Counts to Posteriors

Likelihoods from Counts

A likelihood is a feature count inside a class divided by that class count. The naive assumption is stated here: features are treated as independent after the class is fixed.

highlighted = computed this step

Likelihoods from feature counts

For F1 in class A, the count is 3 out of 4, so the likelihood is 3/4.

P(F1=1∣A)=3/4P(F1=1\mid A)=3/4
Likelihoods from countsFeature likelihoods keep count-over-class-count provenance visible.Naive Bayes from countstotal=6; counts=A:4, B:2feature counts: F1=1: A=3, B=1; F2=1: A=2, B=1query: F1=1, F2=1; smoothing=noneclasscount(c)likelihood count/countA4P(F1=1|A)=3/4P(F2=1|A)=2/4 (=1/2)B2P(F1=1|B)=1/2P(F2=1|B)=1/2naive independence assumed; no smoothing; no logs or decimals

The modeling assumption

Naive Bayes assumes features are conditionally independent once the class is fixed. That is a modeling choice that lets the feature ratios multiply.

features independent given the class\text{features independent given the class}
Likelihoods from countsFeature likelihoods keep count-over-class-count provenance visible.Naive Bayes from countstotal=6; counts=A:4, B:2feature counts: F1=1: A=3, B=1; F2=1: A=2, B=1query: F1=1, F2=1; smoothing=noneclasscount(c)likelihood count/countA4P(F1=1|A)=3/4P(F2=1|A)=2/4 (=1/2)B2P(F1=1|B)=1/2P(F2=1|B)=1/2naive independence assumed; no smoothing; no logs or decimals

Summary

The displayed likelihood ratios are 3/4, 1/2, 1/2, and 1/2.

[3/4,1/2,1/2,1/2][3/4,1/2,1/2,1/2]
Likelihoods from countsFeature likelihoods keep count-over-class-count provenance visible.Naive Bayes from countstotal=6; counts=A:4, B:2feature counts: F1=1: A=3, B=1; F2=1: A=2, B=1query: F1=1, F2=1; smoothing=noneclasscount(c)likelihood count/countA4P(F1=1|A)=3/4P(F2=1|A)=2/4 (=1/2)B2P(F1=1|B)=1/2P(F2=1|B)=1/2naive independence assumed; no smoothing; no logs or decimals