Bayes' theorem — derivation and applications: Foundations

2.7.9 · D1 · Maths › Statistics & Probability — Intermediate › Bayes' theorem — derivation and applications

Parent note ki ek bhi line padhne se pehle, tumhe uske symbols mein fluent hona padega. Yeh page har ek cheez zero se build karta hai, us order mein jisme ek cheez doosri par depend karti hai. Kabhi bhi aisa symbol use nahi karenge jisko pehle draw na kiya ho.

0. Sample space $S$ — poori picture

Picture: ek rectangle. Uske andar har dot ek aisi cheez hai jo ho sakti hai (ek patient, ek bulb, ek coin toss). Rectangle ke bahar kuch nahi hota.

Yeh topic ko kyun chahiye: Bayes' theorem probabilities ke baare mein hai, aur probability hamesha "poore ka kitna fraction?" hoti hai. Bina ek fixed whole ke, "fraction" ka koi matlab nahi.

Figure — Bayes' theorem — derivation and applications

1. Ek event $A$ — whole ka ek slice

Picture: rectangle ke andar ek coloured blob. "Patient sick hai" ( $D$ ) ek blob hai; "test positive aaya" ( $+$ ) ek aur blob hai.

Yeh topic ko kyun chahiye: Bayes ka poora point ek event (ek cause, jaise disease) ko doosre event (the evidence, jaise ek positive test) se relate karna hai. Events is poori theory ke nouns hain.

2. Probability $P (A)$ — slice kitni badi hai?

$0 \leq P (A) \leq 1, P (S) = 1.$

Picture: agar blob $A$ paper ka ek chauthai hissa cover karta hai, toh $P (A) = 0.25$ . Agar sab kuch cover karta hai, toh $P (A) = P (S) = 1$ . Agar kuch nahi cover karta, toh $P (A) = 0$ .

Yeh topic ko kyun chahiye: priors, likelihoods aur posteriors sab probabilities hain — sab "blobs ke areas". Yeh ek reading is poore page ko power deti hai.

3. Complement $\overset{ˉ}{A}$ — har woh cheez jo $A$ NAHI hai

Picture: rectangle ko blob ke bahar shade karo. Milakar, $A$ aur $\overset{ˉ}{A}$ poori sheet ko bina kisi gap aur bina kisi overlap ke tile karte hain.

$P (A) + P (\overset{ˉ}{A}) = 1.$

Yeh topic ko kyun chahiye: medical example mein, ek positive test ya toh ek sick person ( $D$ ) se aata hai ya ek healthy person se ( $\overset{ˉ}{D}$ ). Parent note mein $P (+ ∣ \overset{ˉ}{D})$ bar bar use hota hai — woh " $\overset{ˉ}{D}$ " yahi idea hai.

4. Intersection $A \cap B$ — overlap

Picture: do overlapping blobs; beech ki lens-shaped jagah $A \cap B$ hai.

Yeh topic ko kyun chahiye: Bayes ki poori derivation is baat par hinges karti hai ki $P (A \cap B)$ ek akela number hai jis tak do alag tareekon se pahuncha ja sakta hai. Woh shared overlap parent note ke Step 1 ka pivot hai.

5. Conditional probability $P (A ∣ B)$ — ek slice mein zoom karo

Yeh show ka star hai. Vertical bar $∣$ ko "given" padha jaata hai.

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) \neq = 0.$

Yeh formula KYA karta hai: numerator $P (A \cap B)$ overlap hai; denominator $P (B)$ naya whole hai (blob $B$ tumhara poora universe ban jaata hai).

$P (B)$ se divide KYUN karte hain: kyunki tumne zoom in kiya hai. $B$ ab total area hai, isliye is naye world ke andar probabilities ko $1$ par sum karte rakhne ke liye, tumhe $B$ ki area se divide karke re-scale karna padega.

YEH KAISA DIKHTA HAI: rectangle ko sirf blob $B$ tak shrink karo. Overlap $A \cap B$ ek choti si lens thi; ab chote blob $B$ ke against measure karne par woh ek bada fraction ban jaati hai. Woh re-scaling hi division hai.

Yeh topic ko kyun chahiye: Bayes' theorem conditional probabilities se bani hai — likelihood $P (B ∣ A)$ aur posterior $P (A ∣ B)$ dono is form mein hain. Parent page par baaki sab kuch is ek definition ke aas paas bookkeeping hai.

