I have some problems understanding the definition of Gibbs sampling.
Let us take into consideration a bivariate distribution \begin{equation} \pi(x_1,x_2): S \subset \mathcal{R^2} \rightarrow \mathcal{R} \end{equation}
If I understand correctly, in order to implement the Gibbs sampling on $\pi(x_1,x_2)$, I should be able to have the full analytical expression of the conditionals with respect to $x_1$ and $x_2$. Let us call them $\pi(x_1|x_2)$ and $\pi(x_2|x_1)$ respectively.
Once I have these expressions, I can sample directly -- i.e. numerically -- from them, one at the time, in the way it is stated by the Gibbs algorithm.
There might be cases in which, the normalization for the conditional distributions might not be so easy to calculate. In this case, I would be driven to perform a Metropolis-Hastings sampling instead of direct sampling on the conditionals.
In the case of direct sampling, at each step $k$ I would extract a random value -- for instance $\tilde{x_2}$ -- for a conditional. I would then accept it as the $k$-th value of my chain $x_2^{(k)}$ with regards to its normalized probability $\pi(\tilde{x}_2|x_1^{(k-1)})$.
When the normalization factor is hard to calculate, my idea would be that of proposing a "jump" from the current $(k-1)$th value of $x_2$ to the randomly extracted value $\tilde{x}_2$. I would then accept it with the Metropolis-Hastings acceptance ratio for the appropriate conditional. In the case of a simmetric proposal distribution this ratio would for example be $\alpha = \frac{\pi(\tilde{x}_2|x_1^{(k-1)})}{\pi(x_2^{k-1}|x_1^{(k-1)})}$.
If I were to sample on the conditionals in this way, would the resulting algorithm still be considered an implementation of the Gibbs algorithm?
