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贝叶斯工作流

建模、诊断与后验预测

贝叶斯篇

贝叶斯工作流程

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

案例

我们用ames房屋价格，演示贝叶斯数据分析的工作流程

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

1) 数据探索和准备

R 
rawdf <- readr::read_rds("./demo_data/ames_houseprice.rds") 
rawdf

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 ?)
R 
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)
R 
df %>%
  ggplot(aes(x = lot_area, y = sale_price)) +
  geom_point(colour = "blue") +
  geom_smooth(method = lm, se = FALSE, formula = "y ~ x")
R 
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))

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

R 
stan_program <- &quot;
data {
  int&lt;lower=1&gt; n;           
  int&lt;lower=1&gt; n_neighbour;      
  int&lt;lower=1&gt; neighbour[n];     
  vector[n] lot;  
  vector[n] price;  
  
  real alpha_sd;
  real beta_sd;
  int&lt;lower = 0, upper = 1&gt; run_estimation;
}
parameters {
  vector[n_neighbour] alpha;
  real beta;
  real&lt;lower=0&gt; 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);
   }
}
&quot;

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）
R 
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)



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)
  )
R 
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)



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)
  )

4) 模型应用到模拟数据，看参数恢复情况

R 
df_random_draw <- model_only_prior_sd_1 %>% 
  tidybayes::gather_draws(alpha[i], beta, sigma, y_hat[i], n = 1)

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，

R 
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

R 
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()

R 
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)

5) 模型应用到真实数据

应用到真实数据

R 
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)

6) 检查抽样效率和模型收敛情况

  • 检查traceplot
R 
rstan::traceplot(model)
  • 检查neff 和 Rhat
R 
print(model,
  pars = c("alpha", "beta", "sigma"),
  probs = c(0.025, 0.50, 0.975),
  digits_summary = 3
)
  • 检查posterior sample
R 
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

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

7) 模型评估和后验预测检查

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

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

Conclusion

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

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

R 
pacman::p_unload(pacman::p_loaded(), character.only = TRUE)