B Naive Bayesian classifier

The Naive Bayesian Classifier algorithm is a classification algorithm based on calculating the probability of belonging to each class. That is to say that the test data $($to be classified$)$ will be assigned to the most likely class. The probability of belonging to each class is calculated from the learning data as follows:

class$(x)=argma{x}_{{}_k}\{{}_{i}P(\frac{w_i}{{}_k})P({}_k)\}$

Here $P({}_k)$ is the a priori probability of belonging to the class $k.$ in other words its the probability of obtaining a data of class $k$ if we draw a data at random. $P(\frac{x_i}{{}_k})$ is the probability that a data $x$ has the value $x_i$ for the variable $i,$ if we know its class ${}_k.$ Here we will calculate this probability by calculating the distance between the data $x_i$ and each barycenter of the classes $($i$.$e$.$ the class average$),$ divided by the sum distances between this data and each barycenter.

List of useful functions:

mean, sum: calculate the mean and the sum of a list of values.

unique: returns the list of list values, without repeating values.

asarray: transforms a list into a vector.

vector.prod: makes the product of the values of a vector.

naive$\_$bayes.GaussianNB: the naive Bayesian Classifier from sklearn.

Create a $CBN(x,Y)$ function that takes $x$ data and labels as input $Y$ and which returns a label, for each data, predicted from class la more likely according to equation $(1).$ Here again, we take each data, one by one, as test data and we consider all data as training data. It is advisable to first calculate the barycentres and the a priori probabilities $P({}_k)$ for each class, then calculate the conditional probabilities $P(\frac{x_i}{{}_k})$ for each class and each variable.