Coding a multiple integral function in R

Question

With the goal of turning the following into a function, I was wondering how I can write the following double integral in terms of R codes?: ($\bar{x} = \mu$):

Answer 1

Assuming pi0 and pi1 implement your functions $\pi_0$ and $\pi_1$ in a vectorized way, a possible solution is:

integral <- function(n, mu, s, pi0, pi1) {

  C <- (2 * pi)^(-n/2) 
  C * integrate(f = function(sigmavec) sapply(sigmavec, function(sigma) {
    integrate(f = function(delta) {

      exp(-n/2 * ((mu / sigma - delta)^2 + (s / sigma)^2)) * pi1(delta)

    }, lower = -Inf, upper = Inf)$value

  }) * pi0(sigmavec) / (sigmavec^n), lower = 0, upper = Inf)$value

}

# Tests
integral(n = 1, mu = 0, s = 1, pi0 = dnorm, pi1 = dnorm)
# [1] 0.0473819
integral(n = 1, mu = 0, s = 1, pi0 = function(sigma) 1/sigma, pi1 = dcauchy)
# [1] 0.2615783