Bayes : Examples

P(Spam) = 400 / 1000
        = 0.4

P(NotSpam) = 600 / 1000
           = 0.6

Step 2: Calculate Likelihoods

P(Offer | Spam) = 280 / 400
                = 0.7

P(Offer | NotSpam) = 120 / 600
                   = 0.2

Step 3: Calculate Scores

Spam Score

= P(Offer | Spam) × P(Spam)

= 0.7 × 0.4

= 0.28

NotSpam Score

= P(Offer | NotSpam) × P(NotSpam)

= 0.2 × 0.6

= 0.12

Step 4: Compare Scores

Spam Score    = 0.28

NotSpam Score = 0.12

0.28 > 0.12

Prediction

Spam

Example 2: Disease Diagnosis

Problem Statement

In a hospital:

• 300 patients have Flu

• 700 patients do not have Flu

Among Flu patients:

• 240 have Fever

Among Non-Flu patients:

• 70 have Fever

A patient arrives with Fever.

Predict whether the patient has Flu.

Step 1: Prior Probabilities

P(Flu) = 300 / 1000
       = 0.3

P(NoFlu) = 700 / 1000
         = 0.7

Step 2: Likelihoods

P(Fever | Flu) = 240 / 300
               = 0.8

P(Fever | NoFlu) = 70 / 700
                 = 0.1

Step 3: Calculate Scores

Flu Score

= P(Fever | Flu) × P(Flu)

= 0.8 × 0.3

= 0.24

NoFlu Score

= P(Fever | NoFlu) × P(NoFlu)

= 0.1 × 0.7

= 0.07

Step 4: Compare Scores

Flu Score   = 0.24

NoFlu Score = 0.07

0.24 > 0.07

Prediction

Patient has Flu

Example 3: Student Pass Prediction

Problem Statement

A university collected data from 1000 students.

• 700 students passed

• 300 students failed

Among students who passed:

• 630 studied more than 5 hours daily

Among students who failed:

• 60 studied more than 5 hours daily

A new student studies more than 5 hours daily.

Predict whether the student will Pass or Fail.

Step 1: Prior Probabilities

P(Pass) = 700 / 1000
        = 0.7

P(Fail) = 300 / 1000
        = 0.3

Step 2: Likelihoods

P(Study | Pass) = 630 / 700
                = 0.9

P(Study | Fail) = 60 / 300
                = 0.2

Step 3: Calculate Scores

Pass Score

= P(Study | Pass) × P(Pass)

= 0.9 × 0.7

= 0.63

Fail Score

= P(Study | Fail) × P(Fail)

= 0.2 × 0.3

= 0.06

Step 4: Compare Scores

Pass Score = 0.63

Fail Score = 0.06

0.63 > 0.06

Prediction

Pass

Example 4: Weather Prediction

Problem Statement

Historical records show:

• 400 rainy days

• 600 non-rainy days

Among rainy days:

• 300 were cloudy

Among non-rainy days:

• 180 were cloudy

Today is cloudy.

Predict whether it will rain.

Step 1: Prior Probabilities

P(Rain) = 400 / 1000
        = 0.4

P(NoRain) = 600 / 1000
          = 0.6

Step 2: Likelihoods

P(Cloudy | Rain) = 300 / 400
                 = 0.75

P(Cloudy | NoRain) = 180 / 600
                   = 0.30

Step 3: Calculate Scores

Rain Score

= P(Cloudy | Rain) × P(Rain)

= 0.75 × 0.4

= 0.30

NoRain Score

= P(Cloudy | NoRain) × P(NoRain)

= 0.30 × 0.6

= 0.18

Step 4: Compare Scores

Rain Score   = 0.30

NoRain Score = 0.18

0.30 > 0.18

Prediction

Rain

Example 5: Loan Approval

Problem Statement

A bank has data for 1000 customers.

• 800 loans approved

• 200 loans rejected

Among approved customers:

• 560 have high salaries

Among rejected customers:

• 40 have high salaries

A new customer has a high salary.

Predict whether the loan should be approved.

Step 1: Prior Probabilities

P(Approve) = 800 / 1000
           = 0.8

P(Reject) = 200 / 1000
          = 0.2

Step 2: Likelihoods

P(HighSalary | Approve) = 560 / 800
                        = 0.7

P(HighSalary | Reject) = 40 / 200
                       = 0.2

Step 3: Calculate Scores

Approve Score

= P(HighSalary | Approve) × P(Approve)

= 0.7 × 0.8

= 0.56

Reject Score

= P(HighSalary | Reject) × P(Reject)

= 0.2 × 0.2

= 0.04

Step 4: Compare Scores

Approve Score = 0.56

Reject Score = 0.04

0.56 > 0.04

Prediction

Loan Approved

Example 6: Naive Bayes with Multiple Features

Problem Statement

An email contains two words:

• Offer

• Free

Given:

P(Spam) = 0.4

P(NotSpam) = 0.6

P(Offer | Spam) = 0.7

P(Free | Spam) = 0.8

P(Offer | NotSpam) = 0.2

P(Free | NotSpam) = 0.1

Predict whether the email is Spam.

Step 1: Calculate Spam Score

Spam Score

= P(Spam)
  × P(Offer | Spam)
  × P(Free | Spam)

= 0.4 × 0.7 × 0.8

= 0.224

Step 2: Calculate NotSpam Score

NotSpam Score

= P(NotSpam)
  × P(Offer | NotSpam)
  × P(Free | NotSpam)

= 0.6 × 0.2 × 0.1

= 0.012

Step 3: Compare Scores

Spam Score    = 0.224

NotSpam Score = 0.012

0.224 > 0.012

Prediction

Spam

Example 7: Disease Diagnosis — Prediction is No Flu

• 300 patients have Flu
• 700 patients do not have Flu

• 60 have Body Pain

• 350 have Body Pain

P(Flu) = 300 / 1000
       = 0.3

P(NoFlu) = 700 / 1000
         = 0.7

P(BodyPain | Flu) = 60 / 300
                  = 0.2

P(BodyPain | NoFlu) = 350 / 700
                    = 0.5

Flu Score

= P(BodyPain | Flu) × P(Flu)

= 0.2 × 0.3

= 0.06

NoFlu Score

= P(BodyPain | NoFlu) × P(NoFlu)

= 0.5 × 0.7

= 0.35

Flu Score   = 0.06

NoFlu Score = 0.35

0.35 > 0.06

Prediction