# Consistency of Probability Decision Rules and Its Inference in Probability Decision Table

Authors: Zheng Pei; Li Zou; Hamid Reza Karimi; Peng Shi
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

In most synthesis evaluation systems and decision-making systems, data are represented by objects and attributes of objects with a degree of belief. Formally, these data can be abstracted by the form (objects; attributes; P), where P represents a kind degree of belief between objects and attributes, such that, P is a basic probability assignment. In the paper, we provide a kind of probability information system to describe these data and then employ rough sets theory to extract probability decision rules. By extension of Dempster-Shafer evidence theory, we can get probabilities of antecedents and conclusion of probability decision rules. Furthermore, we analyze the consistency of probability decision rules. Based on consistency of probability decision rules, we provide an inference method to finish inference of probability decision rules, which can be used to decide the class of a new object x′. The conclusion points out that the inference method of the paper not only deals with precise information, but also imprecise
or uncertain information as well.


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## Body

## 1. Introduction

Processing uncertain or incomplete information may be placed in the sphere of artificial intelligence. The term “reasoning with uncertain or incomplete information” in the narrow sense means the way of representing a partial information that is available to a user about a fragment of reality and the way of processing such an information. In the broader sense, it is used to denote the interdisciplinary sphere of research concerned with the search for methods of modeling uncertain or incomplete knowledge. Those methods can refer to any application domain and any level of knowledge; one seeks way of representing both object-level knowledge and meta-level knowledge, the latter being the knowledge about the former [1–5]. Nowadays, the main tools of processing uncertain or incomplete information are fuzzy set theory, probability theory, possibility theory, Dempster-Shafer evidence theory, and rough sets theory. The main advantage of fuzzy set theory is that the fuzzy set framework provides a lot of combination operators, which allows the user to adapt the processing scheme to the specificity of the data at hand [6–17]. The probabilistic theory has solid mathematic theory, but its inference cannot model uncertain measurement [18–20]. Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning, because variability of observations should be distinguished from incomplete information [21, 22]. Dempster-Shafer evidence theory has the ability to deal with ignorance and missing information [23]. In particular, it provides explicit estimations of imprecision and conflict between information from different sources. Indeed, probability theory may be seen as a limit of Dempster-Shafer evidence theory when it is assumed that there is no imprecision, and that only certainty has to be taken into account [24–27]. By using indistinguishability relations, rough sets theory can model and handle incomplete information and uncertain knowledge discovered from information system [2, 28–30].In most synthesis evaluation and decision-making systems, data reflect the relation between objects and attributes with a degree of belief, that is, an object has an attribute with a degree of belief. Informally, the information system or decision tables with a degree of belief are called probability information system or probability decision tables. In this paper, we focus on probability decision tables, we analyze the consistency of probability decision rules which are extracted from the probability decision tables, and provide an inference method based on probability decision rules. The organization of this paper is as follows. In Section2, we make briefly a review of extension of Dempster-Shafer evidence theory (DSEV). In Section 3, we provide a kind of probability information system and probability decision tables to represent objects and attributes with a degree of belief, and it is shown that our probability information system is extension of classical information system and a special case of interval-valued information system. In Section 4, we discuss how to extract probability decision rules from a probability decision table. In Section 5, we discuss consistency of probability decision rules. In Section 6, we provide a method to finish inference of probability decision rules. We conclude in Section 7.


