Editing Naive Bayes classifier (section)

===Gaussian naive Bayes===
When dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a [[Normal distribution|normal]] (or Gaussian) distribution. For example, suppose the training data contains a continuous attribute, '''<math>x</math>'''. The data is first segmented by the class, and then the mean and [[Variance#Estimating the variance|variance]] of <math>x</math> is computed in each class. Let <math>\mu_k</math> be the mean of the values in <math>x</math> associated with class <math>C_k</math>, and let <math>\sigma^2_k</math> be the [[Bessel's correction|Bessel corrected variance]] of the values in <math>x</math> associated with class <math>C_k</math>. Suppose one has collected some observation value <math>v</math>. Then, the probability ''density'' of <math>v</math> given a class <math>C_k</math>, i.e., <math>p(x=v \mid C_k)</math>, can be computed by plugging <math>v</math> into the equation for a [[normal distribution]] parameterized by <math>\mu_k</math> and <math>\sigma^2_k</math>. Formally,
<math display="block">
p(x=v \mid C_k) = \frac{1}{\sqrt{2\pi\sigma^2_k}}\,e^{ -\frac{(v-\mu_k)^2}{2\sigma^2_k} }
</math>

Another common technique for handling continuous values is to use binning to [[Discretization of continuous features|discretize]] the feature values and obtain a new set of Bernoulli-distributed features. Some literature suggests that this is required in order to use naive Bayes, but it is not true, as the discretization may [[Discretization error|throw away discriminative information]].<ref name="idiots"/>

Sometimes the distribution of class-conditional marginal densities is far from normal. In these cases, [[kernel density estimation]] can be used for a more realistic estimate of the marginal densities of each class. This method, which was introduced by John and Langley,<ref name="john95"/> can boost the accuracy of the classifier considerably.<ref name="piryonesi2020">{{Cite journal |last1=Piryonesi |first1=S. Madeh |last2=El-Diraby |first2=Tamer E. |date=2020-06-01 |title=Role of Data Analytics in Infrastructure Asset Management: Overcoming Data Size and Quality Problems |journal=Journal of Transportation Engineering, Part B: Pavements |volume=146 |issue=2 |pages=04020022 |doi=10.1061/JPEODX.0000175 |s2cid=216485629}}</ref><ref name="hastie01">{{Cite book |last=Hastie, Trevor. |title=The elements of statistical learning : data mining, inference, and prediction : with 200 full-color illustrations |date=2001 |publisher=Springer |others=Tibshirani, Robert., Friedman, J. H. (Jerome H.) |isbn=0-387-95284-5 |location=New York |oclc=46809224}}</ref>