Intelligent Systems II · TEE · papers
Q4(b) Naive Bayes product-purchase classification [10]
Final / re-examination · 28 Apr 2025
Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.
| Age | Income Level | Previous Purchase | Buy Product |
|---|---|---|---|
| Young | Low | No | No |
| Middle-aged | Medium | Yes | Yes |
| Senior | High | No | No |
| Young | Medium | Yes | Yes |
| Senior | Low | No | No |
| Middle-aged | High | Yes | Yes |
| Young | Low | Yes | Yes |
| Senior | Medium | No | No |
A company wants to predict whether a customer will buy a product, based on the following features: Age (Young, Middle-aged, Senior), Income Level (Low, Medium, High), Previous Purchase (Yes, No). The output is Buy Product (Yes, No). Given a new customer with the following attributes: Age=Young, Income Level=High, Previous Purchase=No. Use Naive Bayes to predict whether this customer will buy the product.
Answer
Given
Eight training rows; query=(Young, High, No). No smoothing instruction is printed.
To find
The two class scores and the larger-posterior class.
Formula and why
LaTeX: s_c=P(c)P(Young\mid c)P(High\mid c)P(PreviousNo\mid c)Naive Bayes assumes conditional independence of these features given the class. The evidence denominator is common to both classes, so compare the unnormalized scores first.
Step 1. Count the classes
Yes occurs in rows 2,4,6,7:4 rows. No occurs in rows 1,3,5,8:4 rows.
LaTeX: P(Yes)=4/8=1/2,\quad P(No)=4/8=1/2Step 2. Count each query feature within each class
| Query feature | Count among Yes /4 | Count among No /4 |
|---|---|---|
| Young | 2/4 | 1/4 |
| High income | 1/4 | 1/4 |
| Previous purchase No | 0/4 | 4/4 |
The denominator is the class count 4, not the full dataset 8.
Step 3. Multiply the Yes score
LaTeX: s_{Yes}=\frac12\times\frac24\times\frac14\times\frac04=0The zero comes from no training Yes row having Previous Purchase=No.
Step 4. Multiply the No score
LaTeX: s_{No}=\frac12\times\frac14\times\frac14\times\frac44=\frac1{32}=0.03125Step 5. Normalize and predict
LaTeX: P(Yes\mid x)=\frac0{0+0.03125}=0,\quad P(No\mid x)=\frac{0.03125}{0.03125}=1Predicted class: No, under the unsmoothed model. A posterior of 1 here is a consequence of tiny-data maximum-likelihood estimates and a zero conditional count; it is not certain knowledge about a real customer.
Step 6. If smoothing is explicitly required, state the alternative
With add-one smoothing, Age and Income each have 3 categories and Previous Purchase has 2. Keep the class priors 1/2. Then:
LaTeX: s_{Yes}^{(+1)}=\frac12\frac37\frac27\frac16=\frac1{98},\quad s_{No}^{(+1)}=\frac12\frac27\frac27\frac56=\frac5{147}
LaTeX: P_{+1}(Yes\mid x)=3/13,\quad P_{+1}(No\mid x)=10/13The decision is still No. This is a labeled alternative, not a silent change to the printed task.