## 2. Extension of DSEV

In DSEV, probability masses are allocated to subsets of a frame of discernment in contrast to Bayesian probability theory, in which only singletons are carrying probability masses. Such subsets with positive mass are called focal elements, and the family of focal elements is said to be the basic probability assignment. So, the DSEV can be seen to be generalization of the classic probability theory. Formally, the DSEV concerns itself with belief structures, which can be defined as follows: letX={x1,x2,…,xn} be a finite set of elements, and let m be a measure on the subsets of X such that (1) 0≤m(A)≤1 for each A⊂X; (2) m(∅)=0; (3) ∑A⊂Xm(A)=1. m is called a basic probability assignment function. Any subset A of X such that m(A)>0 is called a focal element. m and its associated values are called by a belief structure. Two important functions play a significant role in DSEV: Bel(B)=∑A⊂Bm(A) and Pl(B)=∑A∩B≠∅m(A) for any subsets B and A of X. Bel is called a belief function, and Pl is called a plausibility function. Bel(B) measures the total amount of probability that must be distributed amongst the elements of B, and Pl(B) measures the maximal amount of probability that can be distributed among the elements in B. Denoted B¯ is the complement of B, then we have(i)
Pl(B)≥Bel(B),(ii)
Pl(∅)=Bel(∅)=0, Pl(X)=Bel(X)=1,(iii)
Pl(B)=1-Bel(B¯), Bel(B)=1-Pl(B¯),(iv)
Bel(B)+Bel(B¯)≤1, Pl(B)+Pl(B¯)≥1.In the classical probability model, the probability mass function is a mappingP from X to [0,1], which indicates how the probability mass is assigned to the elements. Based on the probability mass function P, the set mapping m′:2X→[0,1] can be induced, where for each A⊂X, we have m′(A)=∑x∈AP(x). Obviously, m′ is a belief structure; furthermore, we have Pl(A)=Bel(A). However, in DSEV, we know probabilities of the focal sets instead of probabilities of each element x∈X; hence, we are not able to calculate the probability P(A) associated with the subsets of X, but instead to use the two measures Bel(A) and Pl(A), corresponding to a lower and upper bound on the unknown P(A), that is, Bel(A)≤P(A)≤Pl(A).In [18], extending belief structure was proposed. Extending belief structure means using a belief structure defined on one frame to obtain a belief structure on another frame. Consider two frames X and Y, whose elements are possible answers to perhaps related questions. We say that an element x∈X is compatible with an element y∈Y if it is possible, relative to our knowledge and opinion, that x is the answer to the question considered by the frame X, and y is the answer to the question considered by the frame Y. Denote this as xRy. If for all x∈X and all y∈Y, we have xRy, then we say that the two questions are independent.LetX and Y be two frames, and X×Y is their Cartesian product, which consists of all pairs (x,y) with x∈X and y∈Y, then the compatibility relation R is a subset of X×Y consisting of all pairs (x,y) for which xRy, that is, (x,y)∈R if xRy. Specially, if X and Y are independent, then R=X×Y. Any compatibility relation R over X×Y can be represented as a multivalued mapping: G:X→2Y such that G(x)={y∣(x,y)∈R}, where R is a compatibility relation. Assume that X and Y are two frames with a compatibility relation R and associated multivalued mapping G. Let BelX be a belief function defined on the frame X. The extension of BelX to Y as BelY can be defined as follows:(2.1)BelY(B)=BelX({x∣(x,y)∈R⟹y∈B})=BelX({x∣G(x)⊂B}).
