I have a univariate random variable $x$ whose density has the following form

$$p(x)\propto \int f(x,y,z) \,\mathrm{d}y \, \mathrm{d}z.$$

And the support for $x$ is bounded, let's say $-1< x < 1$.

I want to directly sample from this density, no MCMC. The bivariate integration could be done using numerically evaluated for a specific $x$ value. I am pretty sure that $p(x)$ is uni-modal but it is not necessarily to be log-concave so adaptive rejection sampling might not work here. Is there any sampling algorithm that works for uni-modal univariate distribution with bounded support?

Any suggestions?

  • $\begingroup$ You can always use straightforward rejection sampling if you can find a constant $c \geq \sup_x p(x)$. However, the computational issue with having to perform a bivariate numerical integration for every proposal step seems pretty important no matter what. Can you write out the three conditional densities and construct a Gibbs sampler? That's almost certain to be a lot faster than bivariate numerical integration at every step. $\endgroup$ – jbowman Oct 1 '17 at 15:50
  • $\begingroup$ Thanks, I can certainly do a data augmentation and Gibbs but I want to avoid doing that. I feel there could be something more efficient given that x is bounded and the distribution is uni-modal. $\endgroup$ – Bayesric Oct 1 '17 at 16:29
  • $\begingroup$ Is $f$ itself a density (or proportional to one)? In that case $p$ would be the marginal density of $x$ and you could sample from it by sampling from $f$ and simply ignoring $y$ and $z$. $\endgroup$ – whuber Oct 1 '17 at 17:29

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