# Q4(a) Naive Bayes and Laplace smoothing [10]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-2024-reexam/q4a

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-2024-reexam--q4a/

Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.

**Question. Q4(a)** [10 marks · 2024-06-25]

(1) Use Naive Bayes on the dataset to predict whether a person buys a computer for Age=Old, Income=Medium, Freelancer=Yes. (2) Explain Laplacian smoothing, whether it is required here and its impact.

| ID | Age | Income | Freelancer | Buys computer |
| --- | --- | --- | --- | --- |
| 1 | Young | High | No | No |
| 2 | Young | High | No | No |
| 3 | Middle Age | High | No | Yes |
| 4 | Old | Medium | No | Yes |
| 5 | Old | Low | Yes | Yes |
| 6 | Old | Low | Yes | No |
| 7 | Middle Age | Low | Yes | Yes |
| 8 | Young | Medium | No | No |
| 9 | Young | Low | Yes | Yes |
| 10 | Young | Medium | Yes | Yes |

### Answer

#### Given

Ten records; query (Old, Medium, Yes). Class counts Yes 6, No 4.

#### To find

Class scores, normalized comparison and smoothing impact.

#### Formula and why it applies

Naive Bayes multiplies the class prior and feature likelihoods, assuming features are conditionally independent given the class. The same evidence denominator cancels when comparing classes.



$$
s(c)=P(c)\prod_j P(x_j\mid c),\qquad P(c\mid x)=\frac{s(c)}{\sum_{k}s(k)}
$$



#### Solution

##### Step 1. Count query-feature matches within each class

| Class | N | Old | Medium income | FreelancerYes |
| --- | --- | --- | --- | --- |
| Yes | 6 | 2 | 2 | 4 |
| No | 4 | 1 | 1 | 1 |

Yes records are 3,4,5,7,9,10; No records 1,2,6,8. Count within the chosen class, not among all ten records.

##### Step 2. Compute unsmoothed Yes score



$$
s(Y)=\frac6{10}\frac26\frac26\frac46=\frac2{45}\approx0.0444444
$$



##### Step 3. Compute unsmoothed No score



$$
s(N)=\frac4{10}\frac14\frac14\frac14=\frac1{160}=0.00625
$$



##### Step 4. Normalize and classify



$$
P(Y\mid x)=\frac{2/45}{2/45+1/160}=\frac{64}{73}\approx0.876712
$$





$$
P(N\mid x)=\frac9{73}\approx0.123288
$$



Predict Yes because its score is larger.

##### Step 5. Explain add-one smoothing

A zero likelihood would make the whole class product zero, even if other features favor the class. Add-one smoothing prevents that by adding one to each category count and adding the number of possible categories to the denominator. No query likelihood is zero here, so smoothing is not needed to rescue a zero score; it still changes the estimates.



$$
P(x_j=v\mid c)=\frac{N_{jvc}+1}{N_c+K_j}
$$



There are 3 age categories,3 income categories and 2 freelancer categories. Retaining empirical priors:



$$
s_{+1}(Y)=\frac6{10}\frac{2+1}{6+3}\frac{2+1}{6+3}\frac{4+1}{6+2}=\frac1{24}\approx0.0416667
$$





$$
s_{+1}(N)=\frac4{10}\frac{1+1}{4+3}\frac{1+1}{4+3}\frac{1+1}{4+2}=\frac8{735}\approx0.0108844
$$



##### Step 6. Compare the smoothed result



$$
P_{+1}(Y\mid x)=\frac{1/24}{1/24+8/735}=\frac{245}{309}\approx0.792880
$$



The predicted class remains Yes, while certainty decreases. Stating whether priors are smoothed matters; here only feature likelihoods were add-one smoothed.

**Final answer.** Unsmoothed predictionYes, posterior 64/73. Add-one feature smoothing still predictsYes, posterior 245/309.