6. Partition — poore ko causes mein kaatna

Picture: rectangle ko bina kisi gap aur overlap ke coloured stripes mein kaato — jaise ek chocolate bar. ${A, \overset{ˉ}{A}}$ sabse simple partition hai (do pieces).

Yeh topic ko kyun chahiye: full Bayes ka denominator, $P (B) = \sum_{i} P (B ∣ A_{i}) P (A_{i})$ , tab hi kaam karta hai jab causes poori duniya ko completely tile karein. Yeh Law of Total Probability hai, aur isko khade rehne ke liye ek partition chahiye.

7. Summation sign $\sum$ — "saare pieces mein add karo"

Picture: ek haath chocolate bar ki har stripe par ek ek karke point karta hai, har ek ki value ek running total mein daalta jaata hai.

$\sum_{i = 1}^{3} x_{i} = x_{1} + x_{2} + x_{3} .$

Yeh topic ko kyun chahiye: do se zyada causes ke saath, $P (B ∣ A_{1}) P (A_{1}) + P (B ∣ A_{2}) P (A_{2}) + \dots$ likhna thaka dene wala hai. $\sum$ ise compress karta hai. Parent ka full-Bayes denominator bas yahi sign kaam karta hua hai.

8. $P (B)$ banana — Law of Total Probability

Ab dekho sab pieces kaise click karte hain. Evidence $B$ (ek blob) partition ke zariye pieces mein kaati jaati hai, har cause ke liye ek:

$P (B) = slice in A_{1} P (B \cap A_{1}) + slice in A_{2} P (B \cap A_{2}) + \dots$

KYUN: kyunki $A_{i}$ poori sheet ko bina overlap ke tile karte hain, blob $B$ non-overlapping slivers mein dice ho jaati hai — slivers ko add karo toh $B$ ka sara hissa wapas mil jaata hai.

Phir har joint ko Step 5 ke formula se rewrite karo ( $P (B \cap A_{i}) = P (B ∣ A_{i}) P (A_{i})$ ):

$P (B) = \sum_{i = 1}^{n} P (B ∣ A_{i}) P (A_{i}) .$

Yahi woh denominator hai jo parent note "expand" karta hai. Ab tum uske har symbol ko samajhte ho.

9. Prior, likelihood, posterior — teen kaam, ek bar

Parent yeh names deta hai. Ab jab tumhare paas $P$ aur $∣$ hai, yeh bas us ek bar ki teen readings hain:

Picture: prior poori sheet dekhta hai; likelihood cause blob mein zoom karta hai; posterior evidence blob mein zoom karta hai. Bayes' theorem woh rule hai jo swap karta hai ki tum kis blob mein zoom karte ho. Dekho Prior and Posterior Distributions ki yeh idea kaise generalise hoti hai.

Prerequisite map

Equipment checklist

Khud ko test karo. Dahini taraf cover karo; tum parent note ke liye tabhi ready ho jab har reveal tumhare apne words se match kare.

P (A)

ka picture ke roop mein kya matlab hai?

Ek rectangle ke andar blob

A

ka area, jiska total area

1

hai.

\overset{ˉ}{A}

kya hai, aur

P (A) + P (\overset{ˉ}{A})

kya hai?

"Not

A

" — blob

A

ke bahar ki har cheez; dono areas

1

mein sum hote hain.

A \cap B

kya region hai?

Woh overlap jahan blobs

A

aur

B

milte hain — ek saath dono mein outcomes.

P (A ∣ B)

ki definition likho aur batao ki

P (B)

se divide kyun karte ho.

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

; divide isliye karte ho kyunki zoom in karne ke baad

B

naya whole ban jaata hai.

P (A ∣ B)

generally

P (B ∣ A)

ke barabar kyun nahi hota?

Upar same overlap, lekin alag denominators (

P (B)

P (A)

Events ko partition banane wali do properties kya hain?

Mutually exclusive (koi overlap nahi) aur exhaustive (poore

S

ko cover karte hain).

\sum_{i = 1}^{n} x_{i}

ka matlab kya hai?

Sum

x_{1} + x_{2} + \dots + x_{n}

Partition

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

ke liye

P (B)

expand karo.

P (B) = \sum_{i = 1}^{n} P (B ∣ A_{i}) P (A_{i})

Prior / likelihood / posterior mein se kaun evidence blob mein zoom karta hai?

Posterior

P (A ∣ B)