Classical & Quantum Subjectivity

Dr Stuart Nettleton
Dr David Boyle
Atturra Ltd.

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf      © Atturra Ltd. and Actionable Certainty Artificial Intelligence 2022

ABSTRACT

Famously, Pierre-Simon Laplace thought that Bayesian logic, which he invented simultaneously with the Rev. Bayes, was of limited value because belief networks were not really beliefs at all and failed to take account of “don’t know” being a realistic alternative to “yes” and “no.” Classical subjective logic was developed in the 1990s and 2000s to combine the opinions of experts taking into account the fuzziness of “don’t know” across extant states such as “yes and no” or “red, green and blue.” Expert opinions were interpreted as a linear pools of non-independent evidence and combined by simple averaging or more sophisticated techniques such as uncertainty weighting or consensus and compromise fusion. The uncertainty expressed by these experts has two components. The first component is measurement noise in their cognitive model that provides a variance to the distribution of probability. This is well-known aleatory or first-order uncertainty. The second component is an expression of the risk that the cognitive model is incomplete or inadequate. It is a quantification of the limited knowledge and Computational Irreducibility known as epistemic or second order uncertainty. When discussing the probability and uncertainty of an opinion, humans inherently refer respectively to the mean of the first-order probability distribution and the proportion of the opinion second order uncertainty.

In classical subjective logic developed in response to Laplace’s criticism, the probability of a state “yes” is the sum of the core belief in “yes” and the allocated second-order uncertainty. The second-order uncertainty is allocated using a belief base rate. This belief base rate is varied, often wildly, to ensure that the components of a belief never become negative. Unfortunately, classical subjective logic proves unsuitable for Bayes or Markov networks of independent evidence and fails to fully address Laplace’s criticism.

We have resolved this impasse by generalising the classical subjective logic paradigm to a quantum subjective logic paradigm. This presentation provides a brief introduction to developments in decision science where hundreds and sometimes thousands of independent professional judgements can now be combined into a single hypothesis assessment. 

In quantum subjective logic, opinions are regarded as quantum variables with two independent observables being probability and second-order uncertainty. The two observable variances satisfies the Heisenberg Uncertainty Principle and the product of the variances and second cumulant are forms of root mean square.

In marked contrast to classical subjective logic, beliefs in quantum subjective logic have no place at all in calculations. Beliefs are merely convenient output superpositions into single number metrics that can be interpreted as fields for confident action. Furthermore, second-order uncertainty is simply allocated in a constant “non-informative prior” proportion, admitting negative beliefs. Negative beliefs only occur in a small number of extreme cases with low probability and high uncertainty. The quantum Kirkwood-Dirac (KD) distribution describes the probability of these probability-uncertainty pairs and allows for the negative probability of some combinations.

We generalise the classical subjective logic paradigm in three ways. Firstly we show that second-order uncertainty in Markov networks is independent of probability. This accords with the way humans intuitively describe an opinion in terms of a probability and a second-order uncertainty (or “risk” element). Secondly we show that combination of second order uncertainties in a Markov network can be achieved using eigenvector techniques, where the uncertainty of a vertex is a function of the uncertainty of the surrounding vertices. Thirdly, we transition from classical subjective logic to quantum subjective logic, within which the former is treated as a special case. 

An additional paper discussing the nature of uncertainty is provided for background information and discussion.

1. Classical Subjective Logic

1.1 Structure of an Opinion

“Subjective Logic - A formalism for reasoning under uncertainty”. [Jøsang, Audun 2016].

In classical Subjective Logic, the probability estimate of an event is distinguished from (and is independent of) the certainty of that probability.

This allows levels of certainty in probabilities to propagate through analysis to output conclusions.

An opinion is a composite function consisting of:

Belief masses;

Uncertainty mass; and

Base rates

A binomial opinion representation (for a binomial random variable in a binary domain) is given by:

(Subscript[ω, x]) = ((Subscript[b, x]), (Subscript[d, x]), (Subscript[u, x]), (Subscript[a, x]))

with the requirement :

(Subscript[b, x]) + (Subscript[d, x]) + (Subscript[u, x]) = 1

where (Subscript[b, x]) = belief mass in support of x being true, (Subscript[d, x]) = disbelief mass in support of x being false, (Subscript[u, x]) = uncertainty mass and (Subscript[a, x]) = base rate (prior probability of x without any additional evidence).

The projected probability of a binomial opinion about value x is defined by:

P(x) = (Subscript[b, x]) + (Subscript[a, x]) (Subscript[u, x])

which becomes:

(Subscript[b, x]) = P(x) - (Subscript[a, x])(Subscript[u, x])

highlighting the independent variables of probability and (un)certainty.

