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

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.

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 Section 2, 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: let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} 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 mapping 𝑃 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′∶2^𝑋→[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.

Let 𝑋 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→2^𝑌 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 Bel_𝑋 be a belief function defined on the frame 𝑋. The extension of Bel_𝑋 to 𝑌 as Bel_𝑌 can be defined as follows:Bel_𝑌(𝐵)=Bel_𝑋𝑥∣𝑥,𝑦∈𝑅⟹𝑦∈𝐵=Bel_𝑋({𝑥∣𝐺(𝑥)⊂𝐵}). (2.1) If {𝐴_𝑖} are the focal elements of Bel_𝑋 where 𝑚_𝑋(𝐴_𝑖)=𝑎_𝑖, then {𝐵_𝑖}, where 𝐵_𝑖=⋃_{𝑥∈𝐴𝑖}𝐺(𝑥), are the focal elements of Bel_𝑌 and 𝑚_𝑌(𝐵_𝑖)=∑_𝑘𝑎_𝑘, over all 𝑘 such that 𝐵_𝑘=𝐵_𝑖. Let 𝐻={𝑥∣𝐺(𝑥)⊂𝐵}, thenBel_𝑌(𝐵)=_{𝐵𝑖⊂𝐵}𝑚_𝑌𝐵_𝑖=^𝑖_{⋃𝑖𝑥∈𝐴𝐺(𝑥)⊂𝐵}𝑚_𝑋𝐴_𝑖,Bel_𝑌(𝐵)=Bel_𝑋(𝐻)=_{𝐴𝑖⊂𝐻}𝑚_𝑋𝐴_𝑖. (2.2) There exist two special cases of the extension: (1) assume that we know Bel_𝑆×𝑇, which is a belief function defined on 𝑆×𝑇, the marginal belief function of Bel_𝑆×𝑇 on 𝑆 (or 𝑇) is defined as the extension of Bel_𝑆×𝑇 to 𝑆 (or 𝑇), denoted as Bel_𝑆 (or Bel_𝑇); (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)Let 𝑋=𝑆×𝑇 and 𝑌=𝑆 (or 𝑇), then we have a multivalued mapping 𝐺∶𝑆×𝑇⟶2^𝑆or𝐺∶𝑆×𝑇⟶2^𝑇, (2.3) in which 𝐺((𝑠,𝑡))={𝑠}or𝐺((𝑠,𝑡))={𝑡}. Thus, we have a belief function Bel_𝑌(𝐵)=Bel_𝑆×𝑇({(𝑠,𝑡)∣𝐺((𝑠,𝑡))⊂𝐵}). (2.4) 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 on 𝑆 (or 𝑇) is Bel_𝑌(𝐵)=Bel_𝑆×𝑇(𝐵×𝑇)orBel_𝑌(𝐵)=Bel_𝑆×𝑇(𝑆×𝐵). (2.5) If {𝐴_𝑖} are focal elements of Bel_𝑆×𝑇 with 𝑚_𝑆×𝑇(𝐴_𝑖)=𝑎_𝑖, we get 𝑚_𝑆 of Bel_𝑆 (or 𝑚_𝑇 of Bel_𝑇) as 𝑚_𝑆(𝐵_𝑖)=𝑎_𝑖(𝑜𝑟𝑚_𝑇(𝐵_𝑖)=𝑎_𝑖), where 𝐵_𝑖=⋃_{(𝑠,𝑡)∈𝐴𝑖}𝐺((𝑠,𝑡)), that is, 𝐵_𝑖 is projection of 𝐴_𝑖 onto 𝑆 (or 𝑇).

Assume that Bel_𝑆 is a belief function on 𝑆, and let Bel_𝑆×𝑇 be the extension of Bel_𝑆 onto 𝑆×𝑇 that we know nothing about the answer on 𝑇. In this case, we call Bel_𝑆×𝑇 the minimal extension of Bel_𝑆; in particular, we assume that every answer in 𝑇 is compatible with any answer in 𝑆; thus, compatible function is𝐺∶𝑆⟶2^𝑆×𝑇,𝐺(𝑠)={(𝑠,𝑡)∣∀𝑡∈𝑇}. (2.6) So, we have extension of Bel_𝑆 onto 𝑆×𝑇 as Bel_𝑆×𝑇(𝐵)=Bel_𝑆({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 Bel_𝑆×𝑇(𝐴×𝑇)=Bel_𝑆(𝐴). Assume that 𝐵≠𝐴×𝑇, then forall𝑠∈𝑆, 𝐺(𝑠)={𝑠}×𝑇⊈𝐵; thus, {𝑠∣𝐺(𝑠)⊂𝐵}=∅, and Bel_𝑆×𝑇(𝐵)=Bel_𝑆(∅)=0. If we have 𝑚_𝑆(𝐴_𝑖)=𝑎_𝑖, we can get Bel_𝑆×𝑇(𝐵_𝑖)=𝑎_𝑖, where 𝐵_𝑖=⋃_{𝑠∈𝐴𝑖}𝐺(𝑠). Since 𝐺(𝑠)={𝑠}×𝑇, then 𝐵_𝑖=𝐴_𝑖×𝑇, that is, 𝐵_𝑖 is the set consisting of each element in 𝐴_𝑖 coupled with each element in 𝑇.

