Bayes' theorem — derivation and applications: Question bank

2.7.9 · D5 · Maths › Statistics & Probability — Intermediate › Bayes' theorem — derivation and applications

Traps se pehle, hum har symbol ko earn karein jo neeche use hua hai, taaki kuch bhi unexplained na lage.

Ab core object jo fire mein hai — the flip, partition upar already define hai: $P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}, P (B) = \sum_{i = 1}^{n} P (B ∣ A_{i}) P (A_{i}) .$ Sum partition $A_{1}, \dots, A_{n}$ par run karta hai: ye evidence $B$ ke har alag cause ke through aane ki probability add karta hai. Neeche har item us line ke ek symbol ko poke karta hai — prior $P (A)$ , likelihood $P (B ∣ A)$ , evidence $P (B)$ , ya bar $∣$ ki direction.

True or false — justify

Jo test

99%

sensitive hai (yaani

P (+ ∣ D) = 0.99

), usmein positive result ka matlab

99%

chance of disease hai.

False. Sensitivity hai

P (+ ∣ D)

, test ka behaviour given truth; patient chahta hai flipped

P (D ∣ +)

, jo prior

P (D)

par bhi depend karta hai — rare disease use

99%

se kaafi neeche kheench deti hai. Area picture mein, left strip patli hai, isliye zyada shaded area right side par hota hai.

Kisi test ko "

99%

accurate" kehna aur uski sensitivity

99%

kehna ek hi baat hai.

False. "Accuracy" overall correct-rate hai, sensitivity aur specificity dono ka blend — prevalence se weighted; ek test

99%

sensitive ho sakta hai lekin poorly specific, toh uski accuracy different hogi. Ye words interchangeable nahi hain — hamesha pucho kaun sa quality number mean kiya gaya hai.

P (A ∣ B) = P (B ∣ A)

tab hota hai jab

A

aur

B

equally common hon.

True (ek special case ke roop mein). Ye factor

P (A) / P (B)

se differ karte hain, toh jab

P (A) = P (B)

ho toh dono conditionals coincide karte hain — lekin ye base rates ka coincidence hai, koi general law nahi.

Agar

P (A ∣ B) > P (A)

, toh

B

dekhne se

A

zyada believable ho gaya.

True. Posterior prior se upar chali gayi, toh evidence

B

A

ko support kiya; equivalently

P (B ∣ A) > P (B)

, matlab

B

"zyada expected" hai jab

A

hold karta hai.

Independent events ke liye Bayes' theorem phir bhi hold karta hai lekin kuch naya nahi batata.

True. Agar

A, B

independent hain toh

P (A ∣ B) = P (A)

: posterior prior ke barabar hai, toh evidence

B

A

mein belief unchanged rehti hai — dono vertical strips same height par shade hoti hain.

Denominator

P (B)

ko chhod sakte hain kyunki ye cancel ho jaata hai.

False.

P (B)

wahi hai jo posteriors ko sab causes mein

1

sum karne ke liye normalise karta hai; isse drop karo toh probabilities nahi, relative weights milenge. Ye sirf tab "cancel" hota hai jab tum do hypotheses ko ratio ke roop mein compare karo.

Positive test se

16.7%

ka posterior matlab test useless hai.

False. Usne phir bhi belief ko

1%

prior se

16.7%

tak lift kiya — roughly

17

-guna jump. "Useless" tab hota jab posterior

1%

par hi rehta.

Agar prior

0

hai, toh koi bhi evidence nonzero posterior produce nahi kar sakta.

True.

P (A) = 0

numerator

P (B ∣ A) P (A) = 0

bana deta hai; Bayes beliefs update kar sakta hai lekin us hypothesis ko nahi jila sakta jise tumne literally impossible assign kar diya.

Bayes mein "cause" aur "evidence" ke roles swap karna ek equally valid theorem deta hai.

True. Bayes construction mein symmetric hai — tum

P (A ∣ B)

P (B ∣ A)

utni hi aasani se compute kar sakte ho; "cause" aur "evidence" humare interpretive labels hain, fixed algebraic roles nahi.

Spot the error

P (D) = 0.01

aur

P (+ ∣ D) = 0.99

, toh

P (D ∣ +) = 0.99 \times 0.01 = 0.0099

Error: wo product joint

P (D \cap +)

hai — left strip ke andar tall sliver ka area — posterior nahi. Tumhe phir bhi total shaded area

P (+) = 0.0594

se divide karna hoga, jisse

\approx 0.167

milega.

"Disease rare hai, toh

P (+)

compute karte waqt healthy group ko ignore kar denge."

Error: healthy group (right strip) exactly wahi hai jahan false positives rehte hain. Wahan ka short band,

P (+ ∣ \overset{ˉ}{D}) P (\overset{ˉ}{D}) = 0.05 \times 0.99 = 0.0495

, true-positive sliver

0.0099

se paanch guna zyada hai — isse ignore karna zyaatar shaded area discard karna hai.

"Factory Y kam bulbs banati hai, toh defective bulb X se aane ki zyada sambhavna hai."

Error: "defective" par conditioning defect rate se reweight karti hai. Y ka

0.05

vs X ka

0.02

balance flip kar deta hai

P (Y ∣ D) = 0.625 > P (X ∣ D) = 0.375

ki taraf.

"Mere causes hain

A_{1}

='sick',

A_{2}

='healthy',

A_{3}

='tired';

P (B)

ke liye unki likelihoods sum karo."

Error: causes ko partition form karni chahiye (mutually exclusive aur exhaustive). "Tired" dono "sick" aur "healthy" se overlap karta hai, isliye slices double-count ho jaate hain —

P (B)

sahi se normalise nahi hoga.