A multinomial opinion is given by:

(Subscript[ω, x]) = ((Subscript[b, x]), (Subscript[u, x]), (Subscript[a, x]))

with the requirement:

(Subscript[u, x]) + (∑_x)(Subscript[b, x])(x)= 1

where (Subscript[b, x]) = belief mass distribution,  (Subscript[u, x]) = uncertainty mass and (Subscript[a, x]) = base rate distribution over the domain.

The projected probability distribution of multinomial opinions is defined in a similar way (ref eqns (3) and (4)), giving:

(Subscript[b, x])(x) = (Subscript[P, x])(x)- (Subscript[a, x])(x) (Subscript[u, x]) , ∀x  ∈𝕏

b[P_,u_]:=P-a u;

1.2 Dirichlet Bijective Mapping

Opinions are equivalent to Beta or Dirichlet Probability Density Functions (PDFs) under a specific  mapping.

There is a bijective mapping between a binomial opinion and a Beta PDF.

Emerges from the requirement that the projected probability of a binomial opinion must be equal to the expected probability of a Beta PDF.

The projected probability can thus be represented as the mean of a Beta Probability Density Function (PDF).

In a multinomial (multi-dimensional) space, this generalises to a Dirichlet PDF.

The projected probability distribution of a multinomial opinion equates to the expected probability distribution of the Dirichlet PDF.

ClearAll["Global`*"]
styleLabelDir[lbl_]:=Style[lbl,Bold,28];
styleLrgBold[in_]:=Style[in,Bold,40];
styleLrgBlue[in_]:=Style[in,Bold,Blue,40];
styleLrgRed[in_]:=Style[in,Bold,Red,40];
toDirichletαx[beliefs_]:=Module[{W=2,bx,ux,ax,rx,sumrx,αx,αxax},
(*Dirichlet alphas: αx->(((bx+ax ux) W)/ux)*)
bx=Most@#1&@@@beliefs;ux=Last@#1&@@@beliefs;ax=#2&@@@beliefs;
sumrx=Table[If[ux[[i]]>0,W/ux[[i]]-W,∞],{i,Length@ux}];
rx=bx(W+sumrx);αx=rx+W ax;
αxax=W ax;{αx,αxax,ux}];
analyticsOut[belief_]:=Module[{αx,axαx,u,𝒟,mean𝒟,marg𝒟,binPlot,ternPlot,meanLine1,meanLine2,finalPlots,margPlots,meanmarg𝒟,grid,gridData},
{{αx},{axαx},{u}}=toDirichletαx@{belief};
Print[Switch[Length[αx],
2,"Opinion: {{(Subscript[b, 1]), (Subscript[b, 2]), u},{(Subscript[a, 1]), (Subscript[a, 2])} == ",
3,"Opinion: {{(Subscript[b, 1]), (Subscript[b, 2]), (Subscript[b, 3]), u},{(Subscript[a, 1]), (Subscript[a, 2]), (Subscript[a, 3])} == ",
_,"Opinion: {{(Subscript[b, 1]), (Subscript[b, 2]) .. (Subscript[b, n]), u},{(Subscript[a, 1]), (Subscript[a, 2]) ... (Subscript[a, n])} == "],belief];
Print[Style["Probabilities from belief: ",Bold,Blue],belief[[1,;;-2]]+belief[[1,-1]]belief[[2]]];
Print["Dirichlet α's: ",αx];
𝒟=DirichletDistribution[αx];
mean𝒟=Mean@𝒟;
Print["Means of Dirichlet Distribution: ",mean𝒟];
Print[Style["Probabilities from Means of Dirichlet Distribution: ",Bold,Red],{#,1-Total@#}&@mean𝒟//Flatten];
gridData=Transpose@{{styleLrgBold[" Probabilities from:\n"],styleLrgBlue[" Belief "],styleLrgRed[" Means of Dirichlet Distribution "]},{" ",styleLrgBold[belief[[1,;;-2]]+belief[[1,-1]]belief[[2]]],styleLrgBold[{#,1-Total@#}&@mean𝒟//Flatten]}};
grid=Grid[gridData,Alignment->{{Left},{Center}},Frame->All];
Switch[Length[αx],
2,binPlot=Plot[PDF[𝒟,{x}],{x,0,1},PlotRange->All,PlotLabel->styleLabelDir["2D Dirichlet Distribution\n"<>ToString@𝒟](*Style["2D Dirichlet Distribution\n"<>ToString@𝒟,Bold,28]*),ImageSize->{800,650},TicksStyle->Directive[Black, 28]],
3,ternPlot=Plot3D[PDF[𝒟,{x,y}],{x,0,1},{y,0,1},PlotRange->All,MeshFunctions->{#3&},MeshShading->{Lighter[Green,0.5],(*Red*)Green,Lighter[Green,0.5],(*Yellow*)Green},PlotPoints->100,PlotLabel->styleLabelDir["3D Dirichlet Distribution\n"<>ToString@𝒟](*Style["3D Dirichlet Distribution\n"<>ToString@𝒟, Bold, 28]*),ImageSize->{1100,800},TicksStyle->Directive[Black, 28]],
_,Print["Plot unavailable as Belief is beyond 3 dimensions"]];
Print@Switch[Length[αx],
2,Show[binPlot(*,meanLine1*)],
3,Show[ternPlot(*,meanLine2*)]];
marg𝒟=MarginalDistribution[𝒟,#]&/@Range[Length[αx]-1];
meanmarg𝒟=Mean/@marg𝒟;
meanLine1=ListLinePlot[{{meanmarg𝒟[[1]],0},{meanmarg𝒟[[1]],MapThread[PDF,{marg𝒟,meanmarg𝒟}][[1]]}},PlotStyle->Red];
Switch[Length[αx],
3,meanLine2=ListLinePlot[{{meanmarg𝒟[[2]],0},{meanmarg𝒟[[2]],MapThread[PDF,{marg𝒟,meanmarg𝒟}][[2]]}},PlotStyle->Red]];
margPlots=MapIndexed[
Plot[PDF[#1,x],{x,0,1},PlotLabel->styleLabelDir["Marginal Distribution "<>ToString@First@#2<>"\n"<>ToString@#1](*Style["Marginal Distribution "<>ToString@First@#2<>"\n"<>ToString@#1,Bold,28]*),ImageSize->{800,650},TicksStyle->Directive[Black, 28]]&,marg𝒟];
If[Length[αx]==2,finalPlots=Show[margPlots[[1]],meanLine1];Print@finalPlots,finalPlots=MapThread[Show,{margPlots,{meanLine1,meanLine2}}];Print@@finalPlots];
Print[Style["Means of Marginal Distributions: ",Red],Mean/@marg𝒟(*,"\n"*)];
Print["\n",grid,"\n\n"]];
Print@Style["\nBinary Belief {No, Yes};  \n(Subscript[b, 1])=Core belief of No,  \n(Subscript[b, 2])=Core belief of Yes",Bold,44];
analyticsOut@{{0.1,0.6,0.3},{1,1}/2};
Print@Style["\nTriple Belief {Red, Yellow, Blue};  \n(Subscript[b, 1])=Core belief of Red,  \n(Subscript[b, 2])=Core belief of Yellow,  \n(Subscript[b, 3])=Core belief of Blue",Bold,44];
analyticsOut@{{0.1,0.3,0.3,0.3},{1,1,1}/3};

