skowron/Busse interpretation (section 3), the matching of Shafer’s conditionality definition had to be paid for with creating a physical copy of a relational table and updating it, whereas our notion works perfectly without any updates - only selection is used just as in probabilistic conditioning.

4.2 Combination as Relational Join

Now let us discuss the most impressive property of the new interpretation: the Dempster’s rule of combination interpreted as relational join.

• Example 4

Let us consider the decision table EQUIP (tab. 3) and BUILD (tab. 1). We want to combine independent evidence from both tables to support a decision. Let us assume that independence of evidence means that no pair of firms (one from BUILD, one from EQUIP) refuse to cooperate on erecting and equipping an object. How many pairs of firms do we have to finish a set of objects mentioned in the offerings ? The answer lies in the relational table FINISH (tab. 4) obtained as a relational join of BUILD and EQUIP (over the common column D) so that the new decision table has as its decision column D and as its information part I,I2:

[TABLE]

Notice that in BUILD, there were 5 cases with distinct information part, in EQUIP - 3, and in BUEQ there are only 10. We have here $Pl_{FINISH}(\{school,restaurant\})=8/10$ and $Bel_{FINISH}(\{school,restaurant\})=3/10$

You can easily check that $Bel_{FINISH}=Bel_{BUILD}\oplus Bel_{EQUIP}$. $\Diamond$

Generally, we can formulate the theorem:

Theorem 3

If the decision tables DT1(I1,D) and DT2(I2,D) with non-overlapping information parts I1,I2 are combined by relational join operation $DT1\otimes DT2$, implemented as

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yielding table DT12(I,D) with I=I1$\cup$I2, then $Bel_{DT12}=Bel_{DT1}\oplus Bel_{DT2}$.

**Proof: **

This can be demonstrated by considering the ”fate” of records counted on calculation of $m_{i}$. If $R_{1}$ is the set of records counted when calculating $m_{1}(A)$ in DT1, and if $R_{2}$ is the set of records counted when calculating $m_{2}(B)$ in DT2, then upon join only records $\mu=(\mu_{1}[\mbox{\bf I1}],\mu_{2}[\mbox{\bf I2}],\mu_{1}[D])$ with $\mu_{1}\in R_{1}$, $\mu_{2}\in R_{2}$, $\mu_{1}[D]=\mu_{2}[D]\in A\cap B$ will be created, hence they will be counted in support of $m(A\cap B)$. Furthermore, their number will be exactly equal to the product of the number of distinct records in $R_{1}$ times the number of distinct records in $R_{2}$, so that the Dempster formula will be matched perfectly upon normalization.

$\Box$

4.3 Relational Marginalization and Decombination

A further intriguing property, not present in any interpretation of MTE known so far, is the relationship between relational marginalization and MTE factorization (”decombination”) of belief functions.

• Example 5

Notice that BUILD and EQUIP in our example are both in first normal form and the domain of the attribute D is identical in both tables. Therefore we know from elementary properties of relational data tables that marginalization of FINISH over I,D

[TABLE]

is exactly identical with BUILD. and marginalization of FINISH over I2,D

[TABLE]

is exactly identical with EQUIP. $\Diamond$

In general:

Theorem 4

*If the information part **I *** of the decision table DT(I,D) can be split into two such parts I1,I2  that $\mbox{\bf I1}\cup\mbox{\bf I2}=\mbox{\bf I}$ and $\mbox{\bf I1}\cap\mbox{\bf I2}=\emptyset$ and the relation DT is identical with DT1$\otimes$DT2, implemented

[TABLE]

where DT1 and DT2 are DT1=DT[I1,D], DT2=DT[I2,D], implemented

[TABLE]

that is there is a multivariate dependency between I1 and I2 given D, then $Bel_{DT}=Bel_{DT^{\downarrow\mbox{\tiny\bf I1},D}}\oplus Bel_{DT^{\downarrow\mbox{\tiny\bf I2},D}}$.

**Proof: **

Follows directly from theorem 3.

$\Box$

Let consider the unnormalized MTE measures of decision tables $m^{\prime}_{TAB}$, $Bel^{\prime}_{TAB}$, $Pl^{\prime}_{TAB}$, $Q^{\prime}_{TAB}$, such that $f^{\prime}_{TAB}=f_{TAB}\cdot card(TAB^{\downarrow I})$ (card - number of distinct rows, $f$ - $m$ or $Bel$ or $Pl$ or $Q$) and the unnormalized combination operator $\oplus^{\prime}$ such that $Bel^{\prime}_{E_{1},E_{2}}=Bel^{\prime}_{E_{1}}\oplus^{\prime}Bel^{\prime}_{E_{2}}$ is defined as follows: $m^{\prime}_{E_{1},E_{2}}(A)=\sum_{B,C;A=B\cap C}m^{\prime}_{E_{1}}(B)\cdot m^{\prime}_{E_{2}}(C)$.

