bayesian_workflow.Rmd

# 贝叶斯工作流程 {#bayesian-workflow}


## 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


## 案例

我们用[ames房屋价格](https://bookdown.org/wangminjie/R4DS/eda-ames-houseprice.html#eda-ames-houseprice)，演示贝叶斯数据分析的工作流程

```{r, message = FALSE, warning = FALSE}
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, warning=FALSE, message=FALSE}
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);
   }
}
"
```





### 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, warning=FALSE, message=FALSE}
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, message=FALSE, results=FALSE}
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, warning=FALSE, message=FALSE}
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, echo = F, message = F, warning = F, results = "hide"}
pacman::p_unload(pacman::p_loaded(), character.only = TRUE)
```