1.3 Markov Probability Network

In developing algorithms for improved decision confidence, we have utilised reasoning models in the form of graphical Markov networks, combined with the principles of SL and uncertainty combination.

We have discussed SL and the utility of the Dirichlet / Belief parallel. Let us turn now to Markov networks.

In the context of probabilistic graphical models, Markov networks represent joint probability distributions over multiple variables.

A Markov network is an undirected,  cyclic graph.

The nodes or vertices represent the variables

The relationships or edges correspond to a notion of direct probabilistic interaction between the neighbouring variables

The influence between the variables is undirected in nature

Graph[{UndirectedEdge[1,2],UndirectedEdge[2,3],UndirectedEdge[3,4],UndirectedEdge[1,4],UndirectedEdge[3,5],UndirectedEdge[4,6],UndirectedEdge[3,6],UndirectedEdge[2,5]},VertexLabels->Table[i->Style[ToString[((Subscript[v, i])),FormatType->StandardForm]<>" (P="<>ToString[N@(i/10)]<>")",Bold,28],{i,6}],ImageSize->{1200,1200},PlotLabel->Style["Example Markov Network\n",Bold,36]]

When opinions - and particularly uncertainty - need to be included in this framework, a method is needed to combine uncertainties across the network.

Classical Subjective Logic multiplicative uncertainty is unsuitable for extension into Markov networks.

Other mechanisms for introducing and combining uncertainty are warranted.

Graph[{UndirectedEdge[1,2],UndirectedEdge[2,3],UndirectedEdge[3,4],UndirectedEdge[1,4],UndirectedEdge[3,5],UndirectedEdge[4,6],UndirectedEdge[3,6],UndirectedEdge[2,5]},VertexLabels->Table[
i->If[i!=5&&i!=6,
Style[ToString[((Subscript[v, i])),FormatType->StandardForm]<>" (P="<>ToString[N@(i/10)]<>", "<>"U=?)",Bold,28],
Style[ToString[((Subscript[v, i])),FormatType->StandardForm]<>" (P="<>ToString[N@(i/10)]<>", "<>"U="<>ToString[Round[RandomReal[{0.1,0.9}],0.1]]<>")",Red,Bold,28]],
{i,6}],ImageSize->{1200,1200},PlotLabel->Style["Markov Network with Uncertainty Inclusion\n",Bold,36]]

