The following contains the basic principles of decision-theoretic rough sets.
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 \textstyle A=\{a_1,\ldots,a_m\} be a
finite set of \textstyle m possible actions and let \textstyle \Omega=\{w_1,\ldots, w_s\} be a finite set of s states. \textstyle P(w_j\mid[x]) is calculated as the
conditional probability of an object \textstyle x being in state \textstyle w_j given the object description \textstyle [x]. \textstyle \lambda(a_i\mid w_j) denotes the loss, or cost, for performing action \textstyle a_i when the state is \textstyle w_j. The expected loss (conditional risk) associated with taking action \textstyle a_i is given by: : 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 \textstyle A=\{a_P,a_N,a_B\}, where \textstyle a_P, \textstyle a_N, and \textstyle a_B represent the three actions in classifying an object into POS(\textstyle A), NEG(\textstyle A), and BND(\textstyle A) respectively. To indicate whether an element is in \textstyle A or not in \textstyle A, the set of states is given by \textstyle \Omega=\{A,A^c\}. Let \textstyle \lambda(a_\diamond\mid A) denote the loss incurred by taking action \textstyle a_\diamond when an object belongs to \textstyle A, and let \textstyle \lambda(a_\diamond\mid A^c) denote the loss incurred by take the same action when the object belongs to \textstyle A^c.
Loss functions Let \textstyle \lambda_{PP} denote the
loss function for classifying an object in \textstyle A into the POS region, \textstyle \lambda_{BP} denote the loss function for classifying an object in \textstyle A into the BND region, and let \textstyle \lambda_{NP} denote the loss function for classifying an object in \textstyle A into the NEG region. A loss function \textstyle \lambda_{\diamond N} denotes the loss of classifying an object that does not belong to \textstyle A into the regions specified by \textstyle \diamond. Taking individual can be associated with the expected loss \textstyle R(a_\diamond\mid[x])actions and can be expressed as: : \textstyle R(a_P\mid[x]) = \lambda_{PP}P(A\mid[x]) + \lambda_{PN}P(A^c\mid[x]), : \textstyle R(a_N\mid[x]) = \lambda_{NP}P(A\mid[x]) + \lambda_{NN}P(A^c\mid[x]), : \textstyle R(a_B\mid[x]) = \lambda_{BP}P(A\mid[x]) + \lambda_{BN}P(A^c\mid[x]), where \textstyle \lambda_{\diamond P}=\lambda(a_\diamond\mid A), \textstyle \lambda_{\diamond N}=\lambda(a_\diamond\mid A^c), and \textstyle \diamond=P, \textstyle N, or \textstyle B.
Minimum-risk decision rules If we consider the loss functions \textstyle \lambda_{PP} \leq \lambda_{BP} and \textstyle \lambda_{NN} \leq \lambda_{BN} , the following decision rules are formulated (
P,
N,
B): •
P: If \textstyle P(A\mid[x]) \geq \gamma and \textstyle P(A\mid[x]) \geq \alpha, decide POS(\textstyle A); •
N: If \textstyle P(A\mid[x]) \leq \beta and \textstyle P(A\mid[x]) \leq \gamma, decide NEG(\textstyle A); •
B: If \textstyle \beta \leq P(A\mid[x]) \leq \alpha, decide BND(\textstyle A); where, : \alpha = \frac{\lambda_{PN} - \lambda_{BN}}{(\lambda_{BP} - \lambda_{BN}) - (\lambda_{PP}-\lambda_{PN})}, : \gamma = \frac{\lambda_{PN} - \lambda_{NN}}{(\lambda_{NP} - \lambda_{NN}) - (\lambda_{PP}-\lambda_{PN})}, : \beta = \frac{\lambda_{BN} - \lambda_{NN}}{(\lambda_{NP} - \lambda_{NN}) - (\lambda_{BP}-\lambda_{BN})}. The \textstyle \alpha, \textstyle \beta, and \textstyle \gamma values define the three different regions, giving us an associated risk for classifying an object. When \textstyle \alpha > \beta, we get \textstyle \alpha > \gamma > \beta and can simplify (
P,
N,
B) into (
P1,
N1,
B1): •
P1: If \textstyle P(A\mid [x]) \geq \alpha, decide POS(\textstyle A); •
N1: If \textstyle P(A\mid[x]) \leq \beta, decide NEG(\textstyle A); •
B1: If \textstyle \beta , decide BND(\textstyle A). When \textstyle \alpha = \beta = \gamma, we can simplify the rules (P-B) into (P2-B2), which divide the regions based solely on \textstyle \alpha: •
P2: If \textstyle P(A\mid[x]) > \alpha, decide POS(\textstyle A); •
N2: If \textstyle P(A\mid[x]) , decide NEG(\textstyle A); •
B2: If \textstyle P(A\mid[x]) = \alpha, decide BND(\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. ==See also==