%In prior work we have explored a range of data mining methods
%for defect prediction and found that
%classifiers based on Bayes theorem work best for the 
%\fig{data} data~\cite{me07b}
%(for notes on Bayes classifiers, see \fig{bayes}).
%Since that study, we have tried to find  better data mining
%algorithms for defect prediction. To date, we have failed. Our
%recent (as yet, unpublished) experiments have found no additional
%statistically significant improvement from the application of the
%following data mining methods: logistic regression; average
%one-dependence estimators; under- or
%over-sampling~\cite{drummond03}, random forests,
%RIPPER~\cite{cohen95r}, decision tree learning with 
%J48~\cite{quinlan92}, rule learning with OneR~\cite{holte93}
%and bagging~\cite{brieman96}. Only boosting~\cite{FreSch97} on
%discretized data offers a statistically better result than a Bayes
%classifier. However, we cannot recommend boosting: the median
%improvement is quite negligible and  boosting is orders of
%magnitude slower than our simple Bayes classifier.

%Other researchers have also failed to improve our results. For example, Lessmann et al. 
%investigate the statistical difference of the results between 19 learners, including naive Bayes, on the same datasets~\cite{lessmann09}. \fig{lessmann} shows that the simple Bayesian method discussed above ties in
%first place along with 15 other methods.

%
%\begin{figure}
%\begin{center}
%\includegraphics[width=3in]{lessmann.pdf}
%\end{center}
%\caption{Range of ranks seen in 19 learners building defects predictors when, 
%10 times, a random 66\% selection of the data are used for training and
%the remaining data are used for testing. In ranked data, values from one
%method are replaced by their rank in space of all sorted values (so smaller ranks
%mean better performance). In this case, the performance value
%was area under the false positive vs true positive curve (and larger
%values are better). Vertical lines divide the results into regions where
%the results are statistically similar. For example, all the methods
%whose top ranks are 4 to 12 are statistically insignificantly different.
%From~\cite{lessmann09}.
%}\label{fig:lessmann}
%\end{figure}


%
%\begin{figure}[!t]
%\footnotesize
%\begin{tabular}{|p{.95\linewidth}|}\hline
%Bayesian classifiers
%offers a relationship between  fragments of evidence $E_i$,
%a prior probability for
%a class $P(H)$, and a posteriori  probability
%$P(H|E)$. Note that the likelihood $P(H|E)$ is approximated by the product term due to the i.i.d. (independent and identically distributed) assumption: 
%\begin{equation}\label{eq:bayes}
%P(H|E)= \left(\prod_iP(E_i|H)\right) \frac{P(H)}{P(E)}
%\end{equation}
%For example, in our data sets, there are two hypotheses: modules are either
%defective or not; i.e. \mbox{$H\in \left\{ defective,nonDefective \right\}$}.
%Also, if a particular module has $numberofSymbols=27$ and
%$LOC=40$ and was previously classified
%as ``defective'' then
%\[
%\begin{array}{lcl}
%E_1&:& numberOfSymbols=27\\
%E_2& :& LOC=40\\
%H&:&  defective
%\end{array}
%\]
%When building defect predictors,
%the
%posterior probability of each class (``defective'' or ``defect-free'')
%is calculated, given the features
%extracted from a 
%module.
%So, if a data set has 100 modules and 25 of them are faulty, then:
%\[P(defective)=0.25\]
%When testing new data, a module is assigned to the class
%with the higher probability, calculated from \eq{bayes}.

%For
%numeric features, 
%a feature's mean $\mu$ and standard deviation $\sigma$ is used in
%a
%Gaussian  probability
%function~\cite{witten05}:
%\[f(x) =
%1/(\sqrt{2\pi}\sigma)e^{-\frac{(x-\mu)^2}{2\sigma^2}}\] 

%Simple  Bayes classifiers are often called ``naive'' since they assume
%independence of each feature. While this assumption
% simplifies the implementation  
%(frequency
%counts are required only for each feature),
% it is possible
%that
%correlated events are missed by this ``naive'' approach.
%Potentially, this is a significant problem for our kinds of data sets where
%the static code measures are highly correlated (e.g. the number of
%symbols in a module increases linearly with module lines of code). 

%In a previous work, we have investigated variants of Bayesian models considering this issue and observed that the independence assumption of naive Bayes is safe for the Nasa datasets~\cite{turhan07qsic}. In another work, we have also performed extensive feature subset selection (FSS) experiments, where similar prediction performances were obtained using small subsets of features selected by Information Gain~\cite{me07b}. However, we did not observe an increase in the prediction performances when we used whole available data. 
%Furthermore, 
%Domingos and Pazzani show theoretically
%that the independence assumption is a problem in a vanishingly small percent
%of cases~\cite{domingos97optimality}. This explains the repeated
%empirical result that, on average, seemingly naive Bayes classifiers perform as well
%as other seemingly more sophisticated schemes (e.g. see Table~1 in~\cite{domingos97optimality}).

%The Domingos and Pazzani result
%also explains our prior experiments where naive Bayes did not perform
%worse than other learners that continually re-sample the data for dependent
%instances (e.g. decision-tree learners that recurse
%on each ``split'' of the data~\cite{quinlan92}).\\\hline
%\end{tabular}
%\caption{About Bayes classifiers.}\label{fig:bayes}
%\end{figure}