1.4 Eigenvector RMS Uncertainty

Traditional Subjective Logic Multiplicative Uncertainty

Traditional SL opinion multiplication selects the minimum of all component uncertainties in a resultant multiplicative distribution (on the basis of choosing a common uncertainty component across all contributing belief masses)

This progressively minimises or reduces the presence of uncertainty for:

Opinions with multiple component beliefs;

The combination of many opinions; or

The progressive associative combination of opinions

This can also result in variations in underlying base rates across a (non-informative) network, as the constraints of the SL framework are applied

Eigenvector RMS Uncertainty

Eigenvector uncertainty measures the transitive influence of uncertainty at vertices expressed as a weighted sum of uncertainties of its neighbours

A high eigenvector uncertainty means that a vertex is connected to many nodes that have high uncertainties

Analogous to Eigenvector Centrality, which measures the influence of a node in a network and is used in many contexts, such as the Google Page Rank algorithm

Root Mean Square is used as a power function (analogous to standard deviation and Fisher Information) in lieu of linear eigenvalue weighting

Meets the two criteria that should be satisfied for a consistent approach:

Keeps base rates consistent throughout a non-informative network

Supports the assumption that uncertainty is independent of probability

Just one of many potential approaches, but does meet these key criteria

In a random simulation of two simple opinions, the resultant uncertainties calculated using traditional subjective logic opinion multiplication and Eigenvector RMS are comparable (see below)

(*Eigenvector of Uncertainty Weighted Adjacency Matrix*)
edgeUncert={0.3,0.4,0.5,0.6};
(*g=Graph[{"a"\[UndirectedEdge]"b","c"\[UndirectedEdge]"b","b"\[UndirectedEdge]"d","b"\[UndirectedEdge]"e"},EdgeWeight->edgeUncert,VertexLabels->"Name",ImageSize->{1200,1200}];(*Orig*)*)
styleLabel[lbl_]:=Style[lbl,Bold,28];
g=Graph[{1\[UndirectedEdge]2,3\[UndirectedEdge]2,2\[UndirectedEdge]4,2\[UndirectedEdge]5},EdgeWeight->edgeUncert,VertexLabels->Table[i->Placed["Name",Above,styleLabel],{i,5}],EdgeLabels->{1\[UndirectedEdge]2->styleLabel["u = 0.3"],3\[UndirectedEdge]2->styleLabel["u = 0.4"],2\[UndirectedEdge]4->styleLabel["u = 0.5"],2\[UndirectedEdge]5->styleLabel["u = 0.6"]},ImageSize->{1200,1200}];
(*VertexLabels->Table[i->Placed["xxx",Below,rotateLabel],{i,3}]*)
uvec=Insert[edgeUncert,(Subscript[u, 2]),2];
a=Normal@WeightedAdjacencyMatrix@g;
(Subscript[λ, 1])=Max@Eigenvalues@a;
RootMeanSquare@edgeUncert==(Subscript[u, 2])/.Solve[Total[uvec a[[2]]]/(Sqrt[4])==(Subscript[λ, 1])(Subscript[u, 2])][[1]];
Print["g = "];
(*g*)
Show[g,ImageSize->{1500,700}]
Print["(Subscript[u, 2]) = ", (Subscript[u, 2])];
Print["uvec = ",uvec];
Print["a = ",a];
Print["(Subscript[λ, 1]) = ",(Subscript[λ, 1])];
Print["RootMeanSquare@edgeUncert = ",RootMeanSquare@edgeUncert];
Print["(Subscript[u, 2])/.Solve[Total[uvec a[[2]]]/(Sqrt[4])==(Subscript[λ, 1])(Subscript[u, 2])][[1]] = ",(Subscript[u, 2])/.Solve[Total[uvec a[[2]]]/(Sqrt[4])==(Subscript[λ, 1])(Subscript[u, 2])][[1]],"\n"];
gridData2=Transpose@{{styleLrgBold[" Uncertainties from: \n"],styleLrgBlue[" 'Direct' RMS calculation "],styleLrgRed[" Eigenvector RMS calculation "]},{" ",styleLrgBold[RootMeanSquare@edgeUncert],styleLrgBold[(Subscript[u, 2])/.Solve[Total[uvec a[[2]]]/(Sqrt[4])==(Subscript[λ, 1])(Subscript[u, 2])][[1]]]}};
grid2=Grid[gridData2,Alignment->{{Left},{Center}},Frame->All]
Print["\n"];

