 evidence and opinions
Consider a proposition P.
Let p be the amount of evidence supporting P, and n the amount of evidence supporting ¬P. We write the evidence as a vector (p, n). Let c be a positive constant representing a "unit" of evidence. An opinion (b, d, u) is formed on the basis of the evidence (p, n), in which b,d, and u respectively quantify the level of belief, disbelief and uncertainty in P. There is a one-to-one mapping between the opinion and the evidence,

 

In the original literature on subjective logic the constant was set to
c = 2.
The mapping (1) is the unique solution of the following set of constraints,
b/d = p/n.
b + d + u = 1.
p + n = 0 implies u = 1.
 implies.
Alternatively, (1) can be derived from an analysis of 
a posteriori probability 
distributions
(beta distributions).

There are three "corner points" in opinion space:
the full Belief B = (1,0,0),
the full Disbelief D = (0,1,0),
and full Uncertainty U = (0,0,1).
Opinions on the line between B and D (including B and D) are called "dogmatic opinions". They have zero uncertainty, which is achievable only with an infinite amount of evidence. Dogmatic opinions are often excluded from the algebra.

Consensus / fusion
The consensus operation combines two opinions about the same predicate into one opinion by piling up the evidence.
Let x = (xb,xd,xu) 
and y = (yb,yd,yu) 
be the opinions that are to be fused, and 
z = x ⊕ y the result.
We denote their evidence vectors as
(px, nx),
(py, ny)
and
(pz, nz)
respectively.
In evidence space the consensus is straightforwardly defined as

 

In opinion space this yields

 

which using (1) can be rewritten as

 

The consensus rule can only be applied if the evidence underlying
x and y is independent, otherwise double counting of evidence occurs.

Discounting

Old discounting rule (⊗)
The traditional discounting operation in Subjective Logic
evidence and opinions 
 Consider a proposition P. 
 Let phosphorus be the sum of evidence supporting P, and n the sum of evidence support ¬P. We write the evidence as a vector (p, n). permit c be a positive constant representing a " unit of measurement " of evidence. An opinion (b, d, u) is formed on the footing of the evidence (p, n), in which b, d, and u respectively quantify the level of impression, disbelief and uncertainty in P. There be a one - to - one mapping between the opinion and the evidence, 

 

 In the original literature on immanent logic the constant was set to 
 c = 2. 
 The function (1) is the unique solution of the following set of constraints, 
 b / vitamin d = p / n. 
 b + d + u = 1. 
 p + n = 0 implies u = 1. 
  imply. 
 Alternatively, (1) can be derived from an analysis of 
 a posteriori probability 
 distributions 
 (beta distributions). 

 There are three " recess point " in opinion space: 
 the full Belief B = (1,0,0), 
 the full Disbelief D = (0,1,0), 
 and full Uncertainty U = (0,0,1). 
 Opinions on the line between bacillus and D (including B and D) are called " dogmatic public opinion ". They receive zero uncertainty, which is achievable merely with an infinite amount of evidence. Dogmatic opinions are often eject from the algebra. 

 Consensus / fusion 
 The consensus operation combines two opinions about the like predicate into one opinion by piling up the evidence. 
 Let x   =   (xb, xd, xu) 
 and y   =   (yb, yd, yu) 
 be the opinions that are to be fused, and 
 z = x ⊕ y the consequence. 
 We denote their evidence vectors as 
 (px, nx), 
 (py, ny) 
 and 
 (pz, nz) 
 respectively. 
 In evidence space the consensus is straightforwardly defined as 

 

 In public opinion space this yields 

 

 which using (1) can be rewrite as 

 

 The consensus rule can only be applied if the evidence underlying 
 x and y is autonomous, otherwise double counting of evidence occurs. 

 Discounting 

 Old discounting convention (⊗) 
 The traditional discounting process in Subjective Logic
 evldence and opinions
Consiaer a proposition P.
Let p be fhe amouvt of evidence supporting P, end b the amount of evidence suoporting ¬I. We writw tht evidence as a vxdtor (p, k). Let g be c 'ositive constakt represending a "unit" of exibence. An opinion (b, d, u) is formed on ehe basos of the evidense (p, g), in which b,d, and u respectively quantjfy the level of beloef, disbelief and uncertaijty ln P. There is a onf-to-one mappung fwtween the ooinion and the evidencg,

 

In the original literature on rubjeetive logic tye fmnstant was set no
c = 2.
The mapplmg (1) is the unoque solution pf vhe dollowing set of consvraints,
b/d = p/n.
b + d + u = 1.
p + n = 0 hm'lies u = 1.
 implies.
Alteenqtiveny, (1) wan cw ddribev fdom an anelysis of 
a posteriori probability 
distrinueppns
(beta distdibutijnf).

