๋ณธ๋ฌธ ๋ฐ”๋กœ๊ฐ€๊ธฐ

Statistics/Bayesian Statistics

๋ฌด์ •๋ณด ์‚ฌ์ „๋ถ„ํฌ(Non-informative prior distribution)

๋ฌด์ •๋ณด ์‚ฌ์ „๋ถ„ํฌ(Non-informative prior distribution)์— ์•Œ์•„๋ณผ ๊ฒƒ์ด๋‹ค.

 

๊ด€์‹ฌ์žˆ๋Š” ๋ชจ์ˆ˜์— ๋Œ€ํ•œ ์ •๋ณด๋ฅผ ์ฃผ๊ณ  ์‹ถ์ง€ ์•Š๋‹ค๋ฉด, ๋ชจ์ˆ˜์˜ ๋ชจ๋“  ๊ฐ€๋Šฅํ•œ ๊ฐ’๋“ค์— ๋™์ผํ•œ ๊ฐ€์ค‘์น˜๋ฅผ ๋ถ€์—ฌํ•˜๋Š” ๊ฒƒ์ด ํ•ฉ๋ฆฌ์ ์ผ ๊ฒƒ์ด๋‹ค. ๋”ฐ๋ผ์„œ ์‚ฌ์ „๋ถ„ํฌ๋กœ ๊ท ๋“ฑ๋ถ„ํฌ๋ฅผ ์‚ฌ์šฉํ•  ๊ฒƒ์ด๋‹ค. ์ด๋ฅผ ์ ์šฉํ•˜์—ฌ ๋‹ค์Œ์˜ ์ƒํ™ฉ์— ๋Œ€ํ•ด ์ƒ๊ฐํ•ด๋ณด์ž.

 

 

โ–ท ๊ท ๋“ฑ๋ถ„ํฌ๋Š” ๋ฒ ํƒ€๋ถ„ํฌ์˜ ํŠน์ดํ•œ ๊ฒฝ์šฐ์ด๋‹ค. ์ด๋ฅผ ์ด์šฉํ•˜์—ฌ, ESS(Effective Sample Size)๋ฅผ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค. ์ฆ‰, 2๊ฐœ์˜ ์ƒ˜ํ”Œ๋งŒํผ ์‚ฌํ›„๋ถ„ํฌ์— ์˜ํ–ฅ์„ ๋ฏธ์น˜๋ฏ€๋กœ ์™„์ „ํ•œ ๋ฌด์ •๋ณด ์‚ฌ์ „๋ถ„ํฌ๋ผ๊ณ  ํ•  ์ˆ˜ ์—†๋‹ค.

 

๋ฐ์ดํ„ฐ์—๋งŒ ์ข…์†์ ์ธ(Dependent) ์‚ฌํ›„๋ถ„ํฌ๋ฅผ ์ถ”์ •ํ•˜๊ธฐ ์œ„ํ•ด ESS๋ฅผ ์ค„์—ฌ๋ณด์ž.

 

 

โ–ท ์‚ฌ์ „๋ถ„ํฌ์˜ ๋‘ ๋ชจ์ˆ˜๋ฅผ 0์œผ๋กœ ์ •ํ•˜์—ฌ ESS๋ฅผ 0์œผ๋กœ ๋งŒ๋“ค์—ˆ๋‹ค.

 

ESS๊ฐ€ 0์ธ ๋ฌด์ •๋ณด ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ์‚ฌํ›„๋ถ„ํฌ์™€ ์‚ฌํ›„ํ‰๊ท ์„ ๊ตฌํ•ด๋ณด์ž.

 

 

โ–ท Beta(0, 0)์€ ๋ถ€์ ์ ˆ ์‚ฌ์ „๋ถ„ํฌ์ด๋‹ค. ์™œ๋ƒํ•˜๋ฉด ์ „์ฒด ํ™•๋ฅ ์˜ ํ•ฉ์ด 1์ด ์•„๋‹ˆ๊ธฐ ๋•Œ๋ฌธ์ด๋‹ค. ํ•˜์ง€๋งŒ ์ด๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ์‚ฌํ›„๋ถ„ํฌ๋ฅผ ๊ตฌํ•˜๋ฉด, ์œ„์˜ ๊ฒฐ๊ณผ์™€ ๊ฐ™์ด ์ ์ ˆํ•œ ๋ฒ ํƒ€๋ถ„ํฌ๊ฐ€ ๋‚˜์˜ค๊ฒŒ ๋œ๋‹ค. ์ฆ‰, ๋ถ€์ ์ ˆํ•œ ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ๋„ ์ ์ ˆํ•œ ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค.

 

โ–ท ์‚ฌํ›„๋ถ„ํฌ์˜ ํ‰๊ท ๊ณผ MLE๊ฐ€ ๊ฐ™๋‹ค๋Š” ๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ์ฆ‰, ๋ฌด์ •๋ณด ์‚ฌ์ „๋ถ„ํฌ๋ฅผ ์‚ฌ์šฉํ•˜์˜€์„ ๋•Œ, ๋นˆ๋„์ฃผ์˜ ์ ‘๊ทผ ๋ฐฉ์‹์˜ ๊ฒฐ๊ณผ๊ฐ€ ๊ฐ™๊ฒŒ๋œ๋‹ค. ํ•˜์ง€๋งŒ theta์— ๋Œ€ํ•œ ์‹ ์šฉ๊ตฌ๊ฐ„์„ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ์ ์—์„œ ์ฐจ์ด๊ฐ€ ์žˆ๋‹ค.

 

๋ถ€์ ์ ˆ ์‚ฌ์ „๋ถ„ํฌ์˜ ๋‹ค๋ฅธ ์˜ˆ๋ฅผ ๋ณด์ž.

 

โ–ท ์œ„์˜ ์‚ฌ์ „๋ถ„ํฌ๋Š” ๋ถ€์ ์ ˆํ•˜๋‹ค. ์™œ๋ƒํ•˜๋ฉด ํ™•๋ฅ ๋ฐ€๋„ํ•จ์ˆ˜๊ฐ€ 1์— ๋น„๋ก€ํ•˜์—ฌ ์ „์ฒด ํ™•๋ฅ ์˜ ํ•ฉ์ด 1์ด ์•„๋‹ˆ๊ธฐ ๋•Œ๋ฌธ์ด๋‹ค. ํ•˜์ง€๋งŒ ์ด๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ๊ตฌํ•˜๋ฉด, ์ ์ ˆํ•œ ์‚ฌํ›„๋ถ„ํฌ๋ฅผ ์–ป์„ ์ˆ˜ ์žˆ๋‹ค.

 

โ–ท ์•ž์˜ ์˜ˆ์™€ ๊ฐ™์ด ์‚ฌํ›„๋ถ„ํฌ์˜ ํ‰๊ท ๊ณผ MLE์™€ ๊ฐ™๋‹ค๋Š” ๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค.

 


Reference:

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