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45-446: Arguments in subjective logic are subjective opinions about state variables which can take values from a domain (aka state space ), where a state value can be thought of as a proposition which can be true or false. A binomial opinion applies to a binary state variable, and can be represented as a Beta PDF (Probability Density Function). A multinomial opinion applies to a state variable of multiple possible values, and can be represented as

90-766: A X ( x ) = 1 {\displaystyle \sum a_{X}(x)=1\,\!} as well as b X ( x ) , u X , a X ( x ) ∈ [ 0 , 1 ] {\displaystyle b_{X}(x),u_{X},a_{X}(x)\in [0,1]\,\!} . Trinomial opinions can be simply visualised as points inside a tetrahedron , but opinions with dimensions larger than trinomial do not lend themselves to simple visualisation. Dirichlet PDFs are normally denoted as D i r ( p X ; α X ) {\displaystyle \mathrm {Dir} (p_{X};\alpha _{X})\,\!} where p X {\displaystyle p_{X}\,\!}

135-418: A x β = W d x u x + W ( 1 − a x ) {\displaystyle \mathrm {Beta} (p(x);\alpha ,\beta ){\mbox{ where }}{\begin{cases}\alpha &={\frac {Wb_{x}}{u_{x}}}+Wa_{x}\\\beta &={\frac {Wd_{x}}{u_{x}}}+W(1-a_{x})\end{cases}}\,\!} where W {\displaystyle W}

180-517: A x ∈ [ 0 , 1 ] {\displaystyle b_{x},d_{x},u_{x},a_{x}\in [0,1]\,\!} . The characteristics of various opinion classes are listed below. The projected probability of a binomial opinion is defined as P x = b x + a x u x {\displaystyle \mathrm {P} _{x}=b_{x}+a_{x}u_{x}\,\!} . Binomial opinions can be represented on an equilateral triangle as shown below. A point inside

225-465: A Dirichlet PDF (Probability Density Function). Through the correspondence between opinions and Beta/Dirichlet distributions, subjective logic provides an algebra for these functions. Opinions are also related to the belief representation in Dempster–Shafer belief theory . A fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about

270-473: A binomial opinion ω x = ( b x , d x , u x , a x ) {\displaystyle \omega _{x}=(b_{x},d_{x},u_{x},a_{x})\,\!} is the function B e t a ( p ( x ) ; α , β )  where  { α = W b x u x + W

315-504: A conditional relationship between the parent variables and the child variable. [REDACTED] The deduced opinion is computed as ω Z ‖ X Y = ω Z | X Y ⊚ ω X Y {\displaystyle \omega _{Z\|XY}=\omega _{Z|XY}\circledcirc \omega _{XY}} . The joint evidence opinion ω X Y {\displaystyle \omega _{XY}} can be computed as

360-443: A domain (also called state space) e.g. denoted as X {\displaystyle \mathbb {X} } . The values of a domain are assumed to be exhaustive and mutually disjoint, and sources are assumed to have a common semantic interpretation of a domain. The source and variable are attributes of an opinion. Indication of the source can be omitted whenever irrelevant. Let x {\displaystyle x\,\!} be

405-481: A set of 64, or This is significantly greater than the number of legal configurations of the queens, 92. In many games the effective state space is small compared to all reachable/legal states. This property is also observed in Chess , where the effective state space is the set of positions that can be reached by game-legal moves. This is far smaller than the set of positions that can be achieved by placing combinations of

450-438: A state space is the process of enumerating possible states in search of a goal state. The state space of Pacman , for example, contains a goal state whenever all food pellets have been eaten, and is explored by moving Pacman around the board. A search state is a compressed representation of a world state in a state space, and is used for exploration. Search states are used because a state space often encodes more information than

495-631: A state value in a binary domain. A binomial opinion about the truth of state value x {\displaystyle x\,\!} is the ordered quadruple ω x = ( b x , d x , u x , a x ) {\displaystyle \omega _{x}=(b_{x},d_{x},u_{x},a_{x})\,\!} where: These components satisfy b x + d x + u x = 1 {\displaystyle b_{x}+d_{x}+u_{x}=1\,\!} and b x , d x , u x ,