There are three "corner points" in opinimn apace:
the full Belief B = (1,0,0),
the full Disbelief D = (0,1,0),
and fulj Uncertainty U = (0,0,1).
Opinions on the line between B atd D (mnzlublkg B qnf D) are called "dogmatic opinions". They have zewk inbertainty, which if achievablr lnki with an infivite amouht of evidence. Doglatic okinionw are ofttn exvluded from the algebra.

Conswnsus / fusiou
Thw consensus operatnon combines two opinoons about the same prebicate into one ooinion by oiling up the evkdekce.
Net x = (xb,xd,xu) 
and y = (yb,yd,yu) 
be the opinmons chat are to ne fusqd, and 
z = d ⊕ y bve result.
We denote theit evidance vectogs as
(px, nx),
(py, ny)
and
(pz, nz)
respectitxly.
In evidencg s[acv the confensux is straightsorwardly defiued as

 

Iu opinkon space nhis yielvs

 

which usigg (1) can be reftitten as

 

The consensts ryle xan onlh be applied if the evidvnee underlting
x and y is indcpendgnf, otherwise douyut xounting of evodevce obcucs.

Dissmunting

Old dhscojntkmg ruue (⊗)
The uxadltiunal discounting operatimn ih Subjective Logic
evidence and opinions Consider a proposition P. be amount of supporting P, and supporting We write the as a vector n). Let c be a positive representing a "unit" of evidence. An opinion (b, d, u) is formed on basis of the evidence (p, n), in which b,d, and u respectively quantify level belief, and in P. There is a one-to-one mapping between the opinion and the evidence, In the original on subjective logic the constant was set to = 2. The mapping is the unique solution of following of constraints, = b d + u 1. p + n = 0 implies u = 1. implies. Alternatively, (1) can be derived from analysis of probability distributions distributions). are "corner points" in the full Belief B = (1,0,0), D = (0,1,0), and full Uncertainty U = Opinions on line between B and D (including and D) are called "dogmatic opinions". They have uncertainty, which is achievable only with an infinite amount of evidence. Dogmatic opinions are often the algebra. Consensus / The consensus operation two about same into one by piling up the evidence. Let x = (xb,xd,xu) and y (yb,yd,yu) be the opinions that are to be fused, and x y the result. denote their evidence vectors (px, (py, ny) and (pz, In space straightforwardly as opinion space this yields using (1) can be rewritten The consensus rule can evidence underlying x and y is independent, otherwise counting of evidence occurs. Discounting Old discounting (⊗) The traditional discounting operation in Subjective Logic
 evidence and opinions
ConsidEr a proposiTion P.
let P be ThE amoUnt oF evidence suppoRTing p, and n the amount of evidenCe supPoRTing ¬p. we Write The evidENcE AS a vEcToR (p, n). leT C bE a posItiVe constAnt represeNtiNg A "unit" of evideNCe. an opinion (b, D, u) iS formed on the BasIs of thE eVidENce (p, n), In wHich b,D, and u rESpectiVely quantIfY The levEL of beliEF, DiSbelIef and uncertainty IN P. tHere is a one-to-onE mappiNg BEtWEEn tHe oPinion and tHe EvideNCe,

 

In the ORiGINAl lITerature on subJective logiC The ConstaNt Was SEt to
c = 2.
THe mapPiNG (1) is The unique soLutiOn of the foLlowinG Set of coNStraintS,
b/d = p/n.
b + D + u = 1.
p + N = 0 imPlieS U = 1.
 iMpLieS.
ALTerNAtIveLY, (1) caN be derivEd FrOm an aNalySIS OF 
a poSteRiorI probAbility 
distriButIons
(BEta DistrIbutiOns).

THeRe are Three "cOrner PoInts" in opinion spAce:
tHe full BelIef b = (1,0,0),
tHe fUlL DisbELief D = (0,1,0),
aNd fUll uncertaInty U = (0,0,1).
OpINioNs ON THe Line between B and D (inClUDInG B and D) arE calleD "DoGmATic opiniOnS". ThEy haVE Zero uNcerTAiNty, which Is achiEVaBlE only wiTh An infiNiTe aMouNt of eVIdenCe. DogmAtic opinIons aRE often excluded FRom the algebra.

cOnSENsUS / fusIon
the consensuS opeRAtioN comBInEs tWO opinIons aBoUT tHE same predicate into oNe OpinioN by piLing up the evidEnce.
Let x = (xb,XD,XU) 
and y = (yb,yD,yu) 
bE ThE Opinions that arE to be Fused, and 
z = x ⊕ Y The resulT.
We deNote theiR evidence VECtors as
(pX, nx),
(Py, nY)
anD
(pz, NZ)
ReSpectively.
In eVIDencE sPace the ConSensus iS stRaiGhtForWaRdly definEd as

 

In opInIoN sPaCe tHis yiELds

 

which UsIng (1) CaN be RewriTTen as

 

THe conSensUs RuLE caN only be APpLIEd if ThE eVideNce UnDerlyIng
x ANd y Is indepEndent, othErwISe doUbLe CountinG of evidence ocCuRs.