Assume that we have two independent sources of evidence as the location of the special element in 𝑋, 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,…,𝐴_𝑘 and 𝐵₁,…,𝐵_𝑝, respectively, then their combination, denoted by 𝑚=𝑚₁⊕𝑚₂, is a belief structure over 𝑋 such that (1) 𝑚(𝐴≠∅)=∑_{𝐴𝑖∩𝐵𝑗=𝐴}𝑚₁(𝐴_𝑖)𝑚₂(𝐵_𝑗)/(1−𝐾); (2) 𝑚(∅)=0; (3) 𝐾=∑_{𝐴𝑖∩𝐵𝑗=∅}𝑚₁(𝐴_𝑖)𝑚₂(𝐵_𝑗). The focal elements of 𝑚 are all sets 𝐴 such that 𝐴_𝑖∩𝐵_𝑗=𝐴. Because 𝑚 satisfies commutativity and associativity, we have 𝑚=𝑚₁⊕𝑚₂⊕⋯⊕𝑚_𝑟 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→𝑉_𝑎 for 𝑎∈𝐴, where 𝑉_𝑎 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 object 𝑥∈𝑈 and a value 𝑣_𝑎∈𝑉_𝑎 of an attribute 𝑎∈𝐴 is certain, that is, 𝑎∶𝑈→𝑉_𝑎 is a function, and either 𝑎(𝑥)=𝑣_𝑎 or 𝑎(𝑥)≠𝑣_𝑎 is true. In real practice, the relation between objects and attribute values is uncertain, that is, 𝑎(𝑥)=𝑣_𝑎 with a degree of belief 𝑝_{(𝑎(𝑥)=𝑣𝑎)}∈[0,1]. If 𝑝_{(𝑎(𝑥)=𝑣𝑎)}=1, it means that 𝑎(𝑥)=𝑣_𝑎. If 𝑝_{(𝑎(𝑥)=𝑣𝑎)}=0, it means that 𝑎(𝑥)≠𝑣_𝑎. If 𝑝_{(𝑎(𝑥)=𝑣𝑎)}∈(0,1), it means that 𝑎(𝑥)=𝑣_𝑎 with uncertainty. In this paper, we limit the degree of belief in probability of 𝑎(𝑥)=𝑣_𝑎.

Definition 3.1. A triple Ω=(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={𝑃_𝑎∣𝑎∈𝐴}, 𝑃_𝑎∶𝑈×𝐴→[0,1] such that for each 𝑥∈𝑈, ∑_{𝑣𝑎∈𝑉𝑎}𝑃_𝑎(𝑥,𝑣_𝑎)=1, 𝑃_𝑎(𝑥,𝑣_𝑎)=𝑝_{(𝑎(𝑥)=𝑣𝑎)} means that object 𝑥 has 𝑣_𝑎 of 𝑎 with probability 𝑝_{(𝑎(𝑥)=𝑣𝑎)}.

In Definition 3.1, for any fixed object 𝑥∈𝑈, {𝑃_𝑎(𝑥,𝑣_𝑎)∣𝑣_𝑎∈𝑉_𝑎} is a probability density function on 𝑉_𝑎 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, 𝑃_𝑆((1,high))=0.2. For object 1 and residual CO₂, (high, 0) means that object 1 has not high residual CO₂, that is, 𝑃_𝑅((1,high))=0. For object 3 and temperature, (low, 1) means that object 4 has low value, that is, 𝑃_𝑇((1,low))=1.

Table 1 

An example of probability information system.

In a probability information system, for any object 𝑥∈𝑈 and 𝑎∈𝐴, we denoteΛ^𝑥_𝑎=𝑣_𝑎∈𝑉_𝑎∣𝑃_𝑎𝑥,𝑣_𝑎≠0. (3.1) If |Λ^𝑥_𝑎|=1 (|Λ^𝑥_𝑎| is cardinality of Λ^𝑥_𝑎), then there exists unique 𝑣_𝑎∈𝑉_𝑎 such that 𝑃_𝑎((𝑥,𝑣_𝑎))=1 due to ∑_{𝑣𝑎∈𝑉𝑎}𝑃_𝑎(𝑥,𝑣_𝑎)=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 𝑣_𝑎∈𝑉_𝑎 such that 𝑎(𝑥)=𝑣_𝑎. From this point of view, probability information systems are extension of information systems. If |Λ^𝑥_𝑎|>1, then the probability information system becomes an interval-valued information system when all probabilities 𝑃_𝑎((𝑥,𝑣_𝑎))≠0 are not considered, in such case, for each object 𝑥 and 𝑎∈𝐴, we have 𝑎(𝑥)=Λ^𝑥_𝑎. Hence, a probability information system of Definition 3.1 is a special case of interval-valued information system. Deference between them is 𝑎(𝑥)=Λ^𝑥_𝑎 in interval-valued information system, but 𝑎(𝑥)=Λ^𝑥_𝑎 with a probability density function on Λ^𝑥_𝑎 in probability information system.

Based on probability information system, probability decision tables have the form Δ=(𝑈,𝐴∪{𝑏},𝑃), 𝐴 is called conditions, and 𝑏 is called decision. Let 𝑃={𝑃_𝑐∣𝑐∈𝐴∪{𝑏}}, 𝑃_𝑐∶𝑈×(𝐴∪{𝑏})→[0,1] such that for each 𝑥∈𝑈 and 𝑐∈𝐴∪{𝑏}, ∑_{𝑣𝑐∈𝑉𝑐}𝑃^𝑥_𝑐((𝑥,𝑣_𝑐))=1. For simplicity, denote 𝑉_𝑐 and 𝑃_𝑐 as 𝑉_𝑎 and 𝑃_𝑎, respectively, when 𝑐∈𝐴, and 𝑉_𝑐 and 𝑃_𝑐 as 𝑉_𝑏 and 𝑃_𝑏 when 𝑐=𝑏. For any object 𝑥∈𝑈 and 𝑐∈𝐴∪{𝑏}, denote Λ^𝑥_𝑐={𝑣_𝑐∈𝑉_𝑐∣𝑃_𝑐((𝑥,𝑣_𝑐))≠0}. If |Λ^𝑥_𝑐|=1, then probability decision table is reduced to classical decision tables. If |Λ^𝑥_𝑎|>1, then probability decision table becomes interval-valued decision tables when all probabilities 𝑃_𝑎((𝑥,𝑣_𝑎))≠0 are not considered.