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540-438: Is a probability distribution over the state values of X {\displaystyle X} , and α X {\displaystyle \alpha _{X}\,\!} are the strength parameters. The Dirichlet PDF of a multinomial opinion ω X = ( b X , u X , a X ) {\displaystyle \omega _{X}=(b_{X},u_{X},a_{X})\,\!}

585-766: Is distributive over addition, as expressed by ω x ∧ ( y ∪ z ) = ω ( x ∧ y ) ∪ ( x ∧ z ) {\displaystyle \omega _{x\land (y\cup z)}=\omega _{(x\land y)\cup (x\land z)}\,\!} . De Morgan's laws are also satisfied as e.g. expressed by ω x ∧ y ¯ = ω x ¯ ∨ y ¯ {\displaystyle \omega _{\overline {x\land y}}=\omega _{{\overline {x}}\lor {\overline {y}}}\,\!} . Subjective logic allows very efficient computation of mathematically complex models. This

630-490: Is necessary to explore the space. Compressing each world state to only information needed for exploration improves efficiency by reducing the number of states in the search. For example, a state in the Pacman space includes information about the direction Pacman is facing (up, down, left, or right). Since it does not cost anything to change directions in Pacman, search states for Pacman would not include this information and reduce

675-432: Is possible by approximation of the analytically correct functions. While it is relatively simple to analytically multiply two Beta PDFs in the form of a joint Beta PDF , anything more complex than that quickly becomes intractable. When combining two Beta PDFs with some operator/connective, the analytical result is not always a Beta PDF and can involve hypergeometric series . In such cases, subjective logic always approximates

720-456: Is the child variable. The analyst must learn the set of joint conditional opinions ω Z | X Y {\displaystyle \omega _{Z|XY}} in order to apply the deduction operator and derive the marginal opinion ω Z ‖ X Y {\displaystyle \omega _{Z\|XY}} on the variable Z {\displaystyle Z} . The conditional opinions express

765-466: Is the composite tuple ω X = ( b X , u X , a X ) {\displaystyle \omega _{X}=(b_{X},u_{X},a_{X})\,\!} , where b X {\displaystyle b_{X}\,\!} is a belief mass distribution over the possible state values of X {\displaystyle X\,\!} , u X {\displaystyle u_{X}\,\!}

810-509: Is the function D i r ( p X ; α X ) {\displaystyle \mathrm {Dir} (p_{X};\alpha _{X})} where the strength parameters are given by α X ( x ) = W b X ( x ) u X + W a X ( x ) {\displaystyle \alpha _{X}(x)={\frac {Wb_{X}(x)}{u_{X}}}+Wa_{X}(x)\,\!} , where W {\displaystyle W}

855-417: Is the non-informative prior weight, also called a unit of evidence, normally set to W = 2 {\displaystyle W=2} . Let X {\displaystyle X\,\!} be a state variable which can take state values x ∈ X {\displaystyle x\in \mathbb {X} \,\!} . A multinomial opinion over X {\displaystyle X\,\!}

900-441: Is the non-informative prior weight, also called a unit of evidence, normally set to the number of classes. Most operators in the table below are generalisations of binary logic and probability operators. For example addition is simply a generalisation of addition of probabilities. Some operators are only meaningful for combining binomial opinions, and some also apply to multinomial opinion. Most operators are binary, but complement

945-435: Is the uncertainty mass, and a X {\displaystyle a_{X}\,\!} is the prior (base rate) probability distribution over the possible state values of X {\displaystyle X\,\!} . These parameters satisfy u X + ∑ b X ( x ) = 1 {\displaystyle u_{X}+\sum b_{X}(x)=1\,\!} and ∑

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990-872: Is thereby able to derive belief in variable X {\displaystyle X\,\!} . By expressing each trust edge and belief edge as an opinion, it is possible for A {\displaystyle A\,\!} to derive belief in X {\displaystyle X\,\!} expressed as ω X A = ω X [ A ; B ] ⋄ [ A ; C ] = ( ω B A ⊗ ω X B ) ⊕ ( ω C A ⊗ ω X C ) {\displaystyle \omega _{X}^{A}=\omega _{X}^{[A;B]\diamond [A;C]}=(\omega _{B}^{A}\otimes \omega _{X}^{B})\oplus (\omega _{C}^{A}\otimes \omega _{X}^{C})\,\!} . Trust networks can express

