Bayes' Theorem and Its Application to AI Reasoning — Beginner-Friendly Notes

1. The Theorem (one line + one formula)

• $P(H)$ — prior: belief in hypothesis before evidence.
• $P(E|H)$ — likelihood: how likely evidence is if hypothesis true.
• $P(H|E)$ — posterior: belief after seeing evidence (what we want).
• $P(E)$ — evidence normaliser: $P(E)=\Sigma_h P(E|h)P(h)$.

Intuition: Posterior ∝ Prior × Likelihood — evidence updates belief.

2. Derivation in One Step

From $P(H \land E)=P(H|E)P(E)=P(E|H)P(H)$ → rearrange.

3. Mini Example — Spam Filter (AI core use)

Want $P(spam | words)$:

• Prior $P(spam)=0.2$ (20% mails are spam)
• Likelihood $P(words="offer win" | spam)=0.3$, $P(words | ham)=0.01$
• $P(spam|words) = \frac{0.3 \cdot 0.2}{0.3\cdot0.2 + 0.01\cdot0.8} ≈ \frac{0.06}{0.068}=0.88$ → classify as spam.

Same pattern for diagnosis, speech, vision — Bayes turns generative knowledge ($P(E|H)$ easy to model) into diagnostic knowledge ($P(H|E)$ needed).

4. Naive Bayes (most common AI application — preview Module 5)

Assume words independent given class: $P(E1,E2|H)=P(E1|H)\cdot P(E2|H)$ → then

— simple, surprisingly good for text.

5. AI Reasoning Uses

Task	$H$	$E$	What Bayes gives
Diagnosis	disease	symptoms	$P(disease|symptoms)$
Spam	spam	words	$P(spam|words)$
Robot localisation	position	sensor reading	$P(pos|sensor)$
Speech	words spoken	audio	$P(words|audio)$

6. Common Pitfall

$P(E|H)
eq P(H|E)$ — "90% of spam has 'offer'" does NOT mean "90% of 'offer' mails are spam" — base rate matters (see previous topic's 0.15 example).

7. One-Line Summary

Bayes' theorem $P(H|E)=P(E|H)P(H)/P(E)$ inverts likelihood to posterior, letting AI update prior beliefs with evidence and powering classifiers, diagnosis and state estimation.

8. Quick Self-Check

1. Compute posterior if $P(H)=0.01$, $P(E|H)=0.9$, $P(E|
   eg H)=0.05$.
2. Why is $P(E)$ computed by summing over hypotheses?