What may be more surprising, a kind of a reverse theorem holds:

Theorem 5

The information part I of the decision table DT(I,D) can be split into two such parts I1,I2 that $\mbox{\bf I1}\cup\mbox{\bf I2}=I$ and $\mbox{\bf I1}\cap\mbox{\bf I2}=\emptyset$ and $Bel^{\prime}_{DT}=Bel^{\prime}_{DT^{\downarrow\mbox{\tiny\bf I1},D}}\oplus^{\prime}Bel^{\prime}_{DT^{\downarrow\mbox{\tiny\bf I2},D}}$ if and only if the relation DT is identical with DT1$\otimes$DT2, implemented

[TABLE]

where DT1 and DT2 are DT1=DT[I1,D], DT2=DT[I2,D], implemented

[TABLE]

that is there is a multivariate dependency between I1 and I2 given D,

**Proof: **

(An outline.) The if-part parallels exactly theorem 4. We need only pay attention to the fact that we never normalize.

The only-if-part follows from numerical calculations for Q-values of all the tables considered. If a record is counted in DT when calculating $Q^{\prime}_{DT}(SET)$, then it is also counted when calculating both $Q^{\prime}_{DT1}(SET)$ and $Q^{\prime}_{DT2}(SET)$. If we form a join $DT1\cdot DT2$ then $Q^{\prime}_{DT1\cdot DT2}(SET)=Q^{\prime}_{DT1}(SET)\cdot Q^{\prime}_{DT2}(SET)$. And this is the maximum value Q’ can take in DT12. If there is ANY deviation from multivariate dependency concerning records with decision part in SET, then value of $Q^{\prime}_{DT12}(SET)$ is smaller than $Q^{\prime}_{DT1}(SET)\cdot Q^{\prime}_{DT2}(SET)$. This proves our claim.

$\Box$

Remak: We can conclude that Dempster’s rule of combination is equivalent with relational join and the Dempster-Shafer independence of evidence means multivalued dependence of evidence. We can also simulate other rules of combination of evidence. In the above, we assumed that given the decision, we cannot conclude from the information part I1  the value of the information part I2  in DT. This meant qualitative independence. Now let us assume the contrary in another decision table DT’: given the decision d, we can totally predict I2  from I1  for all records r with D(r)=d in DT’ or we can totally predict I1  from I2  for all records r with D(r)=d in DT’. It is immediately clear that in this case for any set of decisions the unnormalized plausibility is calculated as $Pl^{\prime}_{DT^{\prime}}(A)=max(Pl^{\prime}_{{DT^{\prime}}^{\downarrow\mbox{\tiny\bf I1},D}}(A),Pl^{\prime}_{{DT^{\prime}}^{\downarrow\mbox{\tiny\bf I2},D}}(A))$. We can conclude for normalized plausibility that we deal here with the know rule of combination of dependent evidence: $Pl^{\prime}_{DT^{\prime}}(A)=max(\alpha\cdot Pl^{\prime}_{{DT^{\prime}}^{\downarrow\mbox{\tiny\bf I1},D}}(A),(1-\alpha)\cdot Pl^{\prime}_{{DT^{\prime}}^{\downarrow\mbox{\tiny\bf I2},D}}(A))$ where $\alpha$ ranges from 0 to 1 (depending on proportions between the numbers of distinct information parts I1  and I2).

4.4 Multivariate Beliefs and Multidecision Tables

We can extend our consideration to tables with multiple decision variables. In a straight forward way we can extend our definition of MTE measures to such tables and consider multivariate belief distributions (in all the decision variables). It is trivial to see that dropping a decision variables does not diminish the diversity of the information part. Hence dropping a decision variable D$i$ from the set of decision variables D has the same effect as dropping a variable in the belief function. That is for any set B of decision vectors in variables D-{D$i$}: $m_{TAB^{\downarrow{\tiny\bf D}-\{Di\}}}(B)=m_{TAB}^{\downarrow{\tiny\bf D}-\{Di\}}(B)=c\cdot\sum_{A;B=A^{\downarrow{\tiny\bf D}-\{Di\}}}m(A)_{TAB}$ (c - normalizing factor) See table 5 for an example.

The operator of projection $\downarrow$ should be understood as the MTE projection operator applied to a belief function.

Let DTM be a decision table with decision variables D1 and D2. Let the information part consist of two disjoint parts I1 and I2. Let us consider the following views:

[TABLE]

If now the table DTM12 is relationally identical with DTM, then we shall say that the decision variables D1 and D2 are independent in the decision table DTM. It is not surprising that: $Bel_{DTM}=Bel_{DTM}^{\downarrow D1}\oplus Bel_{DTM}^{\downarrow D2}$. This means that independence of decision variables in a decision table implies independence of variables in the corresponding belief function. See table 6 for an example.

Let DTX be a decision table with decision variables D1, D2 and D3. Let the information part consist of two disjoint parts I1 and I2. Let us consider the following views:

[TABLE]

If now the table DTX12 is relationally identical with DTX, then we shall say that the decision variables D1 and D2 are independent given D3 in the decision table DTX. It is not surprising that: $Bel_{DTX}=Bel_{DTX}^{\downarrow D1,D3}\oplus Bel_{DTX}^{\downarrow D2,D3}$. But this means that the variables D1 and D2 are independent given D3 in the belief function $Bel_{DTX}$ in the sense of Shenoy’s VBS [21]. See table 7 for an example.