Let 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛} and 𝐴={𝑎₁,…,𝑎_𝑚}. For each 𝑥_𝑖∈𝑈 and attribute value set 𝑉_𝑎𝑗 of attribute 𝑎_𝑗∈𝐴, 𝑉_𝑎𝑗={𝑣¹_𝑎𝑗,…,𝑣^𝑟𝑗_𝑎𝑗}, denote 𝑃^𝑥𝑖_𝑎𝑗 as a probability density function on 𝑉_𝑎𝑗, and denote 𝑃^𝑥𝑖_𝑏 as a probability density function on decision value set 𝑉_𝑏={𝑣¹_𝑏,…,𝑣^𝑠_𝑏}. Formally, a probability decision table is shown in Table 2.

Table 2 

A probability decision table.

In Table 2, 𝑝^𝑥𝑖_𝑎𝑗(𝑣^𝑟′𝑗_𝑎𝑗) is the probability of “𝑥_𝑖 that has the value 𝑣^𝑟′𝑗_𝑎𝑗 of 𝑎_𝑗", where 𝑟^′_𝑗∈{1,2,…,𝑟_𝑗}, and for each 𝑥_𝑖, ∑^𝑟𝑗_{𝑟′𝑗=1}𝑝^𝑥𝑖_𝑎𝑗(𝑣^𝑟′𝑗_𝑎𝑗)=1. For each 𝑥_𝑖 and 𝑐∈𝐴∪{𝑏}, denote𝑝^𝑥𝑖_𝑎𝑗𝑣^𝑟′𝑗_𝑎𝑗=max𝑝^𝑥𝑖_𝑎𝑗𝑣¹_𝑎𝑗,𝑝^𝑥𝑖_𝑎𝑗𝑣²_𝑎𝑗,…,𝑝^𝑥𝑖_𝑎𝑗𝑣^𝑟𝑗_𝑎𝑗. (3.2)

Then we can get a reduced probability decision table shown in Table 3, which is called the maximum probability decision table in this paper.

Table 3 

The maximum probability decision table of Table 2.

