Bayes' theorem — derivation and applications

2.7.9 · Maths › Statistics & Probability — Intermediate

WHY — humein iska zaroorat kyun hai?

Aksar jo probability hum measure kar sakte hain woh "ulti taraf" hoti hai.

Ek medical test humein $P (positive ∣ disease)$ deta hai — jab koi beemar hota hai to test kaisa behave karta hai.
Lekin patient jaanna chahta hai $P (disease ∣ positive)$ — mera positive result aaya hai, to kya main sach mein beemar hoon?

Ye dono equal nahi hain. Bayes' theorem inke beech ka bridge hai.

WHAT — ingredients kya hain?

Conditional probability ki definition hai: $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) \neq = 0.$

Agar causes $A_{1}, \dots, A_{n}$ mutually exclusive and exhaustive hain (sample space ka ek partition), to koi bhi evidence $B$ sirf unhi mein se kisi ek ke through ho sakti hai: $P (B) = \sum_{i = 1}^{n} P (B ∣ A_{i}) P (A_{i}) .$

Figure — Bayes' theorem — derivation and applications

Worked Example 1 — Classic false-positive shock

Ek disease $1%$ population ko affect karti hai. Ek test $99%$ sensitive hai ( $P (+ ∣ D) = 0.99$ ) aur $95%$ specific ( $P (- ∣ \overset{ˉ}{D}) = 0.95$ , isliye $P (+ ∣ \overset{ˉ}{D}) = 0.05$ ). Tumhara test positive aaya. $P (D ∣ +)$ kya hai?

Step 1 — priors aur likelihoods list karo. $P (D) = 0.01, P (\overset{ˉ}{D}) = 0.99, P (+ ∣ D) = 0.99, P (+ ∣ \overset{ˉ}{D}) = 0.05.$ Kyun? Hum alag karte hain jo hum pehle maante hain (prior) aur test kaisa behave karta hai (likelihood).

Step 2 — positive aane ki total probability. $P (+) = (0.99) (0.01) + (0.05) (0.99) = 0.0099 + 0.0495 = 0.0594.$ Kyun? Positive ya to sach mein beemar insaan se aa sakta hai YA healthy insaan ke false positive se.

Step 3 — Bayes apply karo. $P (D ∣ +) = \frac{0.0099}{0.0594} \approx 0.1667 = 16.7%.$ Surprise kyun? Kyunki disease rare hai, isliye bada healthy group bahut saare false positives produce karta hai jo thode se true positives ko daba dete hain.

Worked Example 2 — Do factories (partition)

Factory X $60%$ bulbs banati hai $2%$ defect rate ke saath; Factory Y $40%$ banati hai $5%$ defect rate ke saath. Ek bulb defective hai. Yeh kaunsi factory se aayi hogi?

Step 1 — priors: $P (X) = 0.6, P (Y) = 0.4$ . Likelihoods: $P (D ∣ X) = 0.02, P (D ∣ Y) = 0.05$ .

Step 2 — evidence: $P (D) = (0.02) (0.6) + (0.05) (0.4) = 0.012 + 0.020 = 0.032.$ Kyun? Dono factories ke upar sum karo (woh saari bulbs ka partition karti hain).

Step 3 — posteriors: $P (X ∣ D) = \frac{0.012}{0.032} = 0.375, P (Y ∣ D) = \frac{0.020}{0.032} = 0.625.$ Kyun? Bhale hi X zyada bulbs banati hai, Y ki zyada defect rate use ek defective bulb ka zyada likely source bana deti hai.

Forecast-then-Verify

Common Mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek box hai jo "DANGER!" flash karta hai jab bhi koi monster paas hota hai — aur yeh sach mein acha hai, $99$ out of $100$ baar flash karta hai jab monster aata hai. Lekin monsters bahut rare hain. To zyatar baar jab yeh flash karta hai, woh actually ek gilehri hoti hai jisne ise bewaqoof banaya. Panic karne se pehle, tumhe sochna chahiye: "Pehli jagah kitne monsters hain?" Bayes' theorem alarm kitna acha hai aur monsters sach mein kitne rare hain ko mix karke tumhe sahi chance batata hai. Naya clue aaya → apna guess update karo.

Active Recall Flashcards

Conditional probability

P (A ∣ B)

ki definition batao.

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

, for

P (B) \neq = 0

Bayes' theorem apni simplest form mein likho.

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

Bayes' theorem mein "prior" kya hota hai?

P (A)

— evidence dekhne se pehle cause mein belief.

"Likelihood" kya hota hai?

P (B ∣ A)

— evidence ki probability given ki cause sach hai.

Partition

A_{1}, \dots, A_{n}

ke liye

P (B)

ko expand kaise karte hain?

P (B) = \sum_{i} P (B ∣ A_{i}) P (A_{i})

(Law of Total Probability).

99%

-accurate test phir bhi low

P (disease ∣ +)

kyun de sakta hai?

Kyunki low prior (rare disease) ka matlab hai ki bade healthy group ke false positives dominate karte hain.

P (A ∣ B)

aur

P (B ∣ A)

ko swap karne ki fallacy ka naam kya hai?

The prosecutor's fallacy / transposed conditional.

Example 2 mein, defective bulb choti factory Y se zyada likely kyun hai?

Y ki zyada defect rate (0.05) X ke bade share ko outweigh karti hai, jab hum "defective" condition lagaate hain.

Connections

Conditional Probability — woh definition jis par Bayes build hota hai.
Law of Total Probability — denominator $P (B)$ provide karta hai.
Independent Events — special case jahan $P (A ∣ B) = P (A)$ .
Naive Bayes Classifier — machine-learning application.
Prior and Posterior Distributions — continuous generalisation.
Tree Diagrams — joint probabilities ke liye visual bookkeeping.