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数据科学中的 R 语言

第 71 章 贝叶斯工作流程

71.1 贝叶斯工作流程

  1. 数据探索和准备
  2. 全概率模型
  3. 先验预测检查，利用先验模拟响应变量
  4. 模型应用到模拟数据，看参数恢复情况
  5. 模型应用到真实数据
  6. 检查抽样效率和模型收敛情况
  7. 模型评估和后验预测检查
  8. 信息准则与交叉验证，以及模型选择

71.2 案例

我们用ames房屋价格，演示贝叶斯数据分析的工作流程

library(tidyverse)
library(tidybayes)
library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

71.2.1 1) 数据探索和准备 

rawdf <- readr::read_rds("./demo_data/ames_houseprice.rds") 
rawdf
## # A tibble: 1,460 × 81
##       id ms_sub_class ms_zoning lot_frontage lot_area street alley lot_shape
##    <dbl>        <dbl> <chr>            <dbl>    <dbl> <chr>  <chr> <chr>    
##  1     1           60 RL                  65     8450 Pave   <NA>  Reg      
##  2     2           20 RL                  80     9600 Pave   <NA>  Reg      
##  3     3           60 RL                  68    11250 Pave   <NA>  IR1      
##  4     4           70 RL                  60     9550 Pave   <NA>  IR1      
##  5     5           60 RL                  84    14260 Pave   <NA>  IR1      
##  6     6           50 RL                  85    14115 Pave   <NA>  IR1      
##  7     7           20 RL                  75    10084 Pave   <NA>  Reg      
##  8     8           60 RL                  NA    10382 Pave   <NA>  IR1      
##  9     9           50 RM                  51     6120 Pave   <NA>  Reg      
## 10    10          190 RL                  50     7420 Pave   <NA>  Reg      
## # ℹ 1,450 more rows
## # ℹ 73 more variables: land_contour <chr>, utilities <chr>, lot_config <chr>,
## #   land_slope <chr>, neighborhood <chr>, condition1 <chr>, condition2 <chr>,
## #   bldg_type <chr>, house_style <chr>, overall_qual <dbl>, overall_cond <dbl>,
## #   year_built <dbl>, year_remod_add <dbl>, roof_style <chr>, roof_matl <chr>,
## #   exterior1st <chr>, exterior2nd <chr>, mas_vnr_type <chr>,
## #   mas_vnr_area <dbl>, exter_qual <chr>, exter_cond <chr>, foundation <chr>, …

为了简化，我们只关注房屋价格(sale_price)与房屋占地面积(lot_area)和所在地理位置(neighborhood)的关系，这里需要点准备工作

  • 房屋价格与房屋占地面积这两个变量对数化处理 (why ?)
  • 地理位置变量转换因子类型 (why ?)
  • 房屋价格与房屋占地面积这两个变量标准化处理 (why ?)
df <- rawdf %>%
  select(sale_price, lot_area, neighborhood) %>%
  drop_na() %>%
  mutate(
    across(c(sale_price, lot_area), log),
    across(neighborhood, as.factor)
  ) %>%
  mutate(
    across(c(sale_price, lot_area), ~ (.x - mean(.x)) /sd(.x) ),
  )

head(df)
## # A tibble: 6 × 3
##   sale_price lot_area neighborhood
##        <dbl>    <dbl> <fct>       
## 1      0.560   -0.133 CollgCr     
## 2      0.213    0.113 Veenker     
## 3      0.734    0.420 CollgCr     
## 4     -0.437    0.103 Crawfor     
## 5      1.01     0.878 NoRidge     
## 6     -0.384    0.858 Mitchel
df %>%
  ggplot(aes(x = lot_area, y = sale_price)) +
  geom_point(colour = "blue") +
  geom_smooth(method = lm, se = FALSE, formula = "y ~ x")
df %>%
  ggplot(aes(x = lot_area, y = sale_price)) +
  geom_point(colour = "blue") +
  geom_smooth(method = lm, se = FALSE, formula = "y ~ x", fullrange = TRUE) +
  facet_wrap(vars(neighborhood))

71.2.2 2) 数据模型

\[ \begin{align} y_i &\sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i &= \alpha_{j} + \beta * x_i \\ \alpha_j & \sim \operatorname{Normal}(0, 10)\\ \beta & \sim \operatorname{Normal}(0, 10) \\ \sigma &\sim \exp(1) \end{align} \]

