When features are continuous (like height, weight, age), Naive Bayes can't use frequency counts. Instead, we model the likelihood P(xi∣y)P(x_i \mid y) using a probability distribution, most commonly a Gaussian (Normal) distribution.

🔸 Gaussian Naive Bayes Assumption

For each feature xix_i, conditioned on class y=cy = c, we assume the data follows a normal distribution:

P(xi∣y=c)=12πσc2⋅exp⁡(−(xi−μc)22σc2)P(x_i \mid y = c) = \frac{1}{\sqrt{2\pi\sigma_c^2}} \cdot \exp\left( -\frac{(x_i - \mu_c)^2}{2\sigma_c^2} \right)

Where:

• μc\mu_c: mean of feature xix_i for class y=cy = c
• σc2\sigma_c^2: variance of feature xix_i for class y=cy = c

🔹 Steps to Estimate Probabilities

1. Split the training data by class y=cy = c
2. For each feature xix_i:
   • Compute the mean μic\mu_{ic} and variance σic2\sigma^2_{ic} from training data for each class
3. Use the Gaussian PDF to compute likelihoods P(xi∣y=c)P(x_i \mid y = c) at prediction time

✅ Example

Suppose you have a feature "height" and two classes: Male and Female.

Training data:

Step 1: Compute mean and variance per class

• Male:
  • Mean: μ=170+1802=175\mu = \frac{170 + 180}{2} = 175
  • Variance: σ2=(170−175)2+(180−175)22=25\sigma^2 = \frac{(170-175)^2 + (180-175)^2}{2} = 25