"Court ne paaya

P (match ∣ innocent) = 1

in a million, toh

P (innocent ∣ match) = 1

in a million."

Error: prosecutor's fallacy — bar

∣

silently flip ho gaya. Actual posterior possible matches ki prior number (population ka base rate) par depend karta hai, jo guilt ko far from certain bana deta hai.

P (A ∣ B) + P (A ∣ \overset{ˉ}{B}) = 1

, kyunki

B

aur uska complement

\overset{ˉ}{B}

sab kuch cover karte hain."

Error: jo

1

sum karta hai woh hai

P (A ∣ B) + P (\overset{ˉ}{A} ∣ B)

(ek fixed right event ke liye left argument par). Alag events

B

aur

\overset{ˉ}{B}

par conditioning do alag shrunken worlds se probabilities deta hai jo add up karne ki zaroorat nahi.

Why questions

Zyada disease prevalence se wahi test zyada trustworthy kyun ho jaata hai?

Bada prior

P (D)

left strip ko wider banata hai, true-positive sliver ko fixed false-positive band ke relative bada karta hai, isliye posterior fraction chadh jaata hai (

1%

par

16.7%

30%

par

89.5%

tak).

Pehle derivation step mein hum conditional probability "dono taraf se" kyun likhte hain?

Dono directions ek same joint

P (A \cap B)

share karte hain — dono events ka same overlap region; dono likhne se hum us common term ko eliminate kar ke flipped conditional solve kar sakte hain.

Full formula apply karne se pehle causes ko partition kyun form karni chahiye?

Taaki

B

exactly ek cause ke through ho sake, bina gaps ya overlaps ke, jisse

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

B

tak har path ka exact accounting ban sake — strips poore square ko tile karte hain bina double-shading ke.

Bayes ko "updating" rule kyun kaha jaata hai, sirf "identity" kyun nahi?

Tum isse ek prior belief aur fresh datum

B

feed karte ho aur ye revised belief return karta hai; kal ka posterior aaj ka prior ban jaata hai, isliye repeated evidence iteratively update karta hai — continuous prior-and-posterior distributions ke peeche yahi engine hai.

"Naive" independence assumption phir bhi useful classifications kyun deta hai?

Ye assume karta hai features class given conditionally independent hain, jo aksar false hota hai; lekin classes ko rank karne ke liye shared evidence term cancel ho jaata hai, aur chote dependency errors rarely change karte hain ki kaun sa class jeetega.

Do log same evidence se different posteriors kyun update kar sakte hain?

Kyunki unhone alag priors

P (A)

se shuru kiya — alag strip widths; Bayes shared likelihood ko har person ke apne prior se multiply karta hai, isliye starting beliefs ke baare mein honest disagreement answer mein propagate hoti hai.

Edge cases

P (A ∣ B)

kya hai jab

P (B) = 0

Undefined. Zero se division — tum us event par condition nahi kar sakte jo kabhi hota hi nahi, isliye definition

P (A \cap B) / P (B)

ka wahan koi meaning nahi.

Posterior ka kya hota hai jab likelihood

P (B ∣ A) = 0

ho?

Numerator vanish ho jaata hai, isliye

P (A ∣ B) = 0

: agar

A

observed evidence kabhi produce hi nahi karta, toh use observe karna

A

ko rule out kar deta hai.

Agar test perfectly specific hai,

P (+ ∣ \overset{ˉ}{D}) = 0

Tab koi false positives nahi hain — right strip ka shaded band zero height ka hai — toh evidence term sirf

P (+ ∣ D) P (D)

reh jaata hai, poora numerator, aur koi bhi positive

P (D ∣ +) = 1

deta hai: positive ab conclusive hai.

Bayes kya deta hai jab prior aur likelihood opposite directions mein point karein (rare cause, strong evidence)?

Ek tug-of-war: tiny prior ko bahut bade likelihood ratio se overpower kiya ja sakta hai, ya wo survive kar sakta hai — outcome literally unka product hai, isliye dono numbers quote karne zaroori hain, kabhi ek akele nahi.

Posterior kya hai agar evidence har cause ke neeche equally likely ho,

P (B ∣ A_{i})

sab equal?

Evidence uninformative hai; likelihoods normalisation mein cancel ho jaate hain aur har posterior apne prior ke barabar ho jaata hai,

P (A_{k} ∣ B) = P (A_{k})

— koi update nahi hota.

Agar tum Bayes ko overwhelming consistent evidence ke saath baar baar apply karo toh kya hota hai?

Posterior true cause ke liye

1

ki taraf converge karta hai (jab tak uska prior exactly

0

na ho), kyunki har update ek likelihood ratio se multiply karta hai jo use favour karta hai, initial prior ko duba deta hai.

Recall Ek-line self-test

Agar tum har conditional ke liye "given kaun se event?" ka jawab de sako jo tum likhte ho, toh tum yahan ke do sabse deadly traps pehle hi dodge kar chuke ho.

Connections

Bayes' theorem — derivation and applications — wo parent jise ye bank interrogate karta hai.
Conditional Probability — jahan bar $∣$ ki direction ke confusions originate hote hain.
Law of Total Probability — partition sum $P (B) = \sum_{i} P (B ∣ A_{i}) P (A_{i})$ ke peeche ki machinery.
Independent Events — "no update" edge case ko formalise karta hai.
Prior and Posterior Distributions — iterated updating ka continuous version.
Naive Bayes Classifier — applied "why it still works" question.
Tree Diagrams — same bookkeeping ke liye area model ka ek alternative.