4. Probability Decision Rules

In a decision table Δ=(𝑈,𝐴∪{𝑏}), decision rules can be extracted and formalized as follows:𝜏⟶𝑏=𝑖, (4.1) where 𝜏 is a formula, which is generated by some 𝑣_𝑖∈𝑉 finitely using connectives ∨ or ∧, 𝑉=⋃_𝑎∈𝐴𝑉_𝑎, and 𝑖∈𝑉_𝑏. 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 Table 3 without considering probability 𝑝^𝑥i_𝑎𝑗(𝑣^𝑟′𝑗_𝑎𝑗); for simplicity, denote𝑉^′_𝑎𝑗=𝑣_𝑎𝑗∣∃𝑥_𝑖∈𝑈,𝑃^𝑥𝑖_𝑎𝑗𝑣_𝑎𝑗=max𝑃^𝑥𝑖_𝑎𝑗𝑣¹_𝑎𝑗,…,𝑃^𝑥𝑖_𝑎𝑗𝑣^𝑟𝑗_𝑎𝑗,𝑉^′_𝑏=𝑣^𝑘_𝑏∣∃𝑥_𝑖∈𝑈,𝑃^𝑥𝑖_𝑏𝑣^𝑘_𝑏=max𝑃^𝑥𝑖_𝑏𝑣¹_𝑏,…,𝑃^𝑥𝑖_𝑏𝑣^𝑠_𝑏. (4.2) Then attributes of Table 3 can be represented as follows:𝑎_𝑗∶𝑈⟶𝑉^′_𝑎𝑗,𝑎_𝑗𝑥_𝑖=𝑣_𝑎𝑗,𝑏∶𝑈⟶𝑉^′_𝑏,𝑏𝑥_𝑖=𝑣^𝑘_𝑏. (4.3) We can define an equivalence relation on 𝑈 as follows:𝑥_𝑖∼𝑥^′_𝑖iff𝑎_𝑗𝑥_𝑖=𝑎_𝑗𝑥^′_𝑖. (4.4) 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:𝜏^𝑘⟶𝑏=𝑣^𝑘_𝑏, (4.5) where 𝜈^𝑘_𝑏∈𝑉^′_𝑏, 𝜏^𝑘=∧^𝑞_𝑙=1𝑣_𝑎′𝑙, 𝑣_𝑎′𝑙∈𝑉^′_𝑎′𝑙, 𝑎^′_𝑙 and 𝑎^′_𝑙∈{𝑎₁,…,𝑎_𝑚}. Let 𝑈_𝜏𝑘⊆𝑈 is the equivalence class decided by 𝜏^𝑘, 𝑈_𝑣𝑘𝑏⊆𝑈 is the equivalence class decided by 𝑣^𝑘_𝑏. forall𝑥_𝑖∈𝑈_𝜏𝑘 and forall𝑣_𝑎′𝑙∈𝜏^𝑘, 𝑃^𝑥𝑖_𝑎′𝑙(𝑣^𝑟′𝑙_𝑎′𝑙) is the probability of “𝑥_𝑖 has 𝑣^𝑟′𝑙_𝑎′𝑙 of 𝑎^′_𝑙”. Let |𝑈_𝜏𝑘|=𝑇, then we have 𝑇 independent sources of evidence over 𝑈_𝜏𝑘×𝑉_𝑎′𝑙, and belief structure of each source of evidence is as follows:𝑚_𝑠𝑈_𝜏𝑘,𝑣¹_𝑎′𝑙=𝑃^𝑥𝑠_𝑎′𝑙𝑣¹_𝑎′𝑙,…,𝑚_𝑠𝑈_𝜏𝑘,𝑣^𝑟𝑙_𝑎′𝑙=𝑃^𝑥𝑠_𝑎′𝑙𝑣^𝑟𝑙_𝑎′𝑙, (4.6) where 𝑠=1,…,𝑇. Based on (4.6), we can obtain a combined belief structure 𝑚 over 𝑈_𝜏𝑘×𝑉_𝑎′𝑙, that is, 𝑚=𝑚₁⊕⋯⊕𝑚_𝑇 as follows [19]: 𝑃^𝑎′𝑙_𝑈𝜏𝑘=⊕_{𝑥𝑠∈𝑈𝜏𝑘}𝑃^𝑥𝑠_𝑎′𝑙,𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑙_𝑎′𝑙=∏_{𝑥𝑠∈𝑈𝜏𝑘}𝑃^𝑥𝑠_𝑎′𝑙𝑣^𝑟′𝑙_𝑎′𝑙1−𝐾,1−𝐾=_{𝑣𝑟′𝑙𝑎′𝑙∈𝑉𝑎′𝑙𝑥𝑠∈𝑈𝜏𝑘}𝑃^𝑥𝑠_𝑎′𝑙𝑣^𝑟′𝑙_𝑎′𝑙. (4.7)(4.8)(4.9) For attribute 𝑎^′_𝑙, 𝑃^𝑥𝑠_𝑎′𝑙(𝑣_𝑎′𝑙) is the maximum probability that 𝑥_𝑠 has the value 𝑣_𝑎′𝑙 of 𝑎^′_𝑙, so, forall𝑣^𝑟′𝑙_𝑎′𝑙∈𝑉_𝑎′𝑙 and forall𝑥_𝑠∈𝑈_𝜏𝑘, 𝑃^𝑥𝑠_𝑎′𝑙(𝑣_𝑎′𝑙)≥𝑃^𝑥𝑠_𝑎′𝑙(𝑣^𝑟′𝑙_𝑎′𝑙), that is,_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙𝑥_𝑖,𝑣_𝑎′𝑙≥_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙𝑥_𝑖,𝑣^𝑟′𝑙_𝑎′𝑙. (4.10) This means that 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙) is the maximum possibility of all 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑙_𝑎′𝑙). Similarly, we can get the probability of each 𝑣^𝑘′_𝑏 as follows: 𝑃^𝑏_{𝑈𝑣𝑘𝑏}=⊕_{𝑥𝑖∈𝑈𝑣𝑘𝑏}𝑃^𝑥𝑖_𝑏,𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘′_𝑏=∏_{𝑥𝑠∈𝑈𝑣𝑘𝑏}𝑃^𝑥𝑠_𝑏𝑣^𝑘′_𝑏1−𝐾,1−𝐾=_{𝑣𝑘′𝑏∈𝑉𝑏𝑥𝑠∈𝑈𝑣𝑘𝑏}𝑃^𝑥𝑠_𝑏𝑣^𝑘′_𝑏, (4.11)(4.12)(4.13) where 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝜈^𝑘_𝑏) is the maximum possibility of 𝑃^𝑏_{𝑈𝜈𝑘𝑏}(𝜈^𝑘′_𝑏).

According to (4.8) and (4.12), we can get a probability decision rule as follows:𝜏^𝑘,𝑃^𝜏𝑘_𝑈𝜏𝑘⟶𝑏=𝑣^𝑘_𝑏,𝑃^𝑏_{𝑈𝜈𝑘𝑏}𝑣^𝑘_𝑏, (4.14) where (𝜏^𝑘,𝑃^𝜏𝑘_𝑈𝜏𝑘)=∧^𝑞_𝑙=1(𝑣_𝑎′𝑙,𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)), 𝑞≤𝑚. The decision rule (4.5) is a special case of the probability decision rule (4.14), that is, if every 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)=1 and 𝑃^𝑏_{𝑈𝜈𝑘𝑏}(𝑣^𝑘_𝑏)=1. Obviously, for every 𝑎^′_𝑙∈𝐴,^𝑟𝑙_{𝑟′𝑙=1}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑙_𝑎′𝑙=1,^𝑠_𝑘′=1𝑃^𝑏_{𝑈𝜈𝑘𝑏}𝑣^𝑘′_𝑏=1. (4.15)

Theorem 4.1. Let 𝑥_𝑖∈𝑈_𝜏𝑘 if 𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣_𝑎′𝑙))=1, then 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)=1.

