베이지안 선형 회귀
rstanarm으로 배우는 Bayesian 회귀 모델링
Jake Thompson
Psychometrician, ATLAS, University of Kansas
베이지안 방법을 사용하는 이유
- p-값은 모수가 아닌 데이터의 확률에 대한 추론
- 사후 분포: 가능도와 사전 분포의 결합
- 사후 분포 샘플링
- 샘플 요약
- 요약을 사용하여 모수에 대한 추론
rstanarm 패키지
- Stan 확률적 프로그래밍 언어의 인터페이스
- rstanarm은 Stan에 대한 고수준 접근 제공
- 사용자 정의 모델 정의 가능
library(rstanarm)stan_model <- stan_glm(kid_score ~ mom_iq, data = kidiq)
SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
Gradient evaluation took 0.000408 seconds
1000 transitions using 10 leapfrog steps per transition would take
4.08 seconds.
Adjust your expectations accordingly!
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Iteration: 2000 / 2000 [100%] (Sampling)
Elapsed Time: 0.37735 seconds (Warm-up)
0.252244 seconds (Sampling)
0.629594 seconds (Total)
summary(stan_model)
Model Info:
function: stan_glm
family: gaussian [identity]
formula: kid_score ~ mom_iq
algorithm: sampling
priors: see help('prior_summary')
sample: 4000 (posterior sample size)
observations: 434
predictors: 2
Estimates:
mean sd 2.5% 25% 50% 75% 97.5%
(Intercept) 25.7 6.0 13.8 21.6 25.7 30.0 37.0
mom_iq 0.6 0.1 0.5 0.6 0.6 0.7 0.7
sigma 18.3 0.6 17.1 17.9 18.3 18.7 19.5
mean_PPD 86.8 1.2 84.3 85.9 86.8 87.6 89.2
log-posterior -1885.4 1.2 -1888.5 -1886.0 -1885.1 -1884.5 -1884.0
Diagnostics:
mcse Rhat n_eff
(Intercept) 0.1 1.0 4000
mom_iq 0.0 1.0 4000
sigma 0.0 1.0 3827
mean_PPD 0.0 1.0 4000
log-posterior 0.0 1.0 1896
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor
on split chains (at convergence Rhat=1).
rstanarm 요약: 추정값
Estimates:
mean sd 2.5% 25% 50% 75% 97.5%
(Intercept) 25.7 6.0 13.8 21.6 25.7 30.0 37.0
mom_iq 0.6 0.1 0.5 0.6 0.6 0.7 0.7
sigma 18.3 0.6 17.1 17.9 18.3 18.7 19.5
mean_PPD 86.8 1.2 84.3 85.9 86.8 87.6 89.2
log-posterior -1885.4 1.2 -1888.5 -1886.0 -1885.1 -1884.5 -1884.0
- sigma: 오차의 표준 편차
- mean_PPD: 사후 예측 샘플의 평균
- log-posterior: 가능도에 유사한 값
rstanarm 요약: 진단
Diagnostics:
mcse Rhat n_eff
(Intercept) 0.1 1.0 4000
mom_iq 0.0 1.0 4000
sigma 0.0 1.0 3827
mean_PPD 0.0 1.0 4000
log-posterior 0.0 1.0 1896
For each parameter, mcse is Monte Carlo standard error,
n_eff is a crude measure of effective sample size, and
Rhat is the potential scale reduction factor on split chains
(at convergence Rhat=1).
- Rhat: 체인 내 분산과 체인 간 분산을 비교하는 측도
- 1.1 미만이면 수렴을 의미
연습해 봅시다!
rstanarm으로 배우는 Bayesian 회귀 모델링