Clear[a,b,p,ax,bx,px,u,ux,fnTable]
px={{1-(Subscript[p, 1]),(Subscript[p, 1])},{1-(Subscript[p, 2]),(Subscript[p, 2])}};ax={{1-(Subscript[a, 1]),(Subscript[a, 1])},{1-(Subscript[a, 2]),(Subscript[a, 2])}};ux={(Subscript[u, 1]),(Subscript[u, 2])};
bx=MapThread[#1-#2#3&,{px,ax,ux}];
uXYMin=Times@@ux;
uXYMax=Plus@@ux-uXYMin;
uXYCandidates=Flatten[(Outer[Times,##]&@@px-Outer[Times,##]&@@bx)/(Outer[Times,##]&@@ax)];
(*params={(Subscript[p, 1])->0.3,(Subscript[p, 2])->0.9,(Subscript[a, 1])->0.5,(Subscript[a, 2])->0.5};*)
(*The maximum uncertainty is given by the space available*)
uMaxFn[params_]:=Values@GroupBy[Flatten[Solve/@Thread[Flatten[bx/.params]==0]],First->Last,Min]
fnTable[params_]:=Module[{umax},
umax=uMaxFn@params;
TableForm[Flatten[Table[{(Subscript[p, 1])/.params,(Subscript[p, 2])/.params,(Subscript[u, 1]),(Subscript[u, 2]),Min@#,Max@#,RootMeanSquare@#,RootMeanSquare@{(Subscript[u, 1]),(Subscript[u, 2])},
Which[RootMeanSquare@{(Subscript[u, 1]),(Subscript[u, 2])}==0,0,
#==0,"na",
True,RootMeanSquare@{(Subscript[u, 1]),(Subscript[u, 2])}/#]&@RootMeanSquare@#,
Which[RootMeanSquare@{(Subscript[u, 1]),(Subscript[u, 2])}==0,0,
#==0,"na",
True,RootMeanSquare@{(Subscript[u, 1]),(Subscript[u, 2])}/#]&@Max@#
}&@Pick[#,(uXYMin<=#<=uXYMax)&/@#]&@(uXYCandidates/.params)&@(uXYCandidates/.params),{(Subscript[u, 1]),0,umax[[1]],0.1},{(Subscript[u, 2]),0,umax[[2]],0.1}],1],TableHeadings->{None, {"(Subscript[p, 1])","(Subscript[p, 2])","(Subscript[u, 1])","(Subscript[u, 2])","(Subscript[u, min])","(Subscript[u, jmax])","(Subscript[u, jrms])","(Subscript[u, erms])","(Subscript[u, erms])/(Subscript[u, jrms])","(Subscript[u, erms])/(Subscript[u, jmax])"}}]]
compFn[paramsRand_]:=Flatten[(fnTable[#]&/@paramsRand)[[All,1]],1];
legendLabel="Resultant Uncertainty\n";
plotLegends={"SL Multiplicative Minimum","SL Multiplicative Maximum","Eigenvector RMS"};
plotAxesLabels=Style[#,{26}]&/@{"(Subscript[u, 1])","(Subscript[u, 2])","(Subscript[u, res])"};
plotLabel="\nTraditional Multiplicative Min and Max (Subscript[u, res]) and EigenVector RMS (Subscript[u, res])\nfor Random Combinations of (Subscript[p, 1]) & (Subscript[p, 2]) and ";
plotFn3D[compRes_,suffix_]:=ListPlot3D[{#[[All,{3,4,5}]],#[[All,{3,4,6}]],#[[All,{3,4,7}]]}&@compRes,PlotLegends->SwatchLegend[plotLegends,LegendLabel->Style[legendLabel,Gray,Bold]],ImageSize->{1200,1200},AxesLabel->plotAxesLabels,PlotLabel->Style[plotLabel<>suffix,{26,Bold}]]
SeedRandom@123;montecarlo=1000;
paramsRand1=Table[{(Subscript[p, 1])->RandomReal[{0,1}],(Subscript[p, 2])->RandomReal[{0,1}],(Subscript[a, 1])->0.5,(Subscript[a, 2])->0.5},montecarlo];
plotFn3D[compFn@paramsRand1,"with (Subscript[a, 1]) = (Subscript[a, 2]) = 0.5"]

SeedRandom@123;
paramsRand2=Table[{(Subscript[p, 1])->RandomReal[{0,1}],(Subscript[p, 2])->RandomReal[{0,1}],(Subscript[a, 1])->RandomReal[{0,1}],(Subscript[a, 2])->RandomReal[{0,1}]},montecarlo];
plotFn3D[compFn@paramsRand2,"(Subscript[a, 1]) & (Subscript[a, 2])"]

Issues arise in situations where very differing opinions require combination.

Note the regions in the above analysis where only Eigenvector RMS generates an answer.

Classical SL is unsuitable for multiplying opinions (or factors) in Markov Nets. Some considerable issues arise in combining uncertainties:

The Classical Opinion space has "no-go" areas in input beliefs and output beliefs where classical beliefs become negative, so Classical SL can only address part of the belief space

Resultant uncertainties are complicated functions of the input probabilities and input uncertainties, with many cross components. This is prohibitively complicated when combining more than two Opinions and can’t be simplified by iterative combination because Opinion combination is non-associative.

