Estimating the mean = regression on constant term

set.seed(42)
n_0 <- 20
y_0 <- rnorm(n_0, 2.0, 5.0)
cat(paste0('mean:', mean(y_0), '\nstandard error:', sd(y_0)/sqrt(n_0),"\n\n"))
## mean:2.95960009711351
## standard error:1.46756319082706
fake_0 <- data.frame(y_0)
fit_0 <- stan_glm(y_0 ~ 1, data=fake_0,
prior_intercept=NULL, prior=NULL, prior_aux=NULL, refresh =0 )
print(fit_0, detail = FALSE)
##             Median MAD_SD
## (Intercept) 3.0    1.5   
## 
## Auxiliary parameter(s):
##       Median MAD_SD
## sigma 6.8    1.1

Flat priors reproduce classical least squares estimate (more on this when we get to section 9.5)

What about default priors?

fit_0 <- stan_glm(y_0 ~ 1, data=fake_0, refresh =0 )
print(fit_0, detail = FALSE)
##             Median MAD_SD
## (Intercept) 3.0    1.4   
## 
## Auxiliary parameter(s):
##       Median MAD_SD
## sigma 6.7    1.1