Naive Bayes Classifier: A Complete Solved Numerical Example

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Scenario: Medical Symptom Diagnosis

The Objective: Predict whether a patient has a viral infection based on reported clinical symptoms.

Step 1: The Training Data

Before we can classify the new target point, we must analyze our historical data.

Data Point	Fever	Fatigue	Cough	Headache	Diagnosis
P1	High	Yes	Yes	Yes	Viral
P2	High	Yes	No	Yes	Viral
P3	Normal	No	Yes	No	Healthy
P4	Normal	No	No	No	Healthy
P5	High	No	Yes	Yes	Viral
P6	Normal	Yes	No	Yes	Healthy
P7	High	Yes	Yes	No	Viral
P8	Normal	No	Yes	Yes	Healthy
P9	Normal	Yes	Yes	No	Healthy
P10	High	Yes	No	No	Viral
Target	High	Yes	Yes	Yes	?

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Step 2: Prior Probabilities P (Class)

First, we calculate the baseline probability of each class occurring in our entire dataset before looking at any specific symptoms.

Formula:

P(c) = \dfrac{\text{Count of Class } c}{\text{Total Data Points}}

P (Viral)

= 5 / 10
= 0.5000

P (Healthy)

= 5 / 10
= 0.5000

Step 3: Conditional Probabilities P (Feature | Class)

Next, we look at our target point and calculate the "likelihood" of each specific feature appearing, assuming a specific class is true. We do this by isolating the rows for a specific class and counting how many times the feature matches.

Formula:

P(x_i | c) = \dfrac{\text{Count of } x_i \text{ within Class } c}{\text{Total Count of Class } c}

For Class: Viral

P (Fever = High | Viral)= 5/5

P (Fatigue = Yes | Viral)= 4/5

P (Cough = Yes | Viral)= 3/5

P (Headache = Yes | Viral)= 3/5

For Class: Healthy

P (Fever = High | Healthy)= 0/5

P (Fatigue = Yes | Healthy)= 2/5

P (Cough = Yes | Healthy)= 3/5

P (Headache = Yes | Healthy)= 2/5

Step 4: Final Probabilities & Conclusion

Finally, we apply Bayes' Theorem (using the naive independence assumption). We multiply the Prior Probability (Step 2) by all the Conditional Probabilities (Step 3) for each class. The class that yields the highest score is our prediction.

Formula:

P(A|B) = \dfrac{P(B|A) \cdot P(A)}{P(B)}

P (Viral | Target)

= 5/10 * 5/5 * 4/5 * 3/5 * 3/5

= 0.1440

P (Healthy | Target)

= 5/10 * 0/5 * 2/5 * 3/5 * 2/5

= 0.0000

Final Prediction

Because Viral has the highest final calculated probability, the Naive Bayes classifier predicts that the target point belongs to the Viral class.