Proof. By 𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣_𝑎′𝑙))=1, we know that forall𝑣^𝑟′𝑙_𝑎′𝑙∈𝑉_𝑎′𝑙, 𝑣^𝑟′𝑙_𝑎′𝑙≠𝑣_𝑎′𝑙, 𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣^𝑟′𝑙_𝑎′𝑙))=0, so 1−𝐾=∑_{𝑣𝑟′𝑙𝑎′𝑙∈𝑉𝑎′𝑙}∏_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣^𝑟′𝑙_𝑎′𝑙))=∏_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣_𝑎′𝑙)), by (4.8), 𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙=∏_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙𝑥_𝑖,𝑣_𝑎′𝑙1−𝐾=1. (4.16)

Theorem 4.2. Let 𝑥′ be a new object, then _{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙<0 (4.17) if and only if 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪{𝑥′}}(𝑣_𝑎′𝑙)<𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙).

Proof. According to (4.7), we know that 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪{𝑥′}}=⊕_{𝑥𝑖∈𝑈𝜏𝑘∪{𝑥′}}𝑃^𝑥𝑖_𝑎′𝑙=(⊕_{𝑥𝑖∈𝑈𝜏𝑘}𝑃^𝑥𝑖_𝑎′𝑙)⊕𝑃^𝑥′_𝑎′𝑙=𝑃^𝑎′𝑙_𝑈𝜏𝑘⊕𝑃^𝑥′_𝑎′𝑙. So, we have 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪𝑥′}𝑣_𝑎′𝑙−𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙=𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙−𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙=𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙−1=𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙∑^{𝑣𝑟′𝑗𝑎′𝑙≠𝑣𝑎′𝑙}_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙=𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙, (4.18) and this means that if 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪{𝑥′}}(𝑣_𝑎′𝑙)<𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), then 𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙<0, (4.19) that is, ∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}(𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙)−𝑃^𝑥′_𝑎′𝑙(𝑣^𝑟′𝑗_𝑎′𝑙))𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑗_𝑎′𝑙)<0. Converse is obvious.

Corollary 4.3. If 𝑃^𝑥′_𝑎′𝑙((𝑥^′,𝑣_𝑎′𝑙))=max_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}{𝑃^𝑥𝑖_𝑎′𝑙((𝑥_𝑖,𝑣^𝑟′𝑗_𝑎′𝑙))}, then 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪𝑥′}𝑣_𝑎′𝑙≥𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙. (4.20)

According to the maximum possibility, Corollary 4.3 means that the more the elements of 𝑈_𝜏𝑘 are, the more the probabilities of 𝑣_𝑎′𝑙 of 𝜏^𝑘 are. Intuitively, the elements of 𝑈_𝜏𝑘 reflect the degree of belief of a decision rule; the more the elements of 𝑈_𝜏𝑘 are, the more the degree of belief of a decision rule is. Theorem 4.2 means that, sometimes, adding an element to 𝑈_𝜏𝑘 can make the degree of belief of a decision rule decrease. According to the condition of Theorem 4.2, one can see that 𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙) is not the maximal probability. From the probability point of view, 𝑥′ may not be in 𝑈_𝜏𝑘; if 𝑥′ is forced in 𝑈_𝜏𝑘, 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 valuation 𝑣 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 forall𝑣_𝑎′𝑙, 𝑙=1,…,𝑞, 𝑥′ is such that 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪𝑥′}𝑣_𝑎′𝑙≥𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙,𝑃^𝑏_{𝑈𝑣𝑘𝑏∪𝑥′}𝑣^𝑘_𝑏≥𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏, (5.1) 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 Theorem 4.2, we know that_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙≥0,^𝑠_𝑠′=1𝑃^𝑥′_𝑏𝑣^𝑘_𝑏−𝑃^𝑥′_𝑏𝑣^𝑠′_𝑏𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑠′_𝑏≥0 (5.2) if and only if (5.1) is satisfied. Obviously, if 𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙) and 𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏) 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 𝑉_𝑎𝑗, 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 𝑣^{𝑟′𝑗′}_𝑎′𝑙 and 𝑣^𝑘′_𝑏, but there does not exist decision rule such that its antecedents and conclusion match 𝑣^{𝑟′𝑗′}_𝑎′𝑙 and 𝑣^𝑘′_𝑏, 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 ∨𝑃^𝑏_{𝑈𝑣𝑘𝑏∪{𝑥′}}(𝑣^𝑘_𝑏).

Corollary 5.2. A new object 𝑥′ 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 object 𝑥′, let ^𝑠_𝑠′=1𝑃^𝑥′_𝑏𝑣^𝑠′_𝑏𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑠′_𝑏=𝛼_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙 (5.3) if 𝛼≤𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏)/𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙) and 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪{𝑥′}}(𝑣_𝑎′𝑙)≥𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), then 𝑃^𝑏_{𝑈𝑣𝑘𝑏∪{𝑥′}}(𝑣^𝑘_𝑏)≥𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏).

Proof. By 𝑃^𝑎′𝑙_{𝑈𝜏𝑘∪{𝑥′}}(𝑣_𝑎′𝑙)≥𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), we know that _{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑎′𝑙𝑣_𝑎′𝑙−𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙≥0. (5.4) On the other hand, 𝑃^𝑏_{𝑈𝑣𝑘𝑏∪𝑥′}𝑣^𝑘_𝑏−𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏=𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏𝑃^𝑥′_𝑏𝑣^𝑘_𝑏∑^𝑠_𝑠′=1𝑃^𝑥′_𝑏𝑣^𝑠′_𝑏𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑠′_𝑏−1=𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏𝑃^𝑥′_𝑏𝑣^𝑘_𝑏𝛼∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙−1=𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑥′_𝑏𝑣^𝑘_𝑏/𝛼−𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑟′𝑗_𝑎′𝑙𝑃^𝑥′_𝑎′𝑙𝑣^𝑟′𝑗_𝑎′𝑙. (5.5) By 𝛼≤𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏)/𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙), get 𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏)/𝛼≥𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙), so we have ∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}(𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏)/𝛼−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑗_𝑎′𝑙))𝑃^𝑥′_𝑎′𝑙(𝑣^𝑟′𝑗_𝑎′𝑙)≥∑_{𝑣𝑟′𝑗𝑎′𝑙∈𝑉𝑎′𝑙}(𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙)−𝑃^𝑥′_𝑎′𝑙(𝑣^𝑟′𝑗_𝑎′𝑙))𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑗_𝑎′𝑙)≥0, that is, 𝑃^𝑏_{𝑈𝑣𝑘𝑏∪{𝑥′}}(𝑣^𝑘_𝑏)≥𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏).

