illustrate tail sensitivity of Bayes (from McElreath)

bayes_tails.R

post_prior <- function(m_pr, m_lik, sd_pr, sd_lik, df_pr, df_lik,
                       xlim = c(-5, 15), n = 1001, ttl = "",...) {
  xvec <- seq(xlim[1], xlim[2], length.out = n)
  tfun <- function(x,m,s,df, ...) dt((x-m)/s, df)
  prvec <- tfun(xvec, m_pr, sd_pr, df_pr)
  likvec <- tfun(xvec, m_lik, sd_lik, df_lik)
  postvec <- prvec*likvec/(sum(prvec*likvec)*diff(xvec[1:2]))
  matplot(xvec, cbind(prvec, likvec, postvec),
          type = "l",
          lty = c(2,1,1),
          col = c(1,1,2),
          lwd = c(1,1,2),
          ylab = "density", 
         ...)
  title(ttl)
  pmax = max(max(likvec), max(prvec))
  text(0, pmax, "likelihood")
  text(10, pmax, "prior")
}

par(las = 1, bty = "l")
par(mfrow=c(2,2))
post_prior(10, 0, 1, 1, 1000,1000, ttl="prior:N, lik:N") # log = "y")
post_prior(10, 0, 1, 1, 2, 2,  ttl="prior:T, lik:T") #, log = "y")
post_prior(10, 0, 1, 1, 2, 1000, ttl="prior:T, lik:N")
post_prior(10, 0, 1, 1, 1000, 2, ttl="prior:N, lik:T")

Leaving a comment here since I can't find how to edit.

As per https://twitter.com/analisereal/status/1701412100403573008 we should also scale the t-density by (1/s), i.e,

dt((x-m)/s, df)*(1/s)

You are right.
Or we could use a library for probability distributions that takes care of all this for us.
like this:

 pr_prior = tfd.StudentT(df_prior, mean_prior, std_prior)

https://colab.research.google.com/github/gerdm/misc/blob/master/2023-09/bayes-tails.ipynb