If {Ai} are the focal elements of BelX where mX(Ai)=ai, then {Bi}, where Bi=⋃x∈AiG(x), are the focal elements of BelY and mY(Bi)=∑kak, over all k such that Bk=Bi. Let H={x∣G(x)⊂B}, then(2.2)BelY(B)=∑Bi⊂BmY(Bi)=∑⋃x∈AiG(x)⊂BimX(Ai),BelY(B)=BelX(H)=∑Ai⊂HmX(Ai).
There exist two special cases of the extension: (1) assume that we know BelS×T, which is a belief function defined on S×T, the marginal belief function of BelS×T on S (or T) is defined as the extension of BelS×T to S (or T), denoted as BelS (or BelT); (2) the extension of a belief function on the frame S to the frame S×T. In (2.1), two extensions are expressed as follows.(1)
LetX=S×T and Y=S (or T), then we have a multivalued mapping
(2.3)G:S×T⟶2SorG:S×T⟶2T,
in which G((s,t))={s}orG((s,t))={t}. Thus, we have a belief function
(2.4)BelY(B)=BelS×T({(s,t)∣G((s,t))⊂B}).
Furthermore, {(s,t)∣G((s,t))⊂B}=⋃s∈B({s}×T)=B×T, or {(s,t)∣G((s,t))⊂B}=⋃t∈B(S×{t})=S×B.(2)
The marginal belief function onS (or T) is
(2.5)BelY(B)=BelS×T(B×T)orBelY(B)=BelS×T(S×B).
If {Ai} are focal elements of BelS×T with mS×T(Ai)=ai, we get mS of BelS (or mT of BelT) as mS(Bi)=ai(ormT(Bi)=ai), where Bi=⋃(s,t)∈AiG((s,t)), that is, Bi is projection of Ai onto S (or T).Assume thatBelS is a belief function on S, and let BelS×T be the extension of BelS onto S×T that we know nothing about the answer on T. In this case, we call BelS×T the minimal extension of BelS; in particular, we assume that every answer in T is compatible with any answer in S; thus, compatible function is(2.6)G:S⟶2S×T,G(s)={(s,t)∣∀t∈T}.
So, we have extension of BelS onto S×T as BelS×T(B)=BelS({s∣G(s)⊂B}). Assume that B=A×T, then as foralls∈A, we have G(s)∈B, and as foralls∈̅A, we have G(s)⊈B. Hence, {s∣G(s)⊂B}=A, and BelS×T(A×T)=BelS(A). Assume that B≠A×T, then foralls∈S, G(s)={s}×T⊈B; thus, {s∣G(s)⊂B}=∅, and BelS×T(B)=BelS(∅)=0. If we have mS(Ai)=ai, we can get BelS×T(Bi)=ai, where Bi=⋃s∈AiG(s). Since G(s)={s}×T, then Bi=Ai×T, that is, Bi is the set consisting of each element in Ai coupled with each element in T.Assume that we have two independent sources of evidence as the location of the special element inX, which have associated belief structures m1 and m2, respectively. The problem is to find a combined belief structure m over X reflecting the “ANDing” of the two pieces of evidence. Let m1 and m2 be two belief structures on X with focal elements A1,…,Ak and B1,…,Bp, respectively, then their combination, denoted by m=m1⊕m2, is a belief structure over X such that (1) m(A≠∅)=∑Ai∩Bj=Am1(Ai)m2(Bj)/(1-K); (2) m(∅)=0; (3) K=∑Ai∩Bj=∅m1(Ai)m2(Bj). The focal elements of m are all sets A such that Ai∩Bj=A. Because m satisfies commutativity and associativity, we have m=m1⊕m2⊕⋯⊕mr for r belief structures on X. 1-K indicates the normalization factor needed to assure ∑m(A)=1. If K=0, then no normalization is required. If K=1, then we cannot obtain m(A≠∅) based on the above method (1). In this case, the belief structures are completely conflicting, and we need another method for combining evidence [18, 20].