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 ∧^𝑞_𝑙=1(𝑣_𝑎′𝑙,𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙))→𝑏=(𝑣^𝑘_𝑏,𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)). Assume that we have a new element 𝑥′ shown in Table 4, we infer a probability density function on 𝑉_𝑏 and decide the class of 𝑥′ and its degree of belief. The inference process can be rewritten as follows:𝑝∶^𝑞_𝑙=1𝑣_𝑎′𝑙,𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙⟶𝑏=𝑣^𝑘_𝑏,𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏,𝑞∶giveTable4,𝑐∶𝑣^𝑠′_𝑏,𝑃^𝑠′_𝑥′,𝑏. (6.1) In (6.1), the probability decision rule 𝑝 has the universes of 𝑈_𝜏𝑘 of antecedent and 𝑈_𝑣𝑘𝑏 of conclusion. In (6.1), 𝑝 can be rewritten as “if the degree of belief of 𝑣_𝑎′1 is 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙) and … and the degree of belief of 𝑣_𝑎′𝑞 is 𝑃^𝑎′𝑞_𝑈𝜏𝑘(𝑣_𝑎′𝑞), then the degree of belief of 𝑣^𝑘_𝑏 is 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏),” 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 𝑣^𝑠′_𝑏 of 𝑥′.” The single condition probability decision rule of 𝑝 has the form:𝑝∶𝑣_𝑎′𝑙,𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙⟶𝑏=𝑣^𝑘_𝑏,𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏,𝑞∶^𝑟𝑙_𝑖=1𝑝^𝑖_{𝑥′,𝑎′𝑙}=1,𝑐∶𝑣^𝑠′_𝑏,𝑃^𝑠′_𝑥′,𝑏. (6.2) In the paper, domains of antecedents and conclusion of 𝑝 are {(𝑈_𝜏𝑘,𝑣^′_𝑎′𝑙)∣𝑣^′_𝑎′𝑙∈𝑉_𝑎′1} and {(𝑈_𝑣𝑘𝑏,𝑣^′_𝑏)∣𝑣^′_𝑏∈𝑉_𝑏}, respectively. We rewrite probability density functions as follows:𝑓^𝑎′1_𝑈𝜏𝑘∶𝑋,𝑣^′_𝑎′𝑙∣𝑣^′_𝑎′𝑙∈𝑉_𝑎′1⟶[0,1],𝑓^𝑎′1_𝑈𝜏𝑘𝑋,𝑣^′_𝑎′𝑙=𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙,𝑓^𝑏_{𝑈𝑣𝑘𝑏}∶𝑋,𝑣^′_𝑏∣𝑣^′_𝑏∈𝑉_𝑏⟶[0,1],𝑓^𝑏_{𝑈𝑣𝑘𝑏}𝑋,𝑣^′_𝑏=𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^′_𝑏,𝑓^𝑎′𝑙_𝑥′∶𝑋,𝑣^′_𝑎′𝑙∣𝑣^′_𝑎′1∈𝑉_𝑎′𝑙⟶[0,1],𝑓^𝑎′𝑙_𝑥′𝑋,𝑣^′_𝑎′𝑙=𝑝^{𝑣′𝑎′𝑙}_{𝑥′,𝑎′𝑙}. (6.3) Then, (6.1) and (6.2) can be modified as𝑝∶^𝑞_𝑙=1𝑓^𝑎′𝑙_𝑈𝜏𝑘⟶𝑓^𝑏_{𝑈𝑣𝑘𝑏},𝑞∶^𝑞_𝑙=1𝑓^𝑎′𝑙_𝑥′,𝑐∶𝑓^𝑏_𝑥′,𝑝∶𝑓^𝑎′1_𝑈𝜏𝑘⟶𝑓^𝑏_{𝑈𝑣𝑘𝑏},𝑞∶𝑓^𝑎′𝑙_𝑥′,𝑐∶𝑓^𝑏_𝑥′. (6.4)(6.5) Let 𝑓^𝑎′1_𝑈𝜏𝑘=𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣¹_𝑎′𝑙)/(𝑋,𝑣¹_𝑎′𝑙)+⋯+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟𝑙_𝑎′𝑙)/(𝑋,𝑣^𝑟𝑙_𝑎′𝑙) and 𝑓^𝑏_{𝑈𝑣𝑘𝑏}=𝑃^𝑏_{𝑈𝑣1𝑏}(𝑣¹_𝑏)/(𝑋,𝑣¹_𝑏)+⋯+𝑃^𝑏_{𝑈𝑣𝑠𝑏}(𝑣^𝑠_𝑏)/(𝑋,𝑣^𝑠_𝑏), then the logical combinations of 𝑓^𝑎′1_𝑈𝜏𝑘 and 𝑓^𝑏_{𝑈𝑣𝑘𝑏} are given as follows: 𝑓^𝑎′1_𝑈𝜏𝑘⊕𝑓^𝑏_{𝑈𝑣𝑘𝑏}=min{1,𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑙_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑠′_𝑏)}/(𝑣^𝑟′𝑙_𝑎′𝑙,𝑣^𝑠′_𝑏), 𝑓^𝑎′1_𝑈𝜏𝑘→𝑓^𝑏_{𝑈𝑣𝑘𝑏}=min{1,1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑙_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣1𝑏}(𝑣^𝑠′_𝑏)}/(𝑣^𝑟′𝑙_𝑎′𝑙,𝑣^𝑠′_𝑏),𝑓^𝑎′1_𝑈𝜏𝑘×𝑓^𝑏_{𝑈𝑣𝑘𝑏}=min{𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑙_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑠′_𝑏)}/(𝑣^𝑟′𝑙_𝑎′𝑙,𝑣^𝑠′_𝑏), and (𝑓^𝑎′1_𝑈𝜏𝑘)′=(1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑟′𝑙_𝑎′𝑙))/(𝑋,𝑣^𝑟′𝑙_𝑎′𝑙). Then, we can get the following relations on {(𝑋,𝑣^′_𝑎′𝑙)∣𝑣^′_𝑎′𝑙∈𝑉_𝑎′1}×{(𝑋,𝑣^′_𝑏)∣𝑣^′_𝑏∈𝑉_𝑏}:𝑅₁=𝑓^𝑎′1_𝑈𝜏𝑘×𝑓^𝑏_{𝑈𝑣𝑘𝑏},𝑅₂=𝑓^𝑎′1_𝑈𝜏𝑘×𝑓^𝑏_{𝑈𝑣𝑘𝑏}∨𝑓^𝑎′1_𝑈𝜏𝑘^′×𝑓^𝑏_{𝑈𝑣𝑘𝑏}^′𝑅₃,=𝑓^𝑎′1_𝑈𝜏𝑘^′⊕𝑓^𝑏_{𝑈𝑣𝑘𝑏}∧𝑓^𝑎′1_𝑈𝜏𝑘⊕𝑓^𝑏_{𝑈𝑣𝑘𝑏}^′𝑅₄,=𝑓^𝑎′1_𝑈𝜏𝑘⟶𝑓^𝑏_{𝑈𝑣𝑘𝑏}𝑅₅,=𝑓^𝑎′1_𝑈𝜏𝑘⟶𝑓^𝑏_{𝑈𝑣𝑘𝑏}∨𝑓^𝑎′1_𝑈𝜏𝑘^′⟶𝑓^𝑏_{𝑈𝑣𝑘𝑏}^′. (6.6)(6.7)(6.8)(6.9)(6.10) The inference for (6.5) can be expressed as follows:𝑓^𝑏_𝑥′=𝑓^𝑎′𝑙_𝑥′∘𝑅_𝑖, (6.11) where “∘” operator is “max-min.” According to (6.11), we can obtain 𝑓^𝑏_𝑥′=𝑃^𝑏_𝑥′(𝑣¹_𝑏)/(𝑋,𝑣¹_𝑏)+⋯+𝑃^𝑏_𝑥′(𝑣^𝑠_𝑏)/(𝑋,𝑣^𝑠_𝑏); generally, maybe 𝑃=∑^𝑠_𝑠′=1𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏)≠1, however, every 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏)(𝑠^′=1,2,…,𝑠) of 𝑓^𝑏_𝑥′ can be normalized as follows:𝑃^𝑏_𝑥′𝑣^𝑠′_𝑏=𝑃^𝑏_𝑥′𝑣^𝑠′_𝑏𝑃, (6.12) where 𝑃 is normalized factor. By (6.12), we can get the probability density function on {(𝑥^′,𝑣^𝑠′_𝑏)∣𝑣^𝑠′_𝑏∈𝑉_𝑏},𝑓^𝑏_𝑥′=𝑃^𝑏_𝑥′𝑣¹_𝑏𝑥′,𝑣¹_𝑏𝑃+⋯+^𝑏𝑥′𝑣^𝑠_𝑏𝑥′,𝑣^𝑠_𝑏. (6.13) 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 of 𝑥^′.

