4.9.24 · D5 · HinglishProbability Theory & Statistics
Question bank — Bayesian statistics — prior, likelihood, posterior (intro)
4.9.24 · D5· Maths › Probability Theory & Statistics › Bayesian statistics — prior, likelihood, posterior (intro)
Upar ki picture woh master image hai jo har question ke saath carry karni hai: prior curve (pehle ki belief), likelihood curve (data kaise vote karta hai), aur sharpened posterior curve (baad ki belief) — dekho kaise red posterior dono ke beech khicha hua baitha hai.
True or false — justify
Prior koi bhi probability distribution ho sakta hai jo main chahoon, koi data required nahi.
True — prior data se pehle ki belief encode karta hai, isliye ise choose kiya jaata hai (theory, past studies, ya convenience se), current dataset se derive nahi kiya jaata. Lekin "koi bhi" ka matlab tab bhi ek valid distribution hona chahiye jo 1 tak integrate ho.
Likelihood ek probability distribution hai ke upar.
False — ek function hai ka lekin data ki probability hai; iska ke upar 1 tak integrate hona zaroori nahi, isliye ye ke upar distribution nahi hai. Dekho Maximum Likelihood Estimation jo exactly is function ko maximize karta hai.
Agar do hypotheses ka observed data ke liye same likelihood ho, toh posterior unhe identically treat karta hai.
False — equal likelihoods sirf "persuasion" step cancel karte hain; posterior ratio phir prior ratio ke barabar ho jaata hai, isliye jis hypothesis par pehle zyada belief thi woh baad mein bhi jeette hai.
Evidence posterior ki shape ko ke upar affect karta hai.
False — ek single number hai jo se independent hai, isliye ye sirf poori curve ko rescale karta hai taaki 1 tak integrate ho; ye kabhi nahi badlata ki kaunsa doosre se zyada probable hai.
Flat (uniform) prior ka matlab hai "koi assumptions nahi / pure objectivity."
False — mein flat prior, ya jaale transform mein flat nahi hota, isliye "flat" ek specific modelling choice hai, choice ki absence nahi.
Enough data ke saath prior ki choice matter karna band ho jaati hai.
Usually true — jaise data accumulate hota hai likelihood sharply peaked ho jaati hai aur smooth prior par dominate karti hai, isliye alag-alag reasonable priors se posteriors converge ho jaate hain. Ye tab fail hota hai jab prior kisi jagah zero probability assign kare jahan truth ho.
aur ek hi cheez ke do notations hain.
False — ye alag-alag cheez par condition karte hain; 99% sensitive, 95% specific test ke saath lekin . Inhe confuse karna prosecutor's fallacy hai, aur Bayes' theorem precisely woh bridge hai jo conditioning ko flip karta hai.
Posterior dono prior aur likelihood se zyada concentrated (zyada certain) ho sakta hai.
True — do overlapping curves ko multiply karke renormalize karne par ek product milta hai jo typically dono factors se narrower hota hai, kyunki prior aur data ke beech agreement belief ko sharpen karta hai.
Posterior mean hamesha prior mean aur data ke estimate ke strictly beech hota hai.
Beta–Binomial jaale conjugate cases mein generally true — ye ek weighted average hai, isliye prior mean aur sample fraction ke beech baitha hai. Ye endpoint tab touch karta hai jab limiting cases hon (zero data, ya infinitely strong prior/data).
Spot the error
"Mujhe ek positive test mila jo 99% sensitive aur 95% specific hai, toh main 99% likely sick hoon."
Error ko se swap karta hai aur prior ko ignore karta hai; rare disease ke saath, bade healthy group ke false positives true positives se numerically zyada ho jaate hain, jisse far less than 99% milta hai.
"Posterior likelihood prior, toh mujhe kabhi ki zaroorat nahi."
Proportional form sahi shape deta hai, lekin jaesi calibrated number batane ke liye se divide karna zaroori hai. Ise omit karna sirf -values compare karne ke liye theek hai, actual probability report karne ke liye nahi.
"Mera prior tha ke liye; bahut saare data ke baad jo suggest karte hain, posterior finally wahan pahuncha."
Galat — posterior prior times likelihood ke proportional hai, aur zero prior se multiply koi bhi cheez hamesha zero rehti hai. Koi hypothesis jise zero prior mila ho, data se kabhi resurrect nahi ho sakta.
"Maine likelihood compute ki, dekha ki ke upar sum tha, aur conclude kiya ki arithmetic mistake hui."
Koi mistake nahi — likelihood ka ke upar integrate ya sum hokar 1 hona required nahi hai; sirf normalized posterior ko chahiye. "Error" ye expect karna hai ki likelihood ke upar distribution ki tarah behave kare.
"Evidence , lekin maine sirf woh hypotheses sum kiye jo mujhe pasand hain."
Sum sabhi hypotheses par Law of Total Probability ke zariye chalna chahiye (aur continuous mein integral ban jaata hai); kuch drop karne se too small ho jaata hai, toh posterior normalize nahi hota aur probabilities 1 se exceed ho jaati hain.