A resultant binary opinion has two completely different resultant uncertainties when properly preserving an Opinion base rate (of say the non-informative prior {0.5,0.5}) through the multiplication process

These two resultant uncertainties are extremely poorly behaved, which is the result of imposing the Classical SL paradigm. Furthermore, these uncertainties can be at different extremes of 0 and 1. Neither the Maximum nor the Minimum of resultant uncertainties, nor any Mean or RootMean Square combination of these resultant uncertainties provides a satisfactory result.

Classical SL interprets the resultant uncertainties at extremes of 0 and 1 as due to dramatic changes in the Opinion base rate. The reason Classical SL elects to violate conservation of the Opinion base rate is to avoid the negative belief “no-go” area.

Classical SL chooses the smallest of the resultant uncertainties and accepts the implied extreme changes to the Opinion base rate. This approach is unsatisfactory when multiplying opinions (or factors) in large Markov Nets. This is because all uncertainties are forced towards zero, and because extreme base rates have no meaning.

If the resultant uncertainty in the Classical SL paradigm is controlled to minimise the error from the Input base rate, it is discovered that the resultant uncertainty becomes quasi-independent of the input probabilities and that the RootMeanSquare of the input uncertainties provides a very close approximation to the error-controlled resultant uncertainty.

In contrast to Classical SL, the RootMeanSquare of the input uncertainties is suitable for very large Markov Nets and implicitly preserves the Opinion base rate.

In simulations combining binary opinions with widely varying uncertainties, resultant beliefs can only be computed across the full range of uncertainties using RMS Uncertainty combination.

approxUfullPx[pmx11_,pmx21_,umx1_,umx2_]:=((((1-2 (Subscript[pmx, 21])))^2) (Subscript[umx, 2]^2)-2 (((1-2 (Subscript[pmx, 21])))^2) (Subscript[umx, 1]) (Subscript[umx, 2]^2)+(Subscript[umx, 1]^2) (1-4 (Subscript[pmx, 11]) (((-1+(Subscript[umx, 2])))^2)+4 (Subscript[pmx, 11]^2) (((-1+(Subscript[umx, 2])))^2)+2 (Subscript[umx, 2]) (-1+(1+2 (-1+(Subscript[pmx, 21])) (Subscript[pmx, 21])) (Subscript[umx, 2]))))/((((1-2 (Subscript[pmx, 21])))^2) (Subscript[umx, 2])+(Subscript[umx, 1]) (1-4 (Subscript[pmx, 11]) (((-1+(Subscript[umx, 2])))^2)+4 (Subscript[pmx, 11]^2) (((-1+(Subscript[umx, 2])))^2)+(Subscript[umx, 2]) (-4+4 (-1+(Subscript[pmx, 21])) (Subscript[pmx, 21]) (-2+(Subscript[umx, 1]))+(Subscript[umx, 1])+(Subscript[umx, 2]))))/.{(Subscript[pmx, 11])->pmx11,(Subscript[pmx, 21])->pmx21,(Subscript[umx, 1])->umx1,(Subscript[umx, 2])->umx2}//Quiet
ax1=(({1,1})/2);Wx1=2;ax2=(({1,1})/2);Wx2=2;
fnPx2[px12_,px22_]:=#[[2]]/Total@#&@{(1-px12)(1 -px22),px12 px22}
rmsUFn[px1_,ux1_,px2_,ux2_]:=RootMeanSquare[{ux1,ux2}]/;And@@{px1[[1]]>=ax1[[1]]ux1,px1[[2]]>=ax1[[2]]ux1,px2[[1]]>=ax2[[1]]ux2,px2[[2]]>=ax2[[2]]ux2}
approxUfullPxFn[px1_,ux1_,px2_,ux2_]:=(approxUfullPx@@Join[Join@@Most/@{px1,px2},{ux1,ux2}])/;And@@{px1[[1]]>=ax1[[1]]ux1,px1[[2]]>=ax1[[2]]ux1,px2[[1]]>=ax2[[1]]ux2,px2[[2]]>=ax2[[2]]ux2}
fnPlotBx[ux1_,ux2_]:=Plot3D[{fnPx2[px12,px22]-ax1[[2]]rmsUFn[{1-px12,px12},ux1,{1-px22,px22},ux2],fnPx2[px12,px22]-ax1[[2]]approxUfullPxFn[{1-px12,px12},ux1,{1-px22,px22},ux2]},{px12,0,1},{px22,0,1},ImageSize->800,AxesLabel->{styleLabel["px12"],styleLabel["px22"],styleLabel["Belief"]},PlotLabel->styleLabel["Belief as a function of\n {ux1, ux2} = "<>ToString@{ux1,ux2}],PlotLegends->{"Uncertainty: Eigenvector RMS","Uncertainty: Minimum Output Base Rate Deviation"}]
{fnPlotBx[0.2,0.2],fnPlotBx[0.2,0.3],fnPlotBx[0.2,0.5],fnPlotBx[0.1,0.9]}
rmsUFnDB[px1_,ux1_,px2_,ux2_]:=RootMeanSquare[{ux1,ux2}]
fnPlotBxDB[ux1_,ux2_]:=Plot3D[{fnPx2[px12,px22]-ax1[[2]]rmsUFnDB[{1-px12,px12},ux1,{1-px22,px22},ux2],fnPx2[px12,px22]-ax1[[2]]approxUfullPxFn[{1-px12,px12},ux1,{1-px22,px22},ux2]},{px12,0,1},{px22,0,1},ImageSize->800,AxesLabel->{styleLabel["px12"],styleLabel["px22"],styleLabel["Belief"]},PlotLabel->styleLabel["Belief as a function of\n {ux1, ux2} = "<>ToString@{ux1,ux2}],PlotLegends->{"Uncertainty: Eigenvector RMS","Uncertainty: Minimum Output Base Rate Deviation"}]
fnPlotBxDB[0.1,0.9]