如果建立了这样的数学模型，可以马上写出stan代码

stan_program <- "
data {
  int<lower=1> n;           
  int<lower=1> n_neighbour;      
  int<lower=1> neighbour[n];     
  vector[n] lot;  
  vector[n] price;  
  
  real alpha_sd;
  real beta_sd;
  int<lower = 0, upper = 1> run_estimation;
}
parameters {
  vector[n_neighbour] alpha;
  real beta;
  real<lower=0> sigma;
}
model {
  vector[n] mu;  
  
  for (i in 1:n) {
    mu[i] = alpha[neighbour[i]] + beta * lot[i];
  }
  
  alpha ~ normal(0, alpha_sd);
  beta ~ normal(0, beta_sd);
  sigma ~ exponential(1);
  
  if(run_estimation == 1) {
     target += normal_lpdf(price | mu, sigma);
  }
    
}
generated quantities {
   vector[n] log_lik; 
   vector[n] y_hat;
   
   for (j in 1:n) {
     log_lik[j] = normal_lpdf(price | alpha[neighbour[j]] + beta * lot[j], sigma);
     y_hat[j]   = normal_rng(alpha[neighbour[j]] + beta * lot[j], sigma);
   }
}
"

71.2.3 3) 先验预测检查，利用先验模拟响应变量

有个问题，我们这个先验概率怎么来的呢？猜的，因为没有人知道它究竟是什么分布(如果您是这个领域的专家，就不是猜，而叫合理假设)。那到底合不合理，我们需要检验下。这里用到的技术是先验预测检验。怎么做？

  • 首先，模拟先验概率分布
  • 然后，通过先验和模型假定的线性关系，模拟相应的响应变量\(y_i\)（注意，不是真实的数据）
stan_data <- df %>%
  tidybayes::compose_data(
    n_neighbour    = n_distinct(neighborhood),
    neighbour      = neighborhood,
    price          = sale_price,
    lot            = lot_area,
    alpha_sd       = 10, 
    beta_sd        = 10, 
    run_estimation = 0
  )



model_only_prior_sd_10 <- stan(model_code = stan_program, data = stan_data, 
                       chains = 1, iter = 2100, warmup = 2000)
## 
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 8.8e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.88 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:    1 / 2100 [  0%]  (Warmup)
## Chain 1: Iteration:  210 / 2100 [ 10%]  (Warmup)
## Chain 1: Iteration:  420 / 2100 [ 20%]  (Warmup)
## Chain 1: Iteration:  630 / 2100 [ 30%]  (Warmup)
## Chain 1: Iteration:  840 / 2100 [ 40%]  (Warmup)
## Chain 1: Iteration: 1050 / 2100 [ 50%]  (Warmup)
## Chain 1: Iteration: 1260 / 2100 [ 60%]  (Warmup)
## Chain 1: Iteration: 1470 / 2100 [ 70%]  (Warmup)
## Chain 1: Iteration: 1680 / 2100 [ 80%]  (Warmup)
## Chain 1: Iteration: 1890 / 2100 [ 90%]  (Warmup)
## Chain 1: Iteration: 2001 / 2100 [ 95%]  (Sampling)
## Chain 1: Iteration: 2100 / 2100 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 6.271 seconds (Warm-up)
## Chain 1:                0.301 seconds (Sampling)
## Chain 1:                6.572 seconds (Total)
## Chain 1:
## Warning: The largest R-hat is 1.25, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess
dt_wide <- model_only_prior_sd_10 %>% 
  as.data.frame() %>% 
  select(`alpha[5]`, beta) %>% 
  rowwise() %>%
  mutate(
    set = list(tibble(
      x = seq(from = -3, to = 3, length.out = 200),
      y = `alpha[5]` + beta * x
    ))
  )


ggplot() +
  map(
    dt_wide$set,
    ~ geom_line(data = ., aes(x = x, y = y), alpha = 0.2)
  )
stan_data <- df %>%
  tidybayes::compose_data(
    n_neighbour    = n_distinct(neighborhood),
    neighbour      = neighborhood,
    price          = sale_price,
    lot            = lot_area,
    alpha_sd       = 1, 
    beta_sd        = 1, 
    run_estimation = 0
  )