These results mean that analysis of independence and conditional independence of variables in a belief function corresponding to a decision table may serve as an indicator of presence or absence of independence or multivalued dependence in the decision table.

5 Concluding Remarks

A novel case-based interpretation of MTE belief functions which is qualitative in nature and has the potential to represent a number of MTE operations has been presented. The interpretation is based on rough sets (in connection with decision tables), but differs from previous interpretations of this type e.g. [24] in that it counts the diversity rather than frequencies in the decision table. The interpretation has the property that given a definition of the MTE measure of objects in the interpretation domain (decision table) we can perform operations in the interpretation domain (e.g. combining decision tables) and the measure of the resulting object is derivable from measures of component objects via MTE operator (e.g. combination). We demonstrated this property for Dempster rule of combination, marginalization, Shafer’s conditioning, independent variables, Shenoy’s notion of conditional independence of variables. Other known case-based (frequency or probabilistic) interpretations fall short of this property. E.g. in [24] complex rational number arithmetic, unnatural for decision tables, is needed to achieve compatibility of the final decision table with Dempster rule of combination. In [10] (probabilistic interpretation) only lower and upper bounds are found for Dempster rule. In [5] (probability structures interpretation) the belief function obtained from the Dempster rule is a potential, but not necessary result of the corresponding probabilistic structure combination operation. See also [25] for discussion of other interpretations.

As probabilistic (frequency based) models of MTE seem to fail in general, we looked for alternatives. The result of any experiment with multiple outcomes can evaluated along two dimensions: the quantitative and the qualitative one. In most real life cases we are more interested in the quantitative aspects. Sometimes, however, the qualitative side may be of more interest. E.g. legal applications we would treat suspiciously similar testimonies as a single argument in favour or against a hypothesis: we would rely on the count of diversity of arguments rather than on their actual counts. And MTE was claimed to be applicable just in legal reasoning [20]. Therefore we considered just the diversity as a possible alternative for a model of MTE and this idea turned out to very fruitful. The diversity is well handled by relational model of databases. Though at a first glance the count of cases in a relational database may appear identical with counting frequencies of objects / events, the difference starts as soon as we make a projection on the subset of attributes. It turns out that projected frequencies differ significantly from the counts of cases in a projected relational database. Therefore we say that our approach is non-frequency, non-quantitative, that is qualitative one.

The paper presents SQL statements performing the calculations of the MTE measures and the MTE related operations on the decision tables.

The new interpretation may be directly applied in the domain of multiple decision decision tables: independence of decision variables or Shenoy’s conditional independence in the sense of MTE may serve as an indication of possibility of decomposition of the decision table into smaller but equivalent tables. Furthermore it may be applied in the area of Cooperative Query Answering [15]. The problem there is that a query posed to a local relational database system may contain an unknown attribute. But possibly other co-operating db systems know it and may explain it to the queried system in terms of known attributes, shared by the various systems. The uncertainties studied in the decision tables arise here in a natural way and our interpretation may be used to measure these uncertainties in terms of MTE (as diversity of support). Furthermore, if several co-operating systems respond, then the queried system may calculate the overall uncertainty measure using MTE combination of measures of individual responses.

Now we can ask how to understand then a MTE belief function in the light of our experience. One possibility is to consider the belief function as a measure of diversity of support. This is an obvious departure from frequency interpretations proposed by Shafer and others. No mater how frequently the same piece of evidence is presented, it is counted once. This insight may encourage to revise other known interpretations of MTE. In the ”legal” interpretations e.g. [20], the witnesses in favor of a hypotheses should be counted separately, if their statements differ in unimportant details permitting to deduce that their statements are personal and not studied in. In the ”probability of provability” approach [12] not a probability of correctness, but rather the number of distinct valid proofs of a statement should be counted. In the ”possible world semantics” [16] the worlds should not be assigned a probability, but rather distinct possible worlds should be counted that differ in non-essential details. Then the operation of combination of independent evidence in the ”legal” interpretation is just mixing compatible statements of two sets of witnesses (which saw the same event from different perspective) and counting different possible combinations.

In the ”probability of provability” approach the combination would mean putting together conclusion compatible proofs stemming from distinct domains (e.g. macro and micro-physical observations) and counting legal combinations. In the possible worlds semantics we may combine worlds spanned over disjoined sets of dimensions.

Please notice also that the rough-set interpretation sheds some light onto what the concept of ”evidence” may mean. The ”evidence” are just different sets of information attributes and the ”independence” means (deterministic) unpredictability of attribute values of the one set from the other set. This should not be confused with predictability of the decision variable. Nor with stochastic predictability which may be present. In the ”legal” interpretation, independence would be measured with non-predictability of insignificant details.

In the ”provability” interpretation the independence may be measured by mutual non-derivability of the sets of underlying axioms. In the possible world semantics by possibility of putting together projections onto separated sets of dimension axes.

Further studies on interpreting other known MTE operators in the spirit of qualitative interpretation presented in this paper are needed and may reveal new potential applications of MTE to real world problems.