How to generate data from noninformative prior?

Question

If we have $$x|\lambda \sim \frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$ with Jeffereys prior $$\lambda \sim \frac{1}{\lambda}$$ How could we generate data $x_1, \cdots, x_n$ from this mixture distribution? I know that if we have an informative prior we could generate $\lambda$ from that prior first then use the $\lambda$ we just got to generate data $x_1, \cdots, x_n$ . How about the noninformative ones?

Thanks~

Answer 1

Your prior is more than uninformative, it is improper on the positive reals so it cannot be proportional to an actual probability density

But if you put limits on $\lambda$ (e.g. $[10^{-6},10^6]$ or something equally arbitrary) then the distribution would become proper and then, as you say, it would be easy enough to generate values, for example with this R code

upperlimit <- 10^6
lowerlimit <- 10^(-6)
numberoflambdas <- 10
numberofxs <- 5
set.seed(1)
lambdas <- lowerlimit * (upperlimit / lowerlimit) ^ 
           runif(numberoflambdas)
simdat  <- matrix(-log(runif(numberoflambdas*numberofxs)) * lambdas,
                  ncol=numberofxs)

which would produce ten simulated $\lambda$s

> lambdas 
 [1] 1.534984e-03 2.920699e-02 7.485837e+00 7.915776e+04 2.631378e-04
 [6] 6.034985e+04 2.168219e+05 8.503026e+01 3.542979e+01 5.513600e-06

and for each $\lambda$, five $x_i$s

> simdat 
              [,1]         [,2]         [,3]         [,4]         [,5]
 [1,] 2.425279e-03 1.036484e-04 1.119994e-03 3.028487e-04 1.134262e-03
 [2,] 5.064822e-02 4.528535e-02 1.494082e-02 1.271427e-02 4.364036e-03
 [3,] 2.810091e+00 3.205519e+00 5.286114e+00 1.831848e+00 6.178171e+00
 [4,] 7.574152e+04 1.642532e+05 1.330515e+05 4.688764e+04 1.114007e+05
 [5,] 6.882914e-05 3.472578e-04 4.986441e-05 1.671997e-04 6.972114e-04
 [6,] 4.210967e+04 5.743027e+04 2.430703e+04 1.427501e+04 1.392837e+05
 [7,] 7.194524e+04 9.352012e+05 4.994922e+04 8.148090e+05 2.495954e+05
 [8,] 6.910272e-01 8.174126e+01 1.892898e+02 6.290170e+01 5.582716e+01
 [9,] 3.427801e+01 4.946617e+00 1.145669e+01 1.103802e+01 1.461415e+01
[10,] 1.388005e-06 5.942468e-06 4.898804e-06 2.024113e-06 4.958708e-06