Naive Bayes Classification Example

Given the following data:

$$\begin{array}{c|c|c|c|c|} \text{Instance} & \text{A} & \text{B} &\text{C} &\text{Class} \\ \hline \text{1} & 1 & 2 & 1 & 1 \\ \hline \text{2}& 0 & 0 & 1 & 1 \\ \hline \text{3} & 2 & 1 & 2 & 2 \\ \hline \text{4} & 1 & 2 & 1 & 2 \\ \hline \text{5} & 0 & 1 & 2 & 1 \\ \hline \text{6} & 2 & 2 & 2 & 2 \\ \hline \text{7} & 1 & 0 & 1 & 1\\ \hline \end{array}$$

Predit the class label for instance $(A=1, B=2, C=2)$ using naive Bayes classifcation.

Let $C_{1}$ be class $1$ and $C_{2}$ be class $2$. I have so far that the class prior probabilties are:

$$P(C_{1})=\dfrac{4}{7}$$ $$P(C_{2})=\dfrac{3}{7}$$

Using Bayes Theorem: $$P(C{i}|X)=\dfrac{P(X|C{i})P(C_{i})}{P(X)}$$

I know that $P(X|C_{i})=\prod^{n}_{k=1}P(X_{k}|C_{i})$ but not sure how to calculate this.

Where do I go from here to go about answering the question?

For $C_1$, by the assumption of Naive Bayesian Classifier, we have $$P(A = 1, B = 2, C=2 \mid C_1) = P(A = 1 \mid C_1) \cdot P(B = 2 \mid C_1) \cdot P(C = 2 \mid C_1)$$ Take $P(A = 1 \mid C_1)$ as an example. There are $4$ training records of $C_1$, among which there are $2$ records with $A = 1$. Therefore, $P(A = 1 \mid C_1) = \frac{2}{4}$. Similarly, you can calculate $P(B = 2 \mid C_1)$ and $P(C = 2 \mid C_1)$.
It is similar to calculate $P(A = 1, B = 2, C = 2 \mid C_2)$.
• @SophieFiler Exactly. Note that you do not need to consider the term $P(X)$ in your formula since it is contained in both probabilities. – PSPACEhard Apr 24 '17 at 10:30