The posterior distribution is proportional to the likelihood times the prior distribution $p(\theta|D) \propto p(D|\theta)p(\theta)$. Computing $p(D|\theta)p(\theta)$ would give the un-normalised posterior distribution and the normalising constant gives rise to a valid probability distribution of 1. However, the normalising constant is just a scaling factor for the height of the distribution and it does not in any way affect the density of the distribution because the height of the posterior are all scaled proportionately. What is the purpose of techniques like MCMC sampling to estimate the posterior distribution ?
Computing p(D|θ)p(θ) would give the un-normalised posterior
That sounds as if MCMC was an alternative to compute the posterior distribution. However, MCMC starts to shine whenever p(D|θ)p(θ) is to complicated and cannot be computed (with reasonable effort).
Bayes books often start with simple cases where there is a conjugate prior and the result can be computed in closed form. In many real life cases such a conjugate prior does not exist or is for other reasons not feasible. Then Monte Carlos simulation can still produce usable results. It is not primarily a technique to determine scaling factors.
MCMC methods approximate the posterior by drawing samples from it. Using these samples, one can
- form an empirical approximation to the true posterior
- calculate certain statistics like posterior mean/variance/moments
- numerically calculate hard integrals
and so on. So, finding the posterior might be seen a byproduct of what you really want to achieve via MCMC.
Analytically, if you have the unnormalized version of a density, i.e. $p(x)=cq(x)$ where $c$ is unknown, integral calculations (moments) will include $c$ as well: $$\mathbb E[X]=\int xp(x)dx=c\int x q(x)dx$$
Even if you can easily evaluate $\int x q(x)dx$, you still need to find $\int q(x)dx$.