๋ณธ๋ฌธ ๋ฐ”๋กœ๊ฐ€๊ธฐ

Statistics/Bayesian Statistics

์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ(Prior and posterior predictive distribution)

์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ(Prior predictive distribution)์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ(Posterior predictive distribution)์— ๋Œ€ํ•ด ์•Œ์•„๋ณผ ๊ฒƒ์ด๋‹ค. ๋‹ค๋ฃฐ ๋‚ด์šฉ์€ ๋‹ค์Œ๊ณผ ๊ฐ™๋‹ค.

 

1. ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์˜ ์ •์˜

2. ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์˜ ์˜ˆ์ œ

 

1. ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์˜ ์ •์˜

 

 

โ–ท ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ๋Š” ๋ฒ ์ด์ฆˆ ์ •๋ฆฌ๋ฅผ ์ด์šฉํ•˜์—ฌ ๊ตฌํ•˜๋ฉด, ์‚ฌ์ „๋ถ„ํฌ์™€ ๊ฐ€๋Šฅ๋„ ํ•จ์ˆ˜์˜ ๊ณฑ์„ ์ ๋ถ„ํ•œ ํ˜•ํƒœ๋กœ ์ •์˜๋œ๋‹ค. ์ฆ‰, theta์— ๋Œ€ํ•œ ๊ฐ€๋Šฅ๋„ ํ•จ์ˆ˜์˜ ํ‰๊ท ์ด๋ผ ํ•  ์ˆ˜ ์žˆ๋‹ค.

 

โ–ท ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ๋Š” ๋ฒ ์ด์ฆˆ ์ •๋ฆฌ๋ฅผ ์ด์šฉํ•˜์—ฌ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค. ์ด๋•Œ, ์ผ๋ฐ˜์ ์œผ๋กœ  ๊ด€์ธก ๊ฒฐ๊ณผ์ธ x์™€ ํ™•๋ฅ  ๋ณ€์ˆ˜ x tilde์˜ ๊ด€๊ณ„๋Š” ๋…๋ฆฝ์ด๋ผ ๊ฐ€์ •ํ•˜๊ธฐ ๋•Œ๋ฌธ์—, theta์˜ ์‚ฌํ›„๋ถ„ํฌ์™€ ๊ฐ€๋Šฅ๋„ ํ•จ์ˆ˜์˜ ๊ณฑ์„ ์ ๋ถ„ํ•œ ํ˜•ํƒœ๋กœ ์ •์˜๋œ๋‹ค. theta์˜ ์‚ฌํ›„ ๋ถ„ํฌ๊ฐ€ ์‚ฌ์ „ ๋ถ„ํฌ๋ผ๊ณ  ์ƒ๊ฐํ•˜๋ฉด ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ๋ฅผ ๊ตฌํ•˜๋Š” ๊ณผ์ •๊ณผ ๋™์ผํ•˜๋‹ค.

 

โ–ท ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์˜ ์ฐจ์ด๋Š” ์–ด๋–ค ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ์“ฐ๋Š”๊ฐ€์— ๋”ฐ๋ผ ๋ฐœ์ƒํ•œ๋‹ค. ์ด๋Š” ์ •๋ณด๊ฐ€ ์ถ”๊ฐ€๋จ์— ๋”ฐ๋ผ ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ์—…๋ฐ์ดํŠธ ํ•˜์—ฌ ์‚ฌํ›„๋ถ„ํฌ๋ฅผ ๊ตฌํ•˜๊ณ , ์ด ์‚ฌํ›„๋ถ„ํฌ๋Š” ๋‹ค์Œ ์˜ˆ์ธก์˜ ์‚ฌ์ „๋ถ„ํฌ๋กœ ํ™œ์šฉ๋˜๊ธฐ ๋•Œ๋ฌธ์ด๋‹ค. ์ด ๊ณผ์ •์„ ํ†ตํ•ด ์ถ”๊ฐ€๋œ ์ •๋ณด๋ฅผ ์‹ค์‹œ๊ฐ„์œผ๋กœ ๋ฐ˜์˜ํ•  ์ˆ˜ ์žˆ๋Š” ๊ฒƒ์ด ๋ฒ ์ด์ง€์•ˆ ์ ‘๊ทผ์˜ ํŠน์ง•์ด๋‹ค.

 

2. ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์™€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์˜ ์˜ˆ์ œ

 

๋ฌธ์ œ)

 

์‹œํ–‰ํšŸ์ˆ˜๊ฐ€ 10์ธ ์ดํ•ญ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅด๋Š” ํ™•๋ฅ  ๋ณ€์ˆ˜ X๊ฐ€ ์žˆ๋‹ค. ์ดํ•ญ๋ถ„ํฌ์˜ ๋ชจ์ˆ˜๊ฐ€ ๊ท ์ผ๋ถ„ํฌ์„ ๋”ฐ๋ฅผ ๋•Œ, ํ™•๋ฅ  ๋ณ€์ˆ˜ X์˜ ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ๋ฅผ ๊ตฌํ•˜์—ฌ๋ผ.

 

ํ’€์ด)

 

 

โ–ท ์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ๋Š” ๋ชจ๋“  ๊ฒฝ์šฐ์—์„œ 1/11๋กœ ํ™•๋ฅ ์ด ๊ฐ™์€ ๊ฒƒ์œผ๋กœ ๋‚˜ํƒ€๋‚ฌ๋‹ค.

 

๋ฌธ์ œ)

 

๋™์ „์„ ๋˜์กŒ์„ ๋•Œ, ์•ž๋ฉด์ด ๋‚˜์˜ฌ ํ™•๋ฅ ์ด ๊ท ์ผ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅธ๋‹ค. ๋™์ „์„ ๋˜์กŒ๋”๋‹ˆ ์•ž๋ฉด์ด ๋‚˜์™”์„ ๋•Œ, ์ด ๊ฒฐ๊ณผ๋ฅผ ์ด์šฉํ•˜์—ฌ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ๋ฅผ ๊ตฌํ•˜์—ฌ๋ผ. ์ด๋•Œ, ๊ฐ ์‹œํ–‰์€ ๋…๋ฆฝ์ ์ด๋‹ค.

 

ํ’€์ด)

 

 

โ–ท ๋™์ „์˜ ์•ž๋ฉด์ด ๋‚˜์˜จ ๊ฒฐ๊ณผ๊ฐ€ ์‚ฌํ›„์˜ˆ์ธก๋ถ„ํฌ์— ๋ฐ˜์˜๋˜์–ด 1/2(์‚ฌ์ „์˜ˆ์ธก๋ถ„ํฌ์—์„œ์˜ ๋™์ „์ด ์•ž๋ฉด์ด ๋‚˜์˜ฌ ํ™•๋ฅ )์—์„œ 2/3๋กœ ์ฆ๊ฐ€ํ•œ ๊ฒƒ์„ ํ™•์ธํ•  ์ˆ˜ ์žˆ๋‹ค.

 


Reference:

"Prior & Posterior Predictive Distributions," DH.Kim, https://donghwa-kim.github.io/Pred_-baye.html.

"Bayesian Statistics: From Concept to Data Analysis," Coursera, https://www.coursera.org/learn/bayesian-statistics/.