2. Quantum Beliefs

Two views of uncertainty
1. Orthodox Ontological view that uncertainty is both inherent in reality and also due to our lack of knowledge (Epistemic) ... ontological uncertainty happens when we try to observe reality which forces quantum to fall to a random state ... so randomness is in the structure of reality .... Heisenberg and Bell
2. Uncertainty is purely Epistemic ... only due to human’s lack of knowledge or our ignorance of reality ... Einstein & Schrodinger ... and this is also the premise of Chaos Theory

Everything is a quantum system including the weather and the galaxies – all of reality. Even though things can behave classically, deep down these things are quantum and the behaviour only looks classical because one can't observe the quantumness, which is hidden in parts of the continua (Hilbert space) that perhaps you easily can't access.

Borns' Rule describes the statistical nature of reality, which is that quantum states are inherently probabilistic and, together with the wave form of quantum objects, gives rise to observational uncertainty. Events have no pattern, are fortuitous, accidental and vary due to chance, which presents a fundamental limit to observability. However, this does not mean events are random. Events may be influenced by a previous state. Even if the universe is fundamentally probabilistic, chains of cause and effect may continue to occur.

Heisenberg’s Uncertainty Principle provides the limit to observability. It asserts that an observer cannot know a pair of conjugate variables of a physical object at the same time. Three conjugate pairs of great importance in quantum mechanics are:

Position and momentum

Angular orientation and angular momentum

Energy and time

Conjugate Variables Together Borns’ Rule and the Heisenberg Uncertainty Principle mean that, starting from initial conditions, the observer cannot learn or accurately predict the next step in physical events. The observer can measure and know only one of the pair of observable conjugate variables of a quantum system at the same time. It is only by measuring the quantity in question that a particular system can be known. However that measurement in itself causes another quantum object to collide with the physical event and the original objects to falls to one of its states.

Frequency and time is are easily understood pair of conjugate variables. We know a time pulse precisely but there is a high uncertainty about the frequency of a click of sound during this pulse. This is described as a rectangle, where as one side becomes more precisely known the other side has develops uncertainty. There is a minimum amount of uncertainty that’s always preserved: the product in the uncertainty of each of two conjugate variables must always be greater than or equal to half of the reduced Planck constant, ħ/2.

Subjective space of information and meaning are mental models developed by processing observations of the objective space of nature and its physical phenomena. Some quantum principles can be applied to subjective opinions. This remains an area for research that we have addressed.

Subjective probability and uncertainty are a conjugate pair of independent variables defining a mental state.

Quasiprobability distribution is the general quantum case of classical probability distributions of the pairs of subjective probability and uncertainty and represent quantum states in the same way as probability densities represent states in classical statistical mechanics

Rectangle analogy knowing the probability to a very high degree of precision means observing a stable system for a very long time. In contrast, complex social and behavioural systems change very rapidly, for example business, geo-politics and inter-personal relations. Domain models can only be observed for a relatively short time. This short observation time means that the Probability is always associated with a tangibly large uncertainty. Furthermore, the domain models evolve very quickly, which is analogous to quantum time uncertainty, which spreads out with the wavefunction in time-evolution. As time goes on, the uncertainty propagates into the future and leads to a physical state that cannot be arbitrarily well-known.

