Conditional risk

Using the Bayesian decision procedure, the decision-theoretic rough set (DTRS) approach allows for minimum-risk decision making based on observed evidence. Let ${\displaystyle \textstyle A=\{a_{1},\ldots ,a_{m}\}}$ be a finite set of ${\displaystyle \textstyle m}$ possible actions and let ${\displaystyle \textstyle \Omega =\{w_{1},\ldots ,w_{s}\}}$ be a finite set of ${\displaystyle s}$ states. ${\displaystyle \textstyle P(w_{j}\mid [x])}$ is calculated as the conditional probability of an object ${\displaystyle \textstyle x}$ being in state ${\displaystyle \textstyle w_{j}}$ given the object description ${\displaystyle \textstyle [x]}$. ${\displaystyle \textstyle \lambda (a_{i}\mid w_{j})}$ denotes the loss, or cost, for performing action ${\displaystyle \textstyle a_{i}}$ when the state is ${\displaystyle \textstyle w_{j}}$. The expected loss (conditional risk) associated with taking action ${\displaystyle \textstyle a_{i}}$ is given by:

${\displaystyle R(a_{i}\mid [x])=\sum _{j=1}^{s}\lambda (a_{i}\mid w_{j})P(w_{j}\mid [x]).}$

Object classification with the approximation operators can be fitted into the Bayesian decision framework. The set of actions is given by ${\displaystyle \textstyle A=\{a_{P},a_{N},a_{B}\}}$, where ${\displaystyle \textstyle a_{P}}$, ${\displaystyle \textstyle a_{N}}$, and ${\displaystyle \textstyle a_{B}}$ represent the three actions in classifying an object into POS(${\displaystyle \textstyle A}$), NEG(${\displaystyle \textstyle A}$), and BND(${\displaystyle \textstyle A}$) respectively. To indicate whether an element is in ${\displaystyle \textstyle A}$ or not in ${\displaystyle \textstyle A}$, the set of states is given by ${\displaystyle \textstyle \Omega =\{A,A^{c}\}}$. Let ${\displaystyle \textstyle \lambda (a_{\diamond }\mid A)}$ denote the loss incurred by taking action ${\displaystyle \textstyle a_{\diamond }}$ when an object belongs to ${\displaystyle \textstyle A}$, and let ${\displaystyle \textstyle \lambda (a_{\diamond }\mid A^{c})}$ denote the loss incurred by take the same action when the object belongs to ${\displaystyle \textstyle A^{c}}$.

Loss functions

Let ${\displaystyle \textstyle \lambda _{PP}}$ denote the loss function for classifying an object in ${\displaystyle \textstyle A}$ into the POS region, ${\displaystyle \textstyle \lambda _{BP}}$ denote the loss function for classifying an object in ${\displaystyle \textstyle A}$ into the BND region, and let ${\displaystyle \textstyle \lambda _{NP}}$ denote the loss function for classifying an object in ${\displaystyle \textstyle A}$ into the NEG region. A loss function ${\displaystyle \textstyle \lambda _{\diamond N}}$ denotes the loss of classifying an object that does not belong to ${\displaystyle \textstyle A}$ into the regions specified by ${\displaystyle \textstyle \diamond }$.

Taking individual can be associated with the expected loss ${\displaystyle \textstyle R(a_{\diamond }\mid [x])}$actions and can be expressed as:

${\displaystyle \textstyle R(a_{P}\mid [x])=\lambda _{PP}P(A\mid [x])+\lambda _{PN}P(A^{c}\mid [x]),}$

${\displaystyle \textstyle R(a_{N}\mid [x])=\lambda _{NP}P(A\mid [x])+\lambda _{NN}P(A^{c}\mid [x]),}$

${\displaystyle \textstyle R(a_{B}\mid [x])=\lambda _{BP}P(A\mid [x])+\lambda _{BN}P(A^{c}\mid [x]),}$

where ${\displaystyle \textstyle \lambda _{\diamond P}=\lambda (a_{\diamond }\mid A)}$, ${\displaystyle \textstyle \lambda _{\diamond N}=\lambda (a_{\diamond }\mid A^{c})}$, and ${\displaystyle \textstyle \diamond =P}$, ${\displaystyle \textstyle N}$, or ${\displaystyle \textstyle B}$.

Minimum-risk decision rules

If we consider the loss functions ${\displaystyle \textstyle \lambda _{PP}\leq \lambda _{BP}<\lambda _{NP}}$ and ${\displaystyle \textstyle \lambda _{NN}\leq \lambda _{BN}<\lambda _{PN}}$, the following decision rules are formulated (P, N, B):

• P: If ${\displaystyle \textstyle P(A\mid [x])\geq \gamma }$ and ${\displaystyle \textstyle P(A\mid [x])\geq \alpha }$, decide POS(${\displaystyle \textstyle A}$);
• N: If ${\displaystyle \textstyle P(A\mid [x])\leq \beta }$ and ${\displaystyle \textstyle P(A\mid [x])\leq \gamma }$, decide NEG(${\displaystyle \textstyle A}$);
• B: If ${\displaystyle \textstyle \beta \leq P(A\mid [x])\leq \alpha }$, decide BND(${\displaystyle \textstyle A}$);

where,

${\displaystyle \alpha ={\frac {\lambda _{PN}-\lambda _{BN}}{(\lambda _{BP}-\lambda _{BN})-(\lambda _{PP}-\lambda _{PN})}},}$

${\displaystyle \gamma ={\frac {\lambda _{PN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{PP}-\lambda _{PN})}},}$

${\displaystyle \beta ={\frac {\lambda _{BN}-\lambda _{NN}}{(\lambda _{NP}-\lambda _{NN})-(\lambda _{BP}-\lambda _{BN})}}.}$

The ${\displaystyle \textstyle \alpha }$, ${\displaystyle \textstyle \beta }$, and ${\displaystyle \textstyle \gamma }$ values define the three different regions, giving us an associated risk for classifying an object. When ${\displaystyle \textstyle \alpha >\beta }$, we get ${\displaystyle \textstyle \alpha >\gamma >\beta }$ and can simplify (P, N, B) into (P1, N1, B1):

• P1: If ${\displaystyle \textstyle P(A\mid [x])\geq \alpha }$, decide POS(${\displaystyle \textstyle A}$);
• N1: If ${\displaystyle \textstyle P(A\mid [x])\leq \beta }$, decide NEG(${\displaystyle \textstyle A}$);
• B1: If ${\displaystyle \textstyle \beta <P(A\mid [x])<\alpha }$, decide BND(${\displaystyle \textstyle A}$).

When ${\displaystyle \textstyle \alpha =\beta =\gamma }$, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on ${\displaystyle \textstyle \alpha }$:

• P2: If ${\displaystyle \textstyle P(A\mid [x])>\alpha }$, decide POS(${\displaystyle \textstyle A}$);
• N2: If ${\displaystyle \textstyle P(A\mid [x])<\alpha }$, decide NEG(${\displaystyle \textstyle A}$);
• B2: If ${\displaystyle \textstyle P(A\mid [x])=\alpha }$, decide BND(${\displaystyle \textstyle A}$).

Data mining, feature selection, information retrieval, and classifications are just some of the applications in which the DTRS approach has been successfully used.