"Maine pakka karne ke liye same data par twice update kiya, aur mera posterior sharper ho gaya."
Illegitimate — ek observation do baar use karna evidence ko double-count karta hai, certainty ko artificially inflate karta hai. Har independent datum ek baar likelihood mein enter hota hai; ise reuse karna us model ko violate karta hai jisne generate kiya.
"Sabse likely dhundhne ke liye, maine likelihood ka peak liya, jo posterior peak ke barabar hona chahiye."
Sirf tab jab relevant region mein prior flat ho — non-uniform prior peak ko shift kar deta hai. Likelihood peak MLE hai; posterior peak (MAP) generally alag hota hai.
Why questions
Kyon ek test jo 99% sensitive aur 95% specific hai, 1% prevalence par sirf ~17% disease chance deta hai?
Kyunki sick logo se ~100 guna zyada healthy log hain, aur us bade group ka 5% false-positive test karta hai, isliye high accuracy ke bawajood false positives numerically true positives se zyada ho jaate hain.
Hum ko random variable kyun treat karte hain, jabki coin ka bias ek fixed physical fact hai?
Hum claim nahi kar rahe ki physically fluctuate karta hai; distribution fixed value ke baare mein hamari uncertainty encode karta hai. Ye core Bayesian vs frequentist split hai — Bayesians knowledge model karte hain, object ko nahi.
Posterior shape dhundhte waqt binomial coefficient kyun drop kar sakte hain?
Ye mein constant hai, isliye normalizer mein absorb ho jaata hai aur cancel ho jaata hai; ye overall scale affect karta hai, kabhi nahi ki kaunsa favoured hai.
Beta prior ko Binomial likelihood ka "conjugate" kyun kehte hain?
Kyunki Beta prior ko Binomial likelihood se multiply karne par phir se Beta milta hai, posterior ko same family mein rakhta hai. Isse update karna integral karne ki bajaye counts add karne ka kaam ban jaata hai — dekho Beta Distribution.
Strong prior ek single data point se kyun change hone se resist karta hai?
Sharply peaked prior apne peak se door near-zero weight deta hai, isliye modest likelihood se multiply karne par product ka peak barely relocate hota hai. Stubborn priors ko shift karne ke liye overwhelming likelihood chahiye.
Naive Bayes Classifier bahut saari likelihoods ko aapas mein multiply kyun karta hai?
Ye assume karta hai ki features class given hone par conditionally independent hain, isliye joint likelihood per-feature likelihoods ke product mein factorize ho jaati hai — ek modelling simplification jo computation sasta banata hai, bhalai approximately hi sahi.
ko "evidence" aur "marginal likelihood" dono kyun kehte hain?
Ye sabhi par marginalized likelihood hai (discrete case mein sum, continuous case mein integral) — average probability jo model data ko assign karta hai — isliye ye measure karta hai ki poore model ne jo observe hua usse kitna achha predict kiya, models compare karne ke liye useful.
Edge cases
Posterior kya hoga agar prior pehle se saari mass ek par rakh de (ek spike)?
Ye usi par spike rehta hai — certainty ka degenerate prior unmovable hai, kyunki baaki jagah zero probability kisi bhi likelihood se multiply hone par zero rehti hai.
Jab zero data observe karo ( empty) toh posterior ka kya hota hai?
Likelihood constant hai (explain karne ko kuch nahi), isliye posterior prior ke barabar ho jaata hai — no data matlab no update, exactly jaise recipe demand karti hai.
Flat prior aur 0 tosses mein 0 heads ke saath posterior kya hai?
Ye rehta hai, uniform distribution — koi observations nahi matlab prior unchanged pass ho jaata hai, isliye har bias equally plausible rehta hai.
Agar observed data ki probability har hypothesis ke under zero ho, toh Bayes kya deta hai?
, toh formula zero se divide karta hai aur undefined ho jaata hai — ye signal karta hai ki model data generate nahi kar sakta, matlab hypothesis space ya model misspecified hai.
Tum ek coin ek baar flip karte ho aur ek head dekhte ho; ke baare mein kya kehta hai?
par posterior density zero ho jaati hai, correctly rule out karta hai ek coin jo kabhi heads nahi aata — ek head ke under impossible hai, isliye likelihood wahan zero hai.
Infinite data ki limit mein Bayesian posterior mean kis cheez par converge karta hai?
Ye true parameter value par converge karta hai (mild conditions ke under), frequentist MLE se coincide karta hai — prior ka influence khatam ho jaata hai jaise likelihood dominate karta hai.
Recall Jaane se pehle ek-line self-test
Kya tum ek breath mein bol sakte ho ki prior/likelihood/posterior/evidence mein se kaunsa (a) 1 tak integrate nahi hona chahiye, (b) ek single -independent number hai, (c) zero se resurrect ho sakta hai, (d) bina data ke survive karta hai? ::: (a) likelihood; (b) evidence; (c) koi nahi — zero prior zero rehta hai; (d) prior (posterior = prior jab koi data nahi).