## 3. Probability Information Systems

Information systems, sometimes called data tables, attribute value systems, condition action tables, knowledge representation systems, and so forth are used for representing knowledge and have been popularly used in artificial intelligence [2]. Formally, a pair Ω=(U,A) is called information systems. Where U is a nonempty, finite set called the universe and A a nonempty, finite set of attributes, that is, a:U→Va for a∈A, where Va is called the value set of a, elements of U are called objects and interpreted as cases, states, processes, patients, and observations. Attributes are interpreted as features, variables, characteristic, and conditions. As a special case of information systems, a decision table has the form Δ=(U,A∪{b}), where b∈¯A is a distinguished attribute called decision (attribute). The elements of A are called conditions (attributes).The relation between an objectx∈U and a value va∈Va of an attribute a∈A is certain, that is, a:U→Va is a function, and either a(x)=va or a(x)≠va is true. In real practice, the relation between objects and attribute values is uncertain, that is, a(x)=va with a degree of belief p(a(x)=va)∈[0,1]. If p(a(x)=va)=1, it means that a(x)=va. If p(a(x)=va)=0, it means that a(x)≠va. If p(a(x)=va)∈(0,1), it means that a(x)=va with uncertainty. In this paper, we limit the degree of belief in probability of a(x)=va.Definition 3.1.
A tripleΩ=(U,A,P) is called a probability information system, where U is a nonempty, finite set called the universe, A a nonempty, finite set of attributes, and P={Pa∣a∈A}, Pa:U×A→[0,1] such that for each x∈U, ∑va∈VaPa(x,va)=1, Pa(x,va)=p(a(x)=va) means that object x has va of a with probability p(a(x)=va).In Definition3.1, for any fixed object x∈U, {Pa(x,va)∣va∈Va} is a probability density function on Va of a. Hence, a probability information system is also understood by an information system with a probability density function on the value set of each attribute for each object. An example of a probability information system is shown in Table 1, in which, for object 1 and solar energy, (high, 0.2) means that object 1 has high solar energy with probability 0.2, that is, PS((1,high))=0.2. For object 1 and residual CO2, (high, 0) means that object 1 has not high residual CO2, that is, PR((1,high))=0. For object 3 and temperature, (low, 1) means that object 4 has low value, that is, PT((1,low))=1.Table 1
An example of probability information system.
FactSolar energy (S)Volcanic activity (V)ResidualCO2 (R)Temperature (T)1(High, 0.2)(High, 0.6)(High, 0)(High, 0.8)(Medium, 0.8)(Medium, 0.3)(Medium, 0.3)(Medium, 0.2)(Low, 0)(Low, 0.1)(Low, 0.7)(Low, 0)2(High, 0.9)(High, 0.8)(High, 0.7)(High, 0.8)(Medium, 0.1)(Medium, 0.2)(Medium, 0.3)(Medium, 0.1)(Low, 0)(Low, 0)(Low, 0)(Low, 0.1)3(High, 0.1)(High, 0)(High, 0)(High, 0)(Medium, 0.2)(Medium, 0.3)(Medium, 0.2)(Medium, 0)(Low, 0.7)(Low, 0.7)(Low, 0.8)(Low, 1)In a probability information system, for any objectx∈U and a∈A, we denote(3.1)Λax={va∈Va∣Pa((x,va))≠0}.
If |Λax|=1 (|Λax| is cardinality of Λax), then there exists unique va∈Va such that Pa((x,va))=1 due to ∑va∈VaPa(x,va)=1. In this case, the probability information system is reduced to an information system because for each object x and a∈A, there exists only one va∈Va such that a(x)=va. From this point of view, probability information systems are extension of information systems. If |Λax|>1, then the probability information system becomes an interval-valued information system when all probabilities Pa((x,va))≠0 are not considered, in such case, for each object x and a∈A, we have a(x)=Λax. Hence, a probability information system of Definition 3.1 is a special case of interval-valued information system. Deference between them is a(x)=Λax in interval-valued information system, but a(x)=Λax with a probability density function on Λax in probability information system.Based on probability information system, probability decision tables have the formΔ=(U,A∪{b},P), A is called conditions, and b is called decision. Let P={Pc∣c∈A∪{b}}, Pc:U×(A∪{b})→[0,1] such that for each x∈U and c∈A∪{b}, ∑vc∈VcPcx((x,vc))=1. For simplicity, denote Vc and Pc as Va and Pa, respectively, when c∈A, and Vc and Pc as Vb and Pb when c=b. For any object x∈U and c∈A∪{b}, denote Λcx={vc∈Vc∣Pc((x,vc))≠0}. If |Λcx|=1, then probability decision table is reduced to classical decision tables. If |Λax|>1, then probability decision table becomes interval-valued decision tables when all probabilities Pa((x,va))≠0 are not considered.LetU={x1,x2,…,xn} and A={a1,…,am}. For each xi∈U and attribute value set Vaj of attribute aj∈A, Vaj={vaj1,…,vajrj}, denote Pajxi as a probability density function on Vaj, and denote Pbxi as a probability density function on decision value set Vb={vb1,…,vbs}. Formally, a probability decision table is shown in Table 2.Table 2
A probability decision table.
U∖A∪{b}a1⋯amb(va11,pa1x1(va11))⋯(vam1,pamx1(vam1))(vb1,pbx1(vb1))x1⋮⋮⋮⋮(va1r1,pa1x1(va1r1))⋯(vamrm,pamx1(vamrm))(vbs,pbx1(vbs))⋮⋮⋮⋮⋮(va11,pa1xn(va11))⋯(vam1,pamxn(vam1))(vb1,pbxn(vb1))xn⋮⋮⋮⋮(va1r1,pa1xn(va1r1))⋯(vamrm,pamxn(vamrm))(vbs,pbxn(vbs))In Table2, pajxi(vajrj′) is the probability of “xi that has the value vajrj′ of aj", where rj′∈{1,2,…,rj}, and for each xi, ∑rj′=1rjpajxi(vajrj′)=1. For each xi and c∈A∪{b}, denote(3.2)pajxi(vajrj′)=max⁡{pajxi(vaj1),pajxi(vaj2),…,pajxi(vajrj)}.Then we can get a reduced probability decision table shown in Table3, which is called the maximum probability decision table in this paper.Table 3
The maximum probability decision table of Table2.
U∖A∪{b}a1a2⋯ambx1(va1r1′,pa1x1(va1r1′))(va2r2′,pa2x1(va2r2′))⋯(vamrm′,pamx1(vamrm′))(vbs′,pbx1(vbs′))⋮⋮⋮⋮⋮⋮xn(va1r1′,pa1xn(va1r1′))(va2r2′,pa2xn(va2r2′))⋯(vamrm′,pamxn(vamrm′))(vbs′,pbxn(vbs′))


