Navie Bayes

Bayesian Classification

Let the dataset $T$ be:

$$T=\{(x_1,y_1),(x_2,y_2),\dots ,(x_N,y_N)\}$$

• Each input $x_i$ is represented as:$$x_i=(x^1,\dots ,x^n)$$
• Each output $y_i$ belongs to a class $c_k$, where $k=1,\dots ,K$.

The probability of a class $c_k$ given input $x$ is calculated as:

$$P(y=c_k\mid x)=\frac{P(y=c_k)P(x\mid y=c_k)}{\sum _{k}P(y=c_k)P(x\mid y=c_k)}$$$$P(y=c_k|x)\propto P(y=c_k)P(x|y=c_k)$$$$P(y=c_k|x)\propto P(y=c_k)\prod _{j}P(x^j|y=c_k)$$