Theorem 6.1. In (6.11), if 𝑅_𝑖=𝑅₁,𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏), and 𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙} are maximum probabilities, respectively, then 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈v𝑘𝑏}(𝑣^𝑘_𝑏)) is the maximum probability of (6.13).

Proof. By (6.6) and (6.11), we know that 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = max{min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}=max{min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)). Obviously, forall𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1 and forall𝑣^𝑠′_𝑏∈𝑉_𝑏, min𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣_𝑎′𝑙,𝑃^𝑏_{𝑈𝑣𝑘𝑏}𝑣^𝑘_𝑏≥min𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘𝑣^𝑚_𝑎′𝑙,𝑃^𝑏_{𝑈𝑣𝑠′𝑏}𝑣^𝑠′_𝑏, (6.14) so we have 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)≥𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏), that is, 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) is the maximum probability of (6.13).

According to Corollary 4.3, 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 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏), and 𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙} are maximum probabilities, respectively.

Theorem 6.2. If 𝑅_𝑖=R₂,𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)≥0.5, and 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)≥0.5, then 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)) is the maximum probability of (6.13).

Theorem 6.3. If 𝑅_𝑖=𝑅₃,then𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏),1+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)) is the maximum probability.

Theorem 6.4. If 𝑅_𝑖=𝑅₄, then 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)) is the maximum probability of (6.13).