## 4. Probability Decision Rules

In a decision tableΔ=(U,A∪{b}), decision rules can be extracted and formalized as follows:(4.1)τ⟶b=i,
where τ is a formula, which is generated by some vi∈V finitely using connectives ∨ or ∧, V=⋃a∈AVa, and i∈Vb. For more details about decision rules, reader can consult [2, 28–30]. In this section, we discuss some properties and formalization of probability decision rules based on Table 3.Firstly, we can use rough set theory to deal with the information of Table3 without considering probability pajxi(vajrj′); for simplicity, denote(4.2)Vaj′={vaj∣∃xi∈U,Pajxi(vaj)=max⁡{Pajxi(vaj1),…,Pajxi(vajrj)}},Vb′={vbk∣∃xi∈U,Pbxivbk=max⁡{Pbxi(vb1),…,Pbxi(vbs)}}.
Then attributes of Table 3 can be represented as follows:(4.3)aj:U⟶Vaj′,aj(xi)=vaj,b:U⟶Vb′,b(xi)=vbk.
We can define an equivalence relation on U as follows:(4.4)xi~xi′iffaj(xi)=aj(xi′).
Then lower and upper approximation, reduction of attributes, decision rules, consistent decision tables, and so on can be discussed. However, in the paper, the problem that we need to deal with is how to get probability of every decision rule. By using rough set theory, we can get decision rules as follows:(4.5)τk⟶b=vbk,
where νbk∈Vb′, τk=∧l=1qval′, val′∈Val′′, al′ and al′∈{a1,…,am}. Let Uτk⊆U is the equivalence class decided by τk, Uvbk⊆U is the equivalence class decided by vbk. forallxi∈Uτk and forallval′∈τk, Pal′xi(val′rl′) is the probability of “xi has val′rl′ of al′”. Let |Uτk|=T, then we have T independent sources of evidence over Uτk×Val′, and belief structure of each source of evidence is as follows:(4.6)ms(Uτk,val′1)=Pal′xs(val′1),…,ms(Uτk,val′rl)=Pal′xs(val′rl),
where s=1,…,T. Based on (4.6), we can obtain a combined belief structure m over Uτk×Val′, that is, m=m1⊕⋯⊕mT as follows [19]:
(4.7)PUτkal′=⊕xs∈UτkPal′xs,(4.8)PUτkal′(val′rl′)=∏xs∈UτkPal′xs(val′rl′)1-K,(4.9)1-K=∑val′rl′∈Val′∏xs∈UτkPal′xs(val′rl′).
For attribute al′, Pal′xs(val′) is the maximum probability that xs has the value val′ of al′, so, forallval′rl′∈Val′ and forallxs∈Uτk, Pal′xs(val′)≥Pal′xs(val′rl′), that is,(4.10)∏xi∈UτkPal′xi((xi,val′))≥∏xi∈UτkPal′xi((xi,val′rl′)).
This means that PUτkal′(val′) is the maximum possibility of all PUτkal′(val′rl′). Similarly, we can get the probability of each vbk′ as follows:
(4.11)PUvbkb=⊕xi∈UvbkPbxi,(4.12)PUvbkb(vbk′)=∏xs∈UvbkPbxs(vbk′)1-K,(4.13)1-K=∑vbk′∈Vb∏xs∈UvbkPbxs(vbk′),
where PUvbkb(νbk) is the maximum possibility of PUνbkb(νbk′).According to (4.8) and (4.12), we can get a probability decision rule as follows:(4.14)(τk,PUτkτk)⟶b=(vbk,PUνbkb(vbk)),
where (τk,PUτkτk)=∧l=1q(val′,PUτkal′(val′)),  q≤m. The decision rule (4.5) is a special case of the probability decision rule (4.14), that is, if every PUτkal′(val′)=1 and PUνbkb(vbk)=1. Obviously, for every al′∈A,(4.15)∑rl′=1rlPUτkal′(val′rl′)=1,∑k′=1sPUνbkb(vbk′)=1.Theorem 4.1.
Letxi∈Uτk if Pal′xi((xi,val′))=1, then PUτkal′(val′)=1.Proof.
ByPal′xi((xi,val′))=1, we know that forallval′rl′∈Val′, val′rl′≠val′, Pal′xi((xi,val′rl′))=0, so 1-K=∑val′rl′∈Val′∏xi∈UτkPal′xi((xi,val′rl′))=∏xi∈UτkPal′xi((xi,val′)), by (4.8),
(4.16)PUτkal′(val′)=∏xi∈UτkPal′xi((xi,val′))1-K=1.Theorem 4.2.
Letx′ be a new object, then
(4.17)∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)<0
if and only if PUτk∪{x′}al′(val′)<PUτkal′(val′).Proof.
According to (4.7), we know that PUτk∪{x′}al′=⊕xi∈Uτk∪{x′}Pal′xi=(⊕xi∈UτkPal′xi)⊕Pal′x′=PUτkal′⊕Pal′x′. So, we have
(4.18)PUτk∪{x′}al′(val′)-PUτkal′(val′)=PUτkal′(val′)Pal′x′(val′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)-PUτkal′(val′)=PUτkal′(val′)(Pal′x′(val′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)-1)=PUτkal′(val′)∑val′rj′∈Val′val′rj′≠val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)=PUτkal′(val′)∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′),
and this means that if PUτk∪{x′}al′(val′)<PUτkal′(val′), then
(4.19)PUτkal′(val′)∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)<0,
that is, ∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)<0. Converse is obvious.Corollary 4.3.
IfPal′x′((x′,val′))=maxval′rj′∈Val′{Pal′xi((xi,val′rj′))}, then
(4.20)PUτk∪{x′}al′(val′)≥PUτkal′(val′).According to the maximum possibility, Corollary4.3 means that the more the elements of Uτk are, the more the probabilities of val′ of τk are. Intuitively, the elements of Uτk reflect the degree of belief of a decision rule; the more the elements of Uτk are, the more the degree of belief of a decision rule is. Theorem 4.2 means that, sometimes, adding an element to Uτk can make the degree of belief of a decision rule decrease. According to the condition of Theorem 4.2, one can see that Pal′x′(val′) is not the maximal probability. From the probability point of view, x′ may not be in Uτk; if x′ is forced in Uτk, then the degree of belief of a decision rule will decrease.