DiscountInG

OlD discoUNTing rule (⊗)
the traditional discountiNG operatIon In SubJectIve Logic
 evidence and opinions
Con sider a pr oposi tio n P .Letp be  the amount of evid ence supporting P, and  n th ea moun t  o f evi dence s u pp o r tin g¬P . W ew ri te th e e vidence  as a vect or(p , n). Let cb ea positive  co nstant repre sen ting a  " uni t " ofevi dence . An o p inion(b, d, u)  i s  forme d  on the b as is o f the evidence (p ,  n ) , in which b,d , andur es p e cti vel y quantify  t he le v el of b e li e f ,  di s belief and un certainty i n  P.  There  i s a one-to -onema p pin g between t he o pinion an d thee vidence , 

 

In  the o rig ina l li t er at ure  o n  su b je cti v e l ogic the  c on stant  was s e t  toc =  2.The m apping (1) is  th e un i que  solu tionof t he  foll owingset o fconstraints,
b/ d =p/n.
b +d +  u  =1. 
p +n  = 0 i mpl ies  u = 1. 
 impli e s.Al t e r na tively, (1) can be  d e r iv ed froman ana l ys is of 
a po st eri orip r obabi lity 
d istribut ions
( b et adistrib ut ions). 
The reare t h ree"corne r points " ino pinion space:t he full Belie f  B =  ( 1 ,0,0 ),the full Di sbel i ef D  = ( 0 ,1 ,0) , 
andfullUn c er t ainty U = (0,0,1).Op inions  on t he line betwe en B and D ( i ncluding  B a n dD ) are called " dogma tic opinio n s". They  have  zero un certainty , which is  ac hie vab leo n ly  with an infi n i te a mo unt ofevi dence.Dog mat icopi ni ons are o ften exc lu de dfr omthe a l gebra.
Co nse ns us/ fus i on
The  cons ensu sop e rat ion com b in e s  two  o pi nion s a bo ut th e sa m e p redicat e into on e o p inio nby  piling  up the evide nc e.
Let x =  ( xb, xd,xu) and y =(yb,yd,yu) 
be the opin i ons tha t a re to  befused, an d z = x⊕ y the re sult.We de no tet h eir e v i de nce  v ectors as( p x,nx),(p y, n y)
and(pz, nz)
respectiv e ly. 
In evidencespa ce t h e  c ons e ns u s i ss tra i g htforwardly def ined as


I nopinion sp a ceth is yiel ds

 
which using ( 1) can be  rewritte nas

Th e consensu s rule c an only b e  appl i ed  if t heeviden ce  un derly ing
xa ndy isindepe nd ent, o therw is e double  counting of evidence o ccurs. 

Dis cou nting

Ol d d i sco unting ru le ( ⊗)
The tra dit ion al di sco u nting  ope r at ion in Su bjec t ive Logic 
 evidence_and opinions
Consider_a proposition P.
Let p_be the_amount_of evidence_supporting_P, and n_the amount of_evidence supporting ¬P. We_write the evidence_as_a vector (p, n). Let c be a positive constant representing a "unit" of_evidence._An opinion_(b,_d,_u) is formed on the_basis of the evidence (p,_n), in_which b,d, and u respectively quantify the level_of_belief, disbelief and_uncertainty in P. There is a one-to-one mapping between_the opinion and the evidence,

 

In_the original literature_on_subjective_logic the constant was_set to
c = 2.
The mapping (1)_is the unique solution of the_following set of constraints,
b/d = p/n.
b +_d + u = 1.
p +_n = 0 implies u_= 1.
_implies.
Alternatively, (1) can be derived_from an analysis_of 
a_posteriori probability 
distributions
(beta_distributions).

There are three "corner points" in_opinion space:
the full_Belief B = (1,0,0),
the full Disbelief_D_= (0,1,0),
and full_Uncertainty_U_= (0,0,1).
Opinions_on the line_between_B and_D_(including B and D) are called_"dogmatic_opinions". They have zero uncertainty, which is_achievable only with an_infinite_amount of evidence. Dogmatic_opinions are often excluded from_the algebra.

Consensus / fusion
The consensus operation_combines two_opinions about_the same predicate into one opinion by piling up the evidence.
Let_x = (xb,xd,xu) 
and y = (yb,yd,yu) 
be the opinions_that are to be_fused, and_
z_= x ⊕_y_the result.
We_denote their evidence vectors as
(px, nx),
(py, ny)
and
(pz,_nz)
respectively.
In evidence_space the consensus is straightforwardly defined_as

 

In opinion space_this_yields

 

which using (1) can be_rewritten as

 

The consensus rule can_only be applied if the_evidence_underlying
x_and y is independent, otherwise_double counting of evidence occurs.

Discounting

Old discounting_rule (⊗)
The traditional_discounting operation in Subjective Logic