at the leaf is I   Let E

$-\sum_{i}^{}$
Let $\omega=\mathbb{E}$. 

$$\begin{array}{r l r l}{P_{i}\ x}&{{}}&{{}}&{{}-{\frac{E}{\sum_{i}m_{i}}},}\end{array}$$
E, + i m, Then using a Laplace estimator

$$\begin{array}{r l r l r l}{P_{i}\,\,\,x}&{{}}&{{}}&{{}{}}&{{}{}-\,\,P_{i}\,\,\,x}&{{}}&{{}}&{{}}\\ {\ }&{}&{}&{{}}&{{}}&{{}}\end{array}\times{\frac{m_{i}}{E}},\,\,{\mathrm{for~}}i\,\,\,/\,\,I.$$

Note that only pruned trees and default settings of C
are used in our experiments NB: Let P i x  be the posterior probability of class i  given instance x  Then

P x
Note that NB uses a Laplacian estimate for estimating the conditional probabilities for each nominal attribute to compute P i l x For each continuous valued attribute a normal distribution is assumed in which case the conditional probabilities can be conveniently represented entirely in terms of the mean and variance of the observed values for each class

IB1: Suppose p nearest neighbors are used  denote them by { y s, x s , s instance x We use p in the experiments Then
* [15] M. C. Gonzalez-Garcia, M. C. Gonzalez-Garcia, M.  
$$\begin{array}{r l}{P_{i}\ x}&{{}\quad{\frac{\sum_{s=1}^{p}f\ y_{s}\ /d\ x,x_{s}}{\sum_{s=1}^{p}\ /d\ x,x_{s}}},}\end{array}$$
$${\mathrm{where~}}\ f\ \ y_{s}$$
$${\mathrm{if~}}i\qquad y_{s}{\mathrm{~and~}}$$
 otherwise  and d is the Euclidean distance function In all three learning algorithms  the predicted class of the level    model  given an instance x is that I for which

$$P_{i}\;\;x\;\;\;\;\;\;P_{i}\;\;x\;\;\;\mathrm{for~all}\;\;i\;\;/\;I\,.$$

## 2.2 Level-1 Generalizers

We compare the effect of four different learning algorithms as the level   generalizer  C
IB using p nearest neighbors   1 NB  and a multi response linear regression algorithm MLR  Only the last needs further explanation MLR is an adaptation of a least squares linear regression algorithm that Breiman a used in regression settings  Any classification problem with real valued attributes can be transformed into a multi response regression problem  If the original classification problem has I classes it is converted into I separate regression problems where the problem for class has instances with responses equal to one when they have class and zero otherwise The input to MLR is level data and we need to consider the situation for the model M ' where the attributes are probabilities separately from that for the model M  where