model_only_prior_sd_1 <- stan(model_code = stan_program, data = stan_data, 
                       chains = 1, iter = 2100, warmup = 2000)
## 
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 3.8e-05 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.38 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:    1 / 2100 [  0%]  (Warmup)
## Chain 1: Iteration:  210 / 2100 [ 10%]  (Warmup)
## Chain 1: Iteration:  420 / 2100 [ 20%]  (Warmup)
## Chain 1: Iteration:  630 / 2100 [ 30%]  (Warmup)
## Chain 1: Iteration:  840 / 2100 [ 40%]  (Warmup)
## Chain 1: Iteration: 1050 / 2100 [ 50%]  (Warmup)
## Chain 1: Iteration: 1260 / 2100 [ 60%]  (Warmup)
## Chain 1: Iteration: 1470 / 2100 [ 70%]  (Warmup)
## Chain 1: Iteration: 1680 / 2100 [ 80%]  (Warmup)
## Chain 1: Iteration: 1890 / 2100 [ 90%]  (Warmup)
## Chain 1: Iteration: 2001 / 2100 [ 95%]  (Sampling)
## Chain 1: Iteration: 2100 / 2100 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 6.057 seconds (Warm-up)
## Chain 1:                0.299 seconds (Sampling)
## Chain 1:                6.356 seconds (Total)
## Chain 1:
## Warning: The largest R-hat is 1.14, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess
## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess
dt_narrow <- model_only_prior_sd_1 %>% 
  as.data.frame() %>% 
  select(`alpha[5]`, beta) %>% 
  rowwise() %>%
  mutate(
    set = list(tibble(
      x = seq(from = -3, to = 3, length.out = 200),
      y = `alpha[5]` + beta * x
    ))
  )


ggplot() +
  map(
    dt_narrow$set,
    ~ geom_line(data = ., aes(x = x, y = y), alpha = 0.2)
  )

71.2.4 4) 模型应用到模拟数据，看参数恢复情况 

df_random_draw <- model_only_prior_sd_1 %>% 
  tidybayes::gather_draws(alpha[i], beta, sigma, y_hat[i], n = 1)
## Warning: 
## In tidybayes::gather_draws(): The `n` argument is a deprecated alias for `ndraws`.
## Use the `ndraws` argument instead.
## See help("tidybayes-deprecated").
true_parameters <- df_random_draw %>% 
  filter(.variable %in% c("alpha", "beta", "sigma")) %>%
  mutate(parameters = if_else(is.na(i), .variable, str_c(.variable, "_", i)))


y_sim <- df_random_draw %>% 
  filter(.variable == "y_hat") %>% 
  pull(.value)

模拟的数据y_sim，导入模型作为响应变量，

stan_data <- df %>%
  tidybayes::compose_data(
    n_neighbour    = n_distinct(neighborhood),
    neighbour      = neighborhood,
    price          = y_sim,      ##  这里是模拟数据
    lot            = lot_area,
    alpha_sd       = 1, 
    beta_sd        = 1, 
    run_estimation = 1
  )

model_on_fake_dat <- stan(model_code = stan_program, data = stan_data)

看参数恢复的如何

model_on_fake_dat %>% 
  tidybayes::gather_draws(alpha[i], beta, sigma) %>% 
  ungroup() %>% 
  mutate(parameters = if_else(is.na(i), .variable, str_c(.variable, "_", i))) %>% 

  ggplot(aes(x = .value)) +
  geom_density() +
  geom_vline(
    data = true_parameters,
    aes(xintercept = .value),
    color = "red"
    ) +
  facet_wrap(vars(parameters), ncol = 5, scales = "free")

如果觉得上面的过程很麻烦，可以直接用bayesplot::mcmc_recover_hist()

posterior_alpha_beta <- 
  as.matrix(model_on_fake_dat, pars = c('alpha', 'beta', 'sigma'))

bayesplot::mcmc_recover_hist(posterior_alpha_beta, true = true_parameters$.value)

71.2.5 5) 模型应用到真实数据

应用到真实数据

stan_data <- df %>%
  tidybayes::compose_data(
    n_neighbour    = n_distinct(neighborhood),
    neighbour      = neighborhood,
    price          = sale_price,      ##  这里是真实数据
    lot            = lot_area,
    alpha_sd       = 1, 
    beta_sd        = 1, 
    run_estimation = 1
  )

model <- stan(model_code = stan_program, data = stan_data)

