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Artificial Intelligence (AI) Machine Learning

NaiveBayes

Lesson 12/19

Naive Bayes

A probability-first classifier. It combines prior belief with evidence likelihoods and predicts the class with highest posterior.

Interactive simulation

Email spam posterior calculator

Adjust priors and evidence in real time.

Prior P(spam)0.30

P(word='offer' | spam)0.78

P(word='meeting' | spam)0.14

Laplace smoothing α1.00

Decision threshold0.50

Posterior P(spam|x)

0.270

Decision

Not spam

Calibration hint

Likely ham

False-alert risk

73%

Mathematical intuition

Bayes theorem in plain language

P(y\mid x)=\frac{P(x\mid y)P(y)}{P(x)}

Read it as: $posterior = evidence\times prior / normalization$ .

P(x\mid y)=\prod_i P(x_i\mid y)

The “naive” assumption treats features as conditionally independent.

Step-by-step workflow

Input → Processing → Learning → Prediction → Output

Input

Tokenized features

Processing

Lookup likelihood tables

Learning

Estimate priors + likelihoods

Prediction

Compute posterior per class

Output

Class + confidence

Code implementation

Beginner Python example

python

import math

# toy Naive Bayes for binary class
def predict(prior_spam, likelihoods_spam, likelihoods_ham):
    num = prior_spam
    den_other = 1 - prior_spam
    for ps, ph in zip(likelihoods_spam, likelihoods_ham):
      num *= ps
      den_other *= ph
    return num / (num + den_other)

p = predict(0.3, [0.78, 0.14], [0.18, 0.72])
print('P(spam|x)=', round(p, 3))

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