#----------------------------------------------------------------
#  rescaled likelihood ("{\cal R}") in the case of background
#  -> effect on a prior distribution of r
#
#  GdA, May 2016
#-----------------------------------------------------------------

# prior, modelled as a lognormal
mlog <- -3
slog <- 1
f0 <- function(r) dlnorm(r, mlog, slog) 

x <- 1
T <- 1
rb <- 1

Rr <- function(r, x, T, rb) exp(-r*T)*(1+r/rb)^x

r <- 10^seq(-4, 1.5, len=201)

plot(r, Rr(r, x, rb, T), ty='l', col='blue', log='xy',
     ylim=c(1e-3, max( c(Rr(r, x, rb, T), f0(r)) ) ) )
points(r, f0(r), ty='l', lty=2, col='red')

# posterior
f <- function(r) Rr(r, x, rb, T) * f0(r)
Z <- integrate (f, min(r), max(r))$value
points(r, f(r)/Z, ty='l', lty=1, col='red')