Why Gibbs sampling?

I am new to Gibbs sampling and sampling in general, so here is a basic question. I am reading this tutorial. Equation (40) is our complicated joint probability and equation (49) the less complicated conditional probability. What is the main obstacle in sampling from (40) that calls for using (49)? After all, for every value of the parameters and variables we can evaluate it using a computational softwar, say R, cannot we?

For completeness I write below formula (40) and (49) :

$\mathrm{P}(\mathbb{C},\mathbf{L},\theta_0,\theta_1;\pmb\mu) =\frac{\Gamma(\gamma_{\pi0}+\gamma_{\pi1})\Gamma(C_1+\gamma_{\pi1})\Gamma(C_0+\gamma_{\pi0})}{\Gamma(\gamma_{\pi1})\Gamma(\gamma_{\pi0}) \Gamma(N+\gamma_{\pi0}+\gamma_{\pi1})}\times\Pi_{i=1}^V\theta_{1,i}^{\mathcal{N}_{\mathbb{C}_1}(i)+\gamma_{\theta_i}-1}\Pi_{i=1}^V\theta_{0,i}^{\mathcal{N}_{\mathbb{C}_0}(i)+\gamma_{\theta_i}-1}.\;\;\;(40)$

And

$\mathrm{P}(\mathbf{L}_j=x|\mathbf{L}^{-j},\mathbb{C}^{-j},\theta;\pmb\mu)=\frac{C_x+\gamma_{\pi_x}-1}{N+\gamma_{\pi1}+\gamma_{\pi0}-1}\Pi_{i=1}^V\theta_{x,i}^{\mathbf{W}_{ji}}.\;\;\;\;\;(49)$

• Please provide the details from the paper that avoid members of this forum to have to read the paper to understand your question. – Xi'an Nov 3 '16 at 21:16
• To know the value of the density at any given parameter is not sufficient to derive a sample from that density, in practice. – Xi'an Nov 3 '16 at 21:19
• You might find this article useful: medium.com/@aliaksei.mikhailiuk/… – Aliaksei Mikhailiuk Apr 27 at 15:46