ベイズ線形回帰

rstanarm で学ぶベイズ回帰モデリング

Jake Thompson

Psychometrician, ATLAS, University of Kansas

ベイズ法を使う理由

p値はデータの確率を推測するものであり、パラメータ値ではない
事後分布: 尤度と事前分布の組み合わせ
- 事後分布からサンプリング
- サンプルの要約
- 要約を用いてパラメータ値を推測

rstanarmパッケージ

Stan確率的プログラミング言語へのインターフェース
rstanarmはStanへの高レベルアクセスを提供
カスタムモデル定義が可能

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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  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の要約: 推定値

 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: 誤差の標準偏差
mean_PPD: 事後予測サンプルの平均
log-posterior: 尤度に類似した指標

rstanarmの要約: 診断

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: チェーン内分散とチェーン間分散の比
1.1未満の値は収束を示す

練習しましょう！

rstanarm で学ぶベイズ回帰モデリング