library(MCMCpack)
library(MASS)
library(ggplot2)
# Generate some data


# parameters for the DGP
mu <- 30
sigma <- sqrt(5)
n <- 50
# data
x <- rnorm(n=n, mean=mu, sd=sigma)
# summary(x)

# ML estimates
xbar <- mean(x)
sdev <- sd
s2 <- var(x)

# Algorithmus 2
# Bayes
S <- 5000 # number of draws
#step 1
sigma2_sim <- rinvgamma(n=S, (n-1)/2, (n-1)*s2/2)
# step2
mu_sim <- rnorm(n=S, mean=xbar, sd= sqrt(sigma2_sim/n))


plot(density(mu_sim))
plot(density(sigma2_sim))

h <- kde2d(mu_sim, sigma2_sim)
contour(h$x,h$y,h$z)
image(h$x,h$y,h$z)