## 5. Consistency of a Probability Decision Rule

From the logic point of view, if there exists a valuationv such that v(a)=1 and v(b)=1, then a→b is satisfiable, otherwise, a→b is not satisfiable. We discuss consistency of probability decision rule similar to satisfiability of a→b.Definition 5.1.
In probability decision rule (4.14), for a new object x′, if forallval′,  l=1,…,q, x′ is such that
(5.1)PUτk∪{x′}al′(val′)≥PUτkal′(val′),PUvbk∪{x′}b(vbk)≥PUvbkb(vbk),
then there exists consistency between x′ and the probability decision rule, and we also call antecedents and conclusion of the probability decision rule consistent. If there does not exist such x′, then we call antecedents and conclusion of the decision rule inconsistent.By Theorem4.2, we know that(5.2)∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)≥0,∑s′=1s(Pbx′(vbk)-Pbx′(vbs′))PUvbkb(vbs′)≥0
if and only if (5.1) is satisfied. Obviously, if Pal′x′(val′) and Pbx′(vbk) are maximum probability assignments, respectively, that is, if they satisfy the condition of Corollary 4.3, then (5.1) is true. So, using maximum probability assignments to get a probability decision rule, its antecedents and conclusion are consistent, and for a new object x′ and its probability assignments on Vaj, if we use maximum probability assignment to decide the class of x′, then x′ and the probability decision rule are consistent. However, sometimes, there exists a case: for a new object x′, according to maximum probability assignment, we get each attribute value val′rj′′ and vbk′, but there does not exist decision rule such that its antecedents and conclusion match val′rj′′ and vbk′, respectively. In this case, using (5.2), we can choose a probability decision rule such that x′ and the probability decision rule are consistent. If there exist more than one decision rule, then x′ is included in the decision rule such that its conclusion is ∨PUvbk∪{x′}b(vbk).Corollary 5.2.
A new objectx′ and decision rule (4.14) are consistent if and only if the probability assignments of x′ are such that (5.2).Theorem 5.3.
For a new objectx′, let
(5.3)∑s′=1sPbx′(vbs′)PUvbkb(vbs′)=α∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)
if α≤Pbx′(vbk)/Pal′x′(val′) and PUτk∪{x′}al′(val′)≥PUτkal′(val′), then PUvbk∪{x′}b(vbk)≥PUvbkb(vbk).Proof.
ByPUτk∪{x′}al′(val′)≥PUτkal′(val′), we know that
(5.4)∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)≥0.
On the other hand,
(5.5)PUvbk∪{x′}b(vbk)-PUvbkb(vbk)=PUvbkb(vbk)(Pbx′(vbk)∑s′=1sPbx′(vbs′)PUvbkb(vbs′)-1)=PUvbkb(vbk)(Pbx′(vbk)α∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′)-1)=PUvbkb(vbk)∑val′rj′∈Val′(Pbx′(vbk)/α-PUτkal′(val′rj′))Pal′x′(val′rj′)∑val′rj′∈Val′PUτkal′(val′rj′)Pal′x′(val′rj′).
By α≤Pbx′(vbk)/Pal′x′(val′), get Pbx′(vbk)/α≥Pal′x′(val′), so we have ∑val′rj′∈Val′(Pbx′(vbk)/α-PUτkal′(val′rj′))Pal′x′(val′rj′)≥∑val′rj′∈Val′(Pal′x′(val′)-Pal′x′(val′rj′))PUτkal′(val′rj′)≥0, that is, PUvbk∪{x′}b(vbk)≥PUvbkb(vbk).


