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of discarding an instance that is an exception rather than an error Indeed Danyluk and note that learning from noisy data is difficult because it is hard to distin Provost guish between noise and exceptions especially if the noise is systematic Ideally the biases of at least one of the learning algorithms will enable it to learn the exception  Therefore one or more of the classifiers that comprise the base level set of detectors can have difficulty capturing a particular exception without causing the exception to be erroneously eliminated from the training data In this case the consensus method will make fewer detection errors than a majority or single algonthm method  Taking a consensus ratherthan a majority vote is a more conservative approach and will result in fewer instances being eliminated from the training data The drawback of a conservative approach is the added risk of retaining bad data In the next section we analyze the probabilities of making identification errors for both retaining bad data and throwing away good data for majority and consensus filters

## 3.3 Identification Errors

In identifying mislabeled instances there are two types of error that can be made  see Figure The first type E occurs when an instance is incorrectly tagged as mislabeled and is subsequently discarded  D   The second type of error  E    occurs when a mislabeled instance M is tagged as correctly labeled In this section we analyze the probability of each of these types of errors for the consensus and majority filter methods

## 3.3.1 Majority Vote

The event of incorrectly tagging a correct instance as mislabeled happens when more than half of the m base level detectors fail to classify the instance correctly  Let P E i the probability that classifier i makes an E  error and for the sake of clarity assume that all m base level classifiers have the same probability of making an E  error that is equal

$$P\ E\qquad\sum_{j\ \ m\ /2}^{j=m}P\ E\ \ _{i}\ \ ^{j}\quad-\ P\ E\ \ _{i}\ \ ^{m-j}\left(\begin{array}{c}{{m}}\\ {{j}}\end{array}\right)$$

If we assume that the errors of the base level classifiers are independent  the to P E ;
probability that a majority vote filter will throw out good data is given by