1035-1095: Is unary, and abduction is ternary. See the referenced publications for mathematical details of each operator. Transitive source combination can be denoted in a compact or expanded form. For example, the transitive trust path from analyst/source A {\displaystyle A\,\!} via source B {\displaystyle B\,\!} to the variable X {\displaystyle X\,\!} can be denoted as [ A ; B , X ] {\displaystyle [A;B,X]\,\!} in compact form, or as [ A ; B ] : [ B , X ] {\displaystyle [A;B]:[B,X]\,\!} in expanded form. Here, [ A ; B ] {\displaystyle [A;B]\,\!} expresses that A {\displaystyle A} has some trust/distrust in source B {\displaystyle B} , whereas [ B , X ] {\displaystyle [B,X]\,\!} expresses that B {\displaystyle B} has an opinion about

1080-466: The distributivity of conjunction over disjunction, expressed as x ∧ ( y ∨ z ) ⇔ ( x ∧ y ) ∨ ( x ∧ z ) {\displaystyle x\land (y\lor z)\Leftrightarrow (x\land y)\lor (x\land z)\,\!} , holds in binary propositional logic. This is no surprise as the corresponding probability operators are also non-distributive. However, multiplication

1125-439: The natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [ N ,  A ,  S ,  G ] where: A state space has some common properties: For example,

1170-552: The Vacuum World has a branching factor of 4, as the vacuum cleaner can end up in 1 of 4 adjacent squares after moving (assuming it cannot stay in the same square nor move diagonally). The arcs of Vacuum World are bidirectional, since any square can be reached from any adjacent square, and the state space is not a tree since it is possible to enter a loop by moving between any 4 adjacent squares. State spaces can be either infinite or finite, and discrete or continuous. The size of

1215-443: The available chess pieces directly on the board. All continuous state spaces can be described by a corresponding continuous function and are therefore infinite. Discrete state spaces can also have ( countably ) infinite size, such as the state space of the time-dependent "counter" system, similar to the system in queueing theory defining the number of customers in a line, which would have state space {0, 1, 2, 3, ...}. Exploring

1260-435: The belief edge from B {\displaystyle B\,\!} to X {\displaystyle X\,\!} . A subjective trust network can for example be expressed as ( [ A ; B ] : [ B , X ] ) ⋄ ( [ A ; C ] : [ C , X ] ) {\displaystyle ([A;B]:[B,X])\diamond ([A;C]:[C,X])\,\!} as illustrated in

1305-630: The corresponding propositional/binary logic operator. Similarly, when the argument opinions are equivalent to traditional probabilities, the result of any subjective logic operator is always equal to that of the corresponding probability operator (when it exists). In case the argument opinions contain degrees of uncertainty, the operators involving multiplication and division (including deduction, abduction and Bayes' theorem) will produce derived opinions that always have correct projected probability but possibly with approximate variance when seen as Beta/Dirichlet PDFs. All other operators produce opinions where

1350-402: The figure below. [REDACTED] The indices 1, 2 and 3 indicate the chronological order in which the trust edges and advice are formed. Thus, given the set of trust edges with index 1, the origin trustor A {\displaystyle A\,\!} receives advice from B {\displaystyle B\,\!} and C {\displaystyle C\,\!} , and

1395-426: The opinions to be used as input opinions to the subjective Bayesian network, as illustrated in the figure below. [REDACTED] Traditional Bayesian network typically do not take into account the reliability of the sources. In subjective networks, the trust in sources is explicitly taken into account. State space (computer science) In computer science , a state space is a discrete space representing

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1440-536: The plots to the right. The numerical values and verbal qualitative descriptions of each opinion are also shown. [REDACTED] The Beta PDF is normally denoted as B e t a ( p ( x ) ; α , β ) {\displaystyle \mathrm {Beta} (p(x);\alpha ,\beta )\,\!} where α {\displaystyle \alpha \,\!} and β {\displaystyle \beta \,\!} are its two strength parameters. The Beta PDF of