## 6. Inference of Probability Decision Rules

Inspired by inference method of Zadeh [13], we provide an inference method of probability decision rules in this section, where the decision rule has the form ∧l=1q(val′,PUτkal′(val′))→b=(vbk,PUvbkb(vbk)). Assume that we have a new element x′ shown in Table 4, we infer a probability density function on Vb and decide the class of x′ and its degree of belief. The inference process can be rewritten as follows:(6.1)p:⋀l=1q(val′,PUτkal′(val′))⟶b=(vbk,PUvbkb(vbk)),q:giveTable4,c:(vbs′,Px′,bs′).
In (6.1), the probability decision rule p has the universes of Uτk of antecedent and Uvbk of conclusion. In (6.1), p can be rewritten as “if the degree of belief of va1′ is PUτkal′(val′) and … and the degree of belief of vaq′ is PUτkaq′(vaq′), then the degree of belief of vbk is PUvbkb(vbk),” q can be rewritten as “the degree of belief of attribute values of x′ is Table 4”; the conclusion is “how many are the degrees of belief of decision attribute values vbs′ of x′.” The single condition probability decision rule of p has the form:(6.2)p:(val′,PUτkal′(val′))⟶b=(vbk,PUvbkb(vbk)),q:∑i=1rlpx′,al′i=1,c:(vbs′,Px′,bs′).
In the paper, domains of antecedents and conclusion of p are {(Uτk,val′′)∣val′′∈Va1′} and {(Uvbk,vb′)∣vb′∈Vb}, respectively. We rewrite probability density functions as follows:(6.3)fa1′̃Uτk:{(X,val′′)∣val′′∈Va1′}⟶[0,1],fa1′̃Uτk(X,val′′)=PUτkal′(val′),fb̃Uvbk:{(X,vb′)∣vb′∈Vb}⟶[0,1],fb̃Uvbk(X,vb′)=PUvbkb(vb′),fal′̃x′:{(X,val′′)∣va1′′∈Val′}⟶[0,1],fal′̃x′(X,val′′)=px′,al′val′′.
Then, (6.1) and (6.2) can be modified as(6.4)p:⋀l=1qfal′̃Uτk⟶fb̃Uvbk,q:⋀l=1qfal′̃x′,c:fx′b̃,(6.5)p:fa1′̃Uτk⟶fb̃Uvbk,q:fal′̃x′,c:fx′b̃.
Let fa1′̃Uτk=PUτkal′(val′1)/(X,val′1)+⋯+PUτkal′(val′rl)/(X,val′rl)  and fb̃Uvbk=PUvb1b(vb1)/(X,vb1)+⋯+PUvbsb(vbs)/(X,vbs), then the logical combinations of fa1′̃Uτk and fb̃Uvbk are given as follows: fa1′̃Uτk⊕fb̃Uvbk=min{1,PUτkal′(val′rl′)+PUvbkb(vbs′)}/(val′rl′,vbs′), fa1′̃Uτk→fb̃Uvbk=min{1,1-PUτkal′(val′rl′)+PUvb1b(vbs′)}/(val′rl′,vbs′),fa1′̃Uτk×fb̃Uvbk=min{PUτkal′(val′rl′),PUvbkb(vbs′)}/(val′rl′,vbs′), and (fa1′̃Uτk)′=(1-PUτkal′(val′rl′))/(X,val′rl′). Then, we can get the following relations on {(X,val′′)∣val′′∈Va1′}×{(X,vb′)∣vb′∈Vb}:(6.6)R1=fa1′̃Uτk×fb̃Uvbk,(6.7)R2=(fa1′̃Uτk×fb̃Uvbk)∨((fa1′̃Uτk)′×(fb̃Uvbk)′),(6.8)R3=((fa1′̃Uτk)′⊕fb̃Uvbk)∧(fa1′̃Uτk⊕(fb̃Uvbk)′),(6.9)R4=fa1′̃Uτk⟶fb̃Uvbk,(6.10)R5=(fa1′̃Uτk⟶fb̃Uvbk)∨((fa1′̃Uτk)′⟶(fb̃Uvbk)′).
The inference for (6.5) can be expressed as follows:(6.11)fx′b̃=fx′al′̃∘Ri,
where “∘” operator is “max-min.” According to (6.11), we can obtain fx′b̃=Px′b̃(vb1)/(X,vb1)+⋯+Px′b̃(vbs)/(X,vbs); generally, maybe P=∑s′=1sPx′b̃(vbs′)≠1, however, every Px′b̃(vbs′)(s′=1,2,…,s) of fx′b̃ can be normalized as follows:(6.12)Px′b(vbs′)=Px′b̃(vbs′)P,
where P is normalized factor. By (6.12), we can get the probability density function on {(x′,vbs′)∣vbs′∈Vb},(6.13)fx′b=Px′b(vb1)(x′,vb1)+⋯+Px′b(vbs)(x′,vbs).
According to the maximum probability of (6.13), we can decide in which class x′ is included and its degree of belief.Table 4
The information ofx′.
a1⋯ambx′(va11,px′,a11)⋯(vam1,px′,am1)⋮⋮⋮⋮(va1r1,px′,a1r1)⋯(vx′,amrm,px′,amrm)Theorem 6.1.
In (6.11), if Ri=R1,PUτkal′(val′),PUvbkb(vbk), and Px′,al′val′ are maximum probabilities, respectively, then Px′b(vbk)=min(Px′,al′val′,PUτkal′(val′),PUvbkb(vbk)) is the maximum probability of (6.13).Proof.
By (6.6) and (6.11), we know that Px′b(vbk) = max{min(Px′,al′val′m,min(PUτkal′(val′m),PUvbkb(vbk)))∣val′m∈Va1′}=max{min(Px′,al′val′m,PUτkal′(val′m),PUvbkb(vbk))∣val′m∈Va1′}=min(Px′,al′val′,PUτkal′(val′),PUvbkb(vbk)). Obviously, forallval′m∈Va1′ and forallvbs′∈Vb,