2.3 Kirkwood-Dirac Distribution

Kirkwood-Dirac (KD) quasiprobability distribution

Intuition from the Bayesian update of a weak measurement: a particle|(Subscript[ϕ, n])> interacts with an observable A to become|(Subscript[ϕ, n+1])> . The particle is pre-selected to have an initial state |ψ> and after the measurement interaction with the observable A of  (OverHat[A])⊗(OverHat[ρ])   (where (OverHat[A]) = ∑ a |a><a| and  (OverHat[ρ]) = |ψ><ψ|) is post-selected on state <f| . The expectation value of the conditional observable is (Subscript[A, W ])= ((<f|A|ψ>)/(<f|ψ>)) where <f|ψ> is the dot product of f and ψ. Decomposing A leads to (Subscript[A, W]) =  ∑ ((<f|a><a|ψ><ψ|f>)/(|<f|ψ>((|)^2)))a . The numerator <f|a><a|ψ><ψ|f> is the Kirkwood-Dirac Quasiprobability Distribution and the denominator is the probability of the finding the particle in the state f when it was initialized in the state of ψ. So  (Subscript[A, W]) is the Bayesian update of the Kirkwood-Dirac distribution (with respect to a and f) conditioned on the final outcome f.

Widely used for representing complex and negative quasiprobabilities, although in many cases there might be just a few complex or negative values

Negative quasiprobabilities focus the energy of the distribution and provide quantum advantage in computation, work extraction and parameter estimation

We assume that all operators on a Hilbert space with finite dimensions d and consider two orthonormal bases,{|(Subscript[a, i])⟩} and {|(Subscript[f, i])⟩}. 
We regard these bases as eigenbases of observables Â = (∑_( i)) (Subscript[a, i ]) |(Subscript[a, i ])⟩ ⟨(Subscript[a, i ])| and   (OverHat[F]) = (∑_( i)) (Subscript[f, i ]) |(Subscript[f, i ])⟩ ⟨(Subscript[f, i ])|.
In terms of these bases, a state (OverHat[ρ]) can be represented by the KD Distribution:
{(Subscript[q, i,j]^(OverHat[ρ])) } ≡ {⟨(Subscript[f, i ])|(Subscript[a, i ])⟩ ⟨ (Subscript[a, i ])| (OverHat[ρ]) |(Subscript[f, i ])⟩ } = {Tr( (Subscript[(OverHat[(∏^( f))]), j]) (Subscript[(OverHat[(∏^(  a))]), i]) (OverHat[ρ]) ) } ,  where (Subscript[(OverHat[(∏^(  a))]), i]) ≡ | (Subscript[a, i ])⟩ ⟨ (Subscript[a, i ])|, etc.
The distribution can be used to calculate expectation values and measurement-outcome probabilities.
{(Subscript[q, i,j]^(OverHat[ρ])) } satisfies some of Kolmogorov’s axioms for joint probability distributions. See Arvidsson-Shukur,David R.M.,Drori,Jacob Chevalier and Nicole Yunger Halpern 2021 (https://arxiv.org/pdf/2009.04468.pdf)

Mathematica Implementation:
aStates={0.3,0.5,0.8,1.0};(*likelihood states*)
evA={0.1,0.3,0.4,0.2};(*probability of likelihood state*)
fStates={0.1,0.4,0.6,0.9};(*certainty states*)
evF={0.2,0.3,0.4,0.1};(*probability of certainty state*)
(*COMPUTATION*)
(*Pauli orthonormal vectors*)
ψxp=(1/(Sqrt[2]))(({{1}, {1}}));ψxm=(1/(Sqrt[2]))(({{1}, {-1}}));ψzp=(({{1}, {0}}));ψzm=(({{0}, {1}}));

{a0,f0}=fnPauliHilbert4/@{{ψzp,ψzm},{ψxp,ψxm}};
{hatA, hatF} = #[[1]] . DiagonalMatrix[#[[2]]] . Inverse@#[[1]] & /@ {{a0, evA}, {f0, evF}};
{a, f} = Eigenvectors /@ {hatA, hatF};
{Πa, Πf} = KroneckerProduct[Conjugate@#, #] & /@ # & /@ {a, f};
ρ=KroneckerProduct[Conjugate@#,#]&@ψ;
qcb[i_, j_] := Tr[Πa[[i]] . Πf[[j]] .ρ]
qcbProb = Table[qcb[i, j], {i, 4}, {j, 4}];

2.2 Quantum Belief Example

In a auto repair shop, some cars take just a few hours to be repaired, for others a part needs to be ordered or the job turns out to be more complicated than anticipated. In our experiment we ask for the workshop manager’s opinion “can all the work can be completed today?” The workshop manager looks over the vehicles and responds something like “I think that’s True but I’m really only willing to be 50/50 held to that.” On another day he might say “I’m not so positive today, probably there’s a 60% chance we’ll get through the work and I’m pretty sure of my estimate this time, say 80%.” A typical bundle of cars is presented each day and the manager’s of the probability and certainty of completing the work in a day is recorded. There are four discrete categories (i.e. states) for each of probability and certainty. At the end of a month a probability distribution for each of probability and certainty is calculated. Classical and quantum expectations for probability, certainty and belief are computed.

Questions