1485-457: The prior probability, is shown as a red pointer along the base line, and the projected probability, P x {\displaystyle \mathrm {P} _{x}\,\!} , is formed by projecting the opinion onto the base, parallel to the base rate projector line. Opinions about three values/propositions X, Y and Z are visualized on the triangle to the left, and their equivalent Beta PDFs (Probability Density Functions) are visualized on

1530-405: The product of independent evidence opinions on X {\displaystyle X\,\!} and Y {\displaystyle Y\,\!} , or as the joint product of partially dependent evidence opinions. The combination of a subjective trust network and a subjective Bayesian network is a subjective network. The subjective trust network can be used to obtain from various sources

1575-532: The projected probabilities and the variance are always analytically correct. Different logic formulas that traditionally are equivalent in propositional logic do not necessarily have equal opinions. For example ω x ∧ ( y ∨ z ) ≠ ω ( x ∧ y ) ∨ ( x ∧ z ) {\displaystyle \omega _{x\land (y\lor z)}\neq \omega _{(x\land y)\lor (x\land z)}\,\!} in general although

1620-638: The reliability of information sources, and can be used to determine subjective opinions about variables that the sources provide information about. Evidence-based subjective logic ( EBSL ) describes an alternative trust-network computation, where the transitivity of opinions (discounting) is handled by applying weights to the evidence underlying the opinions. In the Bayesian network below, X {\displaystyle X\,\!} and Y {\displaystyle Y\,\!} are parent variables and Z {\displaystyle Z\,\!}

1665-405: The result as an opinion that is equivalent to a Beta PDF. Subjective logic is applicable when the situation to be analysed is characterised by considerable epistemic uncertainty due to incomplete knowledge. In this way, subjective logic becomes a probabilistic logic for epistemic-uncertain probabilities. The advantage is that uncertainty is preserved throughout the analysis and is made explicit in

1710-628: The results so that it is possible to distinguish between certain and uncertain conclusions. The modelling of trust networks and Bayesian networks are typical applications of subjective logic. Subjective trust networks can be modelled with a combination of the transitivity and fusion operators. Let [ A ; B ] {\displaystyle [A;B]\,\!} express the referral trust edge from A {\displaystyle A\,\!} to B {\displaystyle B\,\!} , and let [ B , X ] {\displaystyle [B,X]\,\!} express

1755-418: The set of all possible configurations of a "system". It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory . For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are

1800-415: The size of the search space by a factor of 4, one for each direction Pacman could be facing. Standard search algorithms are effective in exploring discrete state spaces. The following algorithms exhibit both completeness and optimality in searching a state space. These methods do not extend naturally to exploring continuous state spaces. Exploring a continuous state space in search of a given goal state

1845-450: The source of belief whenever required. An opinion is usually denoted as ω X A {\displaystyle \omega _{X}^{A}} where A {\displaystyle A\,\!} is the source of the opinion, and X {\displaystyle X\,\!} is the state variable to which the opinion applies. The variable X {\displaystyle X\,\!} can take values from

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1890-420: The state of variable X {\displaystyle X} which is given as an advice to A {\displaystyle A} . The expanded form is the most general, and corresponds directly to the way subjective logic expressions are formed with operators. In case the argument opinions are equivalent to Boolean TRUE or FALSE, the result of any subjective logic operator is always equal to that of

1935-469: The state space for a given system is the number of possible configurations of the space. If the size of the state space is finite, calculating the size of the state space is a combinatorial problem. For example, in the Eight queens puzzle , the state space can be calculated by counting all possible ways to place 8 pieces on an 8x8 chessboard. This is the same as choosing 8 positions without replacement from

1980-478: The triangle represents a ( b x , d x , u x ) {\displaystyle (b_{x},d_{x},u_{x})\,\!} triple. The b , d , u -axes run from one edge to the opposite vertex indicated by the Belief, Disbelief or Uncertainty label. For example, a strong positive opinion is represented by a point towards the bottom right Belief vertex. The base rate, also called

2025-465: The world is true or false. In addition, whenever the truth of a proposition is expressed, it is always done by an individual, and it can never be considered to represent a general and objective belief. These philosophical ideas are directly reflected in the mathematical formalism of subjective logic. Subjective opinions express subjective beliefs about the truth of state values/propositions with degrees of epistemic uncertainty , and can explicitly indicate

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