(6.14)min⁡(Px′,al′val′,PUτkal′(val′),PUvbkb(vbk))≥min⁡(Px′,al′val′m,PUτkal′(val′m),PUvbs′b(vbs′)),
so we have Px′b(vbk)≥Px′b(vbs′), that is, Px′b(vbk) is the maximum probability of (6.13).According to Corollary4.3, Theorem 6.1 shows that the inference conclusion can make x′ and decision rule p consistent. For simplicity, in the rest of this section, we always assume that PUτkal′(val′), PUvbkb(vbk), and Px′,al′val′ are maximum probabilities, respectively.Theorem 6.2.
IfRi=R2,PUτkal′(val′)≥0.5, and PUvbkb(vbk)≥0.5, then Px′b(vbk)=min(Px′,al′val′,PUτkal′(val′),PUvbkb(vbk))  is the maximum probability of (6.13).Theorem 6.3.
IfRi=R3,thenPx′b(vbk)=min(Px′,al′val′,1-PUτkal′(val′)+PUvbkb(vbk),1+PUτkal′(val′)-PUvbkb(vbk)) is the maximum probability.Theorem 6.4.
IfRi=R4, then Px′b(vbk)=min(Px′,al′val′,1-PUτkal′(val′)+PUvbkb(vbk)) is the maximum probability of (6.13).Proof.
By (6.9) and (6.11), we know that Px′b(vbk) = max{ρm,k=min(Px′,al′val′m,min(1,1-PUτkal′(val′m)+PUvbkb(vbk)))∣val′m∈Va1′} = max{ρm,k=min(Px′,al′val′m,1,1-PUτkal′(val′m)+PUvbkb(vbk))∣val′m∈Va1′} = max{ρm,k=min(Px′,al′val′m,1-PUτkal′(val′m)+PUvbkb(vbk))∣val′m∈Va1′}. When val′m≠val′, by PUτkal′(val′m)≤0.5, we know that 1-PUτkal′(val′m)+PUvbkb(vbk)≥1, so ρm,k=Px′,al′val′m≤ρk=min(Px′,al′val′,1-PUτkal′(val′)+PUvbkb(vbk))=Px′b(vbk). Foralls′≠k, ρs′=min(Px′,al′val′,1-PUτkal′(val′)+PUvbkb(vbs′)) and ρm,s′=min(Px′,al′val′m,1-PUτkal′(val′m)+PUvbkb(vbs′))=Px′,al′val′m. obviously, ρk≥max{ρs′,ρm,s′}=Px′b(vbs′), and Px′b(vbk) is the maximum probability.Remark 6.5.
According to (5.2), if Pal′x′(val′) and Pbx′(vbk) are maximum probability assignments, respectively, then the probability decision rule is consistent. Theorems 6.1–6.4 show that we obtain maximum probability. Hence, using R1–R4 ensures the consistency of decision rule.Theorem 6.6.
IfRi=R5, then Px′b(vbk)=min(Px′,al′val′,max(1-PUτkal′(val′)+PUvbkb(vbk),1+PUτkal′(val′)-PUvbkb(vbk))) is the minimum probability of (6.13).Proof.
By (6.10) and (6.11), we know that Px′b(vbk) = max{ρm,k=min(Px′,al′val′m,max(min(1,1-PUτkal′(val′m)+PUvbkb(vbk)),min(1,1+PUτkal′(val′m)-PUvbkb(vbk))))∣val′m∈Va1′}. By PUτkal′(val′m)≤0.5, we know that min(1,1-PUτkal′(val′m)+PUvbkb(vbk))=1, so ρm,k=Px′,al′val′m. ρk = min(Px′,al′val′,max(1-PUτkal′(val′)+PUvbkb(vbk),1+PUτkal′(val′)-PUvbkb(vbk))), so we have Px′b(vbk)=ρk. For Px′b(vbs′), by 1+PUτkal′(val′)-PUvbkb(vbs′)≥1, we get foralls′≠k,Px′b(vbs′)=Px′,al′val′≥ρk=Px′b(vbk), Px′b(vbk) is minimal probability of (6.13).Remark 6.7.
According to (5.2), if Pal′x′(val′) and Pbx′(vbk) are maximum probability assignments, respectively, then the probability decision rule is consistent. If we use R5, Theorem 6.6 show that we obtain minimal probability. Hence, using R5 cannot ensure the consistency of decision rule.


## 7. Conclusion

In this paper, we provide a kind of probability information system and probability decision tables to represent a degree of belief of “objects have attributes,” formally, the kind of probability information system is extension of classical information system and a special case of interval-valued information system. Based on rough set theory, we discuss extraction of probability decision rules. Then we analyse consistency of probability decision rules. Finally, we provide a method to finish inference of probability decision rules.



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*Source: 507857-2012-06-10.xml*