Editing Abductive reasoning (section)

===Subjective logic abduction===
[[Subjective logic]] generalises [[probabilistic logic]] by including degrees of epistemic [[Uncertainty quantification|uncertainty]] in the input arguments, i.e. instead of probabilities, the analyst can express arguments as [[Subjective logic|subjective opinions]]. Abduction in subjective logic is thus a generalization of probabilistic abduction described above.<ref name="Josang2016-SL" /> The input arguments in subjective logic are subjective opinions which can be binomial when the opinion applies to a binary variable or multinomial when it applies to an ''n''-ary variable. A subjective opinion thus applies to a state variable <math>X</math> which takes its values from a domain <math>\mathbf{X}</math> (i.e. a state space of exhaustive and mutually disjoint state values <math>x</math>), and is denoted by the tuple <math>\omega_{X}=(b_{X}, u_{X}, a_{X})\,\!</math>, where <math>b_{X}\,\!</math> is the belief mass distribution over <math>\mathbf{X}</math>, <math>u_{X}\,\!</math> is the epistemic uncertainty mass, and <math>a_{X}\,\!</math> is the [[base rate]] distribution over <math>\mathbf{X}</math>. These parameters satisfy <math>u_{X}+\sum b_{X}(x) = 1\,\!</math> and <math>\sum a_{X}(x) = 1\,\!</math> as well as <math>b_{X}(x),u_{X},a_{X}(x) \in [0,1]\,\!</math>.

Assume the domains <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> with respective variables <math>X</math> and <math>Y</math>, the set of conditional opinions <math>\omega_{X\mid Y}</math> (i.e. one conditional opinion for each value <math>y</math>), and the base rate distribution <math>a_{Y}</math>. Based on these parameters, the subjective [[Bayes' theorem]] denoted with the operator <math>\;\widetilde{\phi}</math> produces the set of inverted conditionals <math>\omega_{Y\tilde{\mid} X}</math> (i.e. one inverted conditional for each value <math>x</math>) expressed by:

:<math>\omega_{Y\tilde{|}X}=\omega_{X|Y}\;\widetilde{\phi\,}\;a_{Y}</math>.

Using these inverted conditionals together with the opinion <math>\omega_{X}</math> subjective [[Deductive reasoning|deduction]] denoted by the operator <math>\circledcirc</math> can be used to abduce the marginal opinion <math>\omega_{Y\,\overline{\|}\,X}</math>. The equality between the different expressions for subjective abduction is given below:

:<math>\begin{align}
\omega_{Y\,\widetilde{\|}\,X} 
&=  \omega_{X\mid Y} \;\widetilde{\circledcirc}\; \omega_{X}\\
&= (\omega_{X\mid Y} \;\widetilde{\phi\,}\; a_{Y}) \;\circledcirc\;\omega_{X}\\
&= \omega_{Y\widetilde{|}X} \;\circledcirc\;\omega_{X}\;.
\end{align}</math>

The symbolic notation for subjective abduction is "<math>\widetilde{\|}</math>", and the operator itself is denoted as "<math>\widetilde{\circledcirc}</math>". The operator for the subjective Bayes' theorem is denoted "<math>\widetilde{\phi\,}</math>", and subjective deduction is denoted "<math>\circledcirc</math>".<ref name="Josang2016-SL">A. Jøsang. ''Subjective Logic: A Formalism for Reasoning Under Uncertainty'', Springer 2016, {{ISBN|978-3-319-42337-1}}</ref>

The advantage of using subjective logic abduction compared to probabilistic abduction is that both aleatoric and epistemic uncertainty about the input argument probabilities can be explicitly expressed and taken into account during the analysis. It is thus possible to perform abductive analysis in the presence of uncertain arguments, which naturally results in degrees of uncertainty in the output conclusions.