71.2.6 6) 检查抽样效率和模型收敛情况

  • 检查traceplot
rstan::traceplot(model)
## 'pars' not specified. Showing first 10 parameters by default.
  • 检查neff 和 Rhat
print(model,
  pars = c("alpha", "beta", "sigma"),
  probs = c(0.025, 0.50, 0.975),
  digits_summary = 3
)
## Inference for Stan model: anon_model.
## 4 chains, each with iter=2000; warmup=1000; thin=1; 
## post-warmup draws per chain=1000, total post-warmup draws=4000.
## 
##             mean se_mean    sd   2.5%    50%  97.5% n_eff  Rhat
## alpha[1]   1.005   0.002 0.149  0.713  1.003  1.291  6883 1.000
## alpha[2]   0.572   0.004 0.397 -0.209  0.577  1.354  8075 0.999
## alpha[3]  -0.103   0.002 0.160 -0.409 -0.105  0.216  5876 0.999
## alpha[4]  -0.688   0.001 0.080 -0.845 -0.688 -0.534  7774 1.000
## alpha[5]   0.002   0.001 0.122 -0.233  0.001  0.237  6923 1.000
## alpha[6]   0.330   0.001 0.051  0.228  0.330  0.432  8714 0.999
## alpha[7]   0.340   0.001 0.085  0.179  0.339  0.511  7270 1.000
## alpha[8]  -0.778   0.001 0.060 -0.895 -0.778 -0.661 10271 0.999
## alpha[9]   0.215   0.001 0.066  0.085  0.215  0.347  8068 1.000
## alpha[10] -1.330   0.001 0.100 -1.522 -1.329 -1.138  9092 0.999
## alpha[11] -0.411   0.002 0.156 -0.708 -0.410 -0.106  6498 1.000
## alpha[12] -0.318   0.001 0.087 -0.490 -0.318 -0.152  9028 0.999
## alpha[13] -0.440   0.000 0.041 -0.520 -0.440 -0.360 10363 0.999
## alpha[14]  1.368   0.001 0.097  1.180  1.367  1.558  8688 0.999
## alpha[15]  0.316   0.002 0.205 -0.092  0.317  0.710  8185 0.999
## alpha[16]  1.412   0.001 0.071  1.266  1.412  1.550  8253 0.999
## alpha[17]  0.101   0.001 0.071 -0.040  0.102  0.241 10103 0.999
## alpha[18] -0.684   0.001 0.056 -0.795 -0.684 -0.575  7970 1.000
## alpha[19] -0.597   0.001 0.069 -0.732 -0.597 -0.462 10452 1.000
## alpha[20]  0.122   0.001 0.081 -0.043  0.121  0.280  9571 0.999
## alpha[21]  0.870   0.001 0.067  0.741  0.870  1.001  7150 0.999
## alpha[22]  1.422   0.001 0.123  1.181  1.419  1.657  9580 0.999
## alpha[23] -0.359   0.001 0.118 -0.596 -0.359 -0.130  8209 1.000
## alpha[24]  0.505   0.001 0.102  0.306  0.507  0.704  7189 0.999
## alpha[25]  0.515   0.002 0.185  0.145  0.516  0.878  9773 0.999
## beta       0.347   0.000 0.021  0.306  0.347  0.389  3960 0.999
## sigma      0.607   0.000 0.011  0.585  0.607  0.630  8316 0.999
## 
## Samples were drawn using NUTS(diag_e) at Tue Jul 18 20:26:14 2023.
## For each parameter, 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).
  • 检查posterior sample
model %>% 
  tidybayes::gather_draws(alpha[i], beta, sigma) %>% 
  ungroup() %>% 
  mutate(parameters = if_else(is.na(i), .variable, str_c(.variable, "_", i))) %>%
  
  ggplot(aes(x = .value, y = parameters)) +
  ggdist::stat_halfeye()

事实上，bayesplot宏包很强大也很好用

bayesplot::mcmc_combo(
  as.array(model),
  combo = c("dens_overlay", "trace"),
  pars = c('alpha[1]', 'beta', 'sigma')
 )

71.2.7 7) 模型评估和后验预测检查 

yrep <- extract(model)[["y_hat"]]

samples <- sample(nrow(yrep), 300)
bayesplot::ppc_dens_overlay(as.vector(df$sale_price), yrep[samples, ])

71.3 Conclusion

71.4 作业

  • 前面的模型只有变化的截距（即不同的商圈有不同的截距）斜率是固定的，要求：增加一个变化的斜率

\[ \begin{align} y_i &\sim \operatorname{Normal}(\mu_i, \sigma) \\ \mu_i &= \alpha_{j} + \beta_{j} * x_i \\ \alpha_j & \sim \operatorname{Normal}(0, 1)\\ \beta_j & \sim \operatorname{Normal}(0, 1) \\ \sigma &\sim \exp(1) \end{align} \]