Bayesian training on rough sets The Bayesian framework can be written as in (Marwala, 2007a,b**; Bishop, 2006):** 

$$P(M\mid D)=\frac{P(D\mid M)\,p(M)}{p(D)}\quad\text{where}\quad M=\begin{cases}w\\ G_{x}\\ N_{r}\\ R\end{cases}\tag{14}$$

The parameter P(M | D) **is the probability of the rough set model given the observed** data, P(D | M ) **is the probability of the data given the assumed rough set model also** 
called the likelihood function, P(M **)is the prior probability of the rough set model and**
P(D) **is the probability of the data and is also called the evidence. The evidence can be** 
treated as the normalisation constant. The likelihood function and the resulting error may be estimated as follows: 

$$P(D\,|\,M\,)=\frac{1}{z_{1}}\exp(-error)=\frac{1}{z_{1}}\exp\bigl{\{}A(w,N_{r},R,G_{x})-1\bigr{\}}\tag{1}$$
$$(15)$$
$$error=\sum_{i}^{L}\sum_{k}^{K}\Biggl{(}t_{ik}-\Biggl{(}f_{out}\Biggl{(}\sum_{j=0}^{M}\gamma_{kj}(G_{x},R)_{kj}f_{inout}\Biggl{(}\sum_{i=0}^{d}w_{ji}^{(1)}x_{i}\Biggr{)}\Biggr{)}\Biggr{)}_{R}\Biggr{)}^{2}\tag{16}$$

Here 1 z is the normalisation constant, L is the number of outputs while K **is the number of** 
training examples. The prior probability in this problem is linked to the concept of reducts, which was explained earlier and it is the prior knowledge that the best rough set model is the one with the minimum number of rules (Nr**) and that the best network is the** 
one whose weights are of the same order of magnitude. Therefore, the prior probability may be written as follows: 

$$P(M)=\frac{1}{z_{2}}\mathrm{exp}\Biggl\{-\alpha N_{r}-\beta\sum w^{2}\Biggr\}$$
P M α r β (17) 
$$(17)$$
