Bayesian classification is based on Baye's Theorem. It is a statistical classifier that predicts class membership probabilities such as the probability that a given tuple belongs to a particular class.

Baye’s Law

P(A/B) = P(B/A) P(A)/ P(B)

• Has high accuracy and speed for large databases.
• Has minimum error rate in comparison to all other classifier.

### Types.

1. Bayesian Belief Networks (Graphical Method)
• Bayesian Belief Network specifies joint conditional probability distributions.
• Bayesian Networks and Probabilistic Network are known as belief network.
• It allows class conditional independencies to be defined between subsets of variables.
• It provides a graphical model of causal relationship on which learning can be performed.
• It represents a set of random variables and their conditional dependencies via a directed acyclic graph

### 2. Naïve Bayesian Classifier

• The Naive Bayes Classifier technique is based on the so-called Bayesian theorem and is particularly suited when the dimensionality of the inputs is high.
• It simplifies the computational complexity.
• Naïve Bayesian Classifier assumes that the effect of an attribute value on a given class is independent of the value of other attributes e. class conditional independence.
• Let D be a training set of tuples and C1, C2,.......... , Cm are their associated classes.
• Given a tuple X, the classifier will predict that X belongs to the class having highest posterior probability conditioned on X i.e. the Naïve Bayesian classifier predicts that tuple x belong to the class Ci if and only if

P(Ci/X) > P(Cj/X) for 1≤ j ≥m, j G i

i.e. P(Ci/X) = P(X/Ci)P(Ci)/ P(X) maximum

P(X) = Constant

P(Ci) => P(C1) = P(C2) =.............. = P(Cm)

So we need to maximize P(X/Ci)

Naïve assumption is class condition independence, = P(x1/Ci) * P (x2/Ci) *.................. * P(xn/Ci)

These probabilities can be calculated from training tuples.