Does Gibbs sampling MCMC place limitations on the posterior?

Question

I'm starting to learn about Gibbs sampling, having so far only worked with Metropolis-Hastings MCMC, and there's something I haven't grasped yet about the way Gibbs sampling works.

To frame the problem, consider the following posterior distribution:

$$ P(\underline{\theta} | \underline{y}) \propto \exp{\left[\sum_i \left( y_i \ln{F(x_i , \underline{\theta})} - F(x_i , \underline{\theta}) \right) \right]} $$ This is a product of Poisson distributions, and $F(x , \underline{\theta})$ is some known, calculable, non-linear function which cannot be expressed as a product over functions of single parameters, i.e. $$ F(x, \underline{\theta}) \ne f_1(x, \theta_1) f_2(x, \theta_2) \dots f_n(x, \theta_n)$$ This fairly typical of a posterior you might find doing physics data analysis, for example.

Is it possible (or appropriate) to use Gibbs sampling with such a posterior?

The univariate conditional distributions for each parameter can be calculated, but only by evaluating the whole posterior and fixing the appropriate parameters - they do not have separate closed forms.

Thanks!

Answer 1

If you can write the joint density $p(\theta|y)$ in closed form, you can deduce $$p(\theta_i|y,\theta_{-i})\propto p(\theta|y)$$ in closed form. At this stage, the practical possibility in using Gibbs depends on the complexity of the above density and on the computing time requirements.

Note that Gibbs sampling is a specific type of Metropolis-Hastings algorithm. There is nothing magical about it, from the probability $1$ of acceptance to the use of the proper conditional densities.