Proof. By (6.9) and (6.11), we know that 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},min(1,1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1} = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},1,1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1} = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}. When 𝑣^𝑚_𝑎′𝑙≠𝑣_𝑎′𝑙, by 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤0.5, we know that 1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)≥1, so 𝜌_𝑚,𝑘=𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙}≤𝜌_𝑘=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))=𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏). Forall𝑠^′≠𝑘, 𝜌_𝑠′=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑠′_𝑏)) and 𝜌_𝑚,𝑠′=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑠′_𝑏))=𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙}. obviously, 𝜌_𝑘≥max{𝜌_𝑠′,𝜌_𝑚,𝑠′}=𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏), and 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) is the maximum probability.

Remark 6.5. According to (5.2), if 𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙) and 𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏) 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. If 𝑅_𝑖=𝑅₅, then 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^b_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏),1+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))) is the minimum probability of (6.13).

Proof. By (6.10) and (6.11), we know that 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(min(1,1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)),min(1,1+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}. By 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤0.5, we know that min(1,1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))=1, so 𝜌_𝑚,𝑘=𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙}. 𝜌_𝑘 = min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)+𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏),1+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))), so we have 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=𝜌_𝑘. For 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏), by 1+𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑠′_𝑏)≥1, we get forall𝑠^′≠𝑘,𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏)=𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙}≥𝜌_𝑘=𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏), 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) is minimal probability of (6.13).

Remark 6.7. According to (5.2), if 𝑃^𝑥′_𝑎′𝑙(𝑣_𝑎′𝑙) and 𝑃^𝑥′_𝑏(𝑣^𝑘_𝑏) 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.

Appendices

A. The Proof of Theorem 6.2

Proof. By (6.7) and (6.11), we know that 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)),min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)))))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}, where index 𝑘 of 𝜌_𝑚,𝑘 shows the 𝑘 column of 𝑅₂, when 𝑣^𝑚_𝑎′𝑙=𝑣_𝑎′𝑙, noted by 𝜌_𝑘. By 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)≥0.5, 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)≥0.5, we get min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))≥min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))). So, 𝜌_𝑘=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)).By for all𝑣^𝑚_𝑎′𝑙≠𝑣_𝑎′𝑙, we have 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)≤1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙), and 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏), so 𝜌_𝑚,𝑘 = min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))) and obviously, 𝜌_𝑘≥𝜌_𝑚,𝑘. So, we have 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)). By for all 𝑣^𝑠′_𝑏≠𝑣^𝑘_𝑏, we have 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏) = max{𝜌_𝑚,𝑠′=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)),min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}. By min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))=𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏), min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))=1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), we have 𝜌_𝑠′=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏),1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙))). Because for all 𝑣^𝑚_𝑎′𝑙≠𝑣_𝑎′𝑙, min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))≤min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))), so 𝜌_𝑚,𝑠′=min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),(1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))=𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙), 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏)=max{𝜌_𝑠′,𝜌_𝑚,𝑠′}, and 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)≥𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏), that is, 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) is the maximum probability of (6.13).

B. The Proof of Theorem 6.3

Proof. By (6.7) and (6.11), we know that 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) = max{𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)),min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)))))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}, where index 𝑘 of 𝜌_𝑚,𝑘 shows the 𝑘 column of 𝑅₂, when 𝑣^𝑚_𝑎′𝑙=𝑣_𝑎′𝑙, noted by 𝜌_𝑘. By 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)≥0.5, 𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)≥0.5, we get min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))≥min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))). So, 𝜌_𝑘=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)). By for all 𝑣^𝑚_𝑎′𝑙≠𝑣_𝑎′𝑙, we have 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)≤1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙), and 𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)≤𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏), so 𝜌_𝑚,𝑘=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),1−𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏))), and obviously, 𝜌_𝑘≥𝜌_𝑚,𝑘. So, we have 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑘𝑏}(𝑣^𝑘_𝑏)). By for all 𝑣^𝑠′_𝑏≠𝑣^𝑘_𝑏, we have 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏) = max{𝜌_𝑚,𝑠′=min(𝑃^{𝑣𝑚𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)),min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))))∣𝑣^𝑚_𝑎′𝑙∈𝑉_𝑎′1}. By min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))=𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏), min((1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))=1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙), we have 𝜌_𝑠′=min(𝑃^{𝑣𝑎′𝑙}_{𝑥′,𝑎′𝑙},max(𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏),1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣_𝑎′𝑙))). Because for all 𝑣^𝑚_𝑎′𝑙≠𝑣_𝑎′𝑙, min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))≤min((1−𝑃^𝑎′𝑙_𝑈𝜏k(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏))), so 𝜌_𝑚,𝑠′=min(𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙),(1−𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙)),(1−𝑃^𝑏_{𝑈𝑣𝑠′𝑏}(𝑣^𝑠′_𝑏)))=𝑃^𝑎′𝑙_𝑈𝜏𝑘(𝑣^𝑚_𝑎′𝑙), 𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏)=max{𝜌_𝑠′,𝜌_𝑚,𝑠′}, and 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏)≥𝑃^𝑏_𝑥′(𝑣^𝑠′_𝑏) that is, 𝑃^𝑏_𝑥′(𝑣^𝑘_𝑏) is the maximum probability of (6.13).

Acknowledgments

This work is partially supported by Sichun Key Laboratory of Intelligent Network Information Processing (SGXZD1002-10), National Natural Science Foundation (61175055, 61105059), Sichuan Key Technology Research and Development Program (2012GZ0019, 2011FZ0051), and Liaoning Excellent Talents in University (LJQ2011116).