Intelligent Systems II · TEE · papers

Q4(b) Naive Bayes: flu [10]

Final examination · 24 Nov 2025

Actual TEE paper · 100 marks · 3 hours · Q1 compulsory + any 4 of Q2–Q7 · every answer verified (key errors flagged)

Question. A hospital wants to predict flu from Fever (Yes/No), Cough (Yes/No) and Temperature Level (Low/Normal/High). Training data: (Yes, Yes, High, Yes), (No, No, Normal, No), (Yes, No, Low, No), (Yes, Yes, High, Yes), (No, Yes, Normal, No), (Yes, Yes, High, Yes), (No, No, Low, No), (Yes, No, Normal, No). Using naive Bayes, predict whether a patient with Fever = Yes, Cough = Yes, Temperature = Normal has flu. [10 marks]

Answer

Given

The full training data are in Step 1. Query: Fever=Yes, Cough=Yes, Temperature=Normal. The main calculation uses unsmoothed counts.

To find

Both class scores and the predicted class.

Formula and why it applies

LaTeX: score(C)=P(C)\prod_jP(a_j\mid C)

C is a class label. Each aⱼ is one query feature value. First count inside a class, then multiply the prior and all three likelihoods.

Solution

Step 1. Write the training rows as a table

Row	Fever	Cough	Temperature	Class
1	Yes	Yes	High	Yes
2	No	No	Normal	No
3	Yes	No	Low	No
4	Yes	Yes	High	Yes
5	No	Yes	Normal	No
6	Yes	Yes	High	Yes
7	No	No	Low	No
8	Yes	No	Normal	No

Step 2. Separate the two class groups

Class	Rows in this group	Class count	Prior
Yes	1, 4, 6	3	3/8 = 0.375
No	2, 3, 5, 7, 8	5	5/8 = 0.625

Step 3. For class Yes, count Fever=Yes

Look only at class Yes rows 1, 4, 6. The matching rows are 1, 4, 6.

LaTeX: P(\text{Fever}=\text{Yes}\mid C=\text{Yes})=\frac{3}{3}=1

Step 4. For class Yes, count Cough=Yes

Look only at class Yes rows 1, 4, 6. The matching rows are 1, 4, 6.

LaTeX: P(\text{Cough}=\text{Yes}\mid C=\text{Yes})=\frac{3}{3}=1

Step 5. For class Yes, count Temperature=Normal

Look only at class Yes rows 1, 4, 6. No row in this class has that feature value.

LaTeX: P(\text{Temperature}=\text{Normal}\mid C=\text{Yes})=\frac{0}{3}=0

Step 6. Multiply the Yes score one factor at a time

LaTeX: P(Yes)=3/8=0.375

LaTeX: 0.375\times1=0.375

LaTeX: 0.375\times1=0.375

LaTeX: 0.375\times0=0

Step 7. For class No, count Fever=Yes

Look only at class No rows 2, 3, 5, 7, 8. The matching rows are 3, 8.

LaTeX: P(\text{Fever}=\text{Yes}\mid C=\text{No})=\frac{2}{5}=0.4

Step 8. For class No, count Cough=Yes

Look only at class No rows 2, 3, 5, 7, 8. The matching rows are 5.

LaTeX: P(\text{Cough}=\text{Yes}\mid C=\text{No})=\frac{1}{5}=0.2

Step 9. For class No, count Temperature=Normal

Look only at class No rows 2, 3, 5, 7, 8. The matching rows are 2, 5, 8.

LaTeX: P(\text{Temperature}=\text{Normal}\mid C=\text{No})=\frac{3}{5}=0.6

Step 10. Multiply the No score one factor at a time

LaTeX: P(No)=5/8=0.625

LaTeX: 0.625\times0.4=0.25

LaTeX: 0.25\times0.2=0.05

LaTeX: 0.05\times0.6=0.03

Step 11. Compare the scores

LaTeX: score(Yes)=0\qquad score(No)=0.03

The larger score belongs to No. Normalise only if a posterior probability is requested: divide each score by their total 0.03.

Final answer. Predict No. Yes score=0; No score=0.03.

The zero Yes score is caused by an absent training combination. If Laplace smoothing is explicitly required, apply it to all feature likelihoods in both classes. That is a different calculation and may change the prediction; do not mix smoothed and unsmoothed sides.

Yaad rakho: Class ke rows alag karo. Har feature ka count usi class ke total se divide. Saare factors multiply karo.