How does a spam filter decide? Naive Bayes multiplies the evidence: start with the prior odds, then for each clue (each word) multiply in how much more likely it is under ‘spam’ than ‘not’. The ‘naive’ part — pretending the clues are independent — is usually false, yet it works remarkably well and is blazing fast. Slide the evidence and watch the posterior tip.
Naive Bayes applies Bayes’ theorem with a strong independence assumption: P(class | features) ∝ P(class)·∏ᵢ P(featureᵢ | class). It picks the class maximizing that product — the prior times each feature’s class-conditional likelihood. The ‘naive’ conditional-independence assumption is usually violated, yet the classifier is fast, needs little data, and is a strong baseline (spam filtering, text categorization). Working in log-space avoids underflow. A fail-loud self-check throws unless evidence favouring A yields class A and evidence favouring B yields B. ◆ real machine learning, node-verified.
Independence is an approximation (correlated features get double-counted), so calibrated PROBABILITIES can be poor even when the ARGMAX class is right; zero-frequency features need smoothing. The Bayes-rule arithmetic is exact.