Naive Bayes

#Computers
Models the probability of observing data $\displaystyle x$ and that the class of our target is $\displaystyle c$. Generative model that aims to maximize the joint likelihood $\displaystyle \sum_{n}\log p(x_{n},y_{n})$.

$\displaystyle P(X=x,Y=c)=\pi_{c}\prod_{k}\theta^{z_{k}}_{ck},X \in \mathbb{R}^{D},Y\in [C]$

• Joint probability of getting your data

$\displaystyle (\pi_{c}^{},\theta_{ck}^{})=\text{argmax} \sum_{n}\log \pi_{y_{n}}+\sum_{n,k}z_{nk}\log \theta_{y_{n}k}$

$\displaystyle \pi_{C}^{*}=\frac{\text{# of data points labeled C}}{N}$

$\displaystyle \theta_{ck}^{*}= \frac{\text{# of times word k shows up in data points labeled c}}{\text{# words in data points labeled c}}$