If we consider a set of parameters $\boldsymbol{\theta}$ that we want to infer we can use Bayes theorem, we start from a prior distribution and we update it using a set of observed data:
$\begin{equation} p(\boldsymbol{\theta}|Y_O)=\dfrac{p(Y_O|\boldsymbol{\theta})p(\boldsymbol{\theta})}{p(Y_O)} \end{equation}$
$Y_O$ is a set of measured data.
Instead of computing the full posterior density function we can use Bayes estimators to get the optimal values of the parameters. For example, a mean squared cost function error Bayes estimator will give the following prediction:
$\hat{\boldsymbol{\theta}}=\int {\boldsymbol{\theta}}p({\boldsymbol{\theta}}|Y_O)d{\boldsymbol{\theta}}$
The above equation is not easy to compute, it's a multidimensional integral and it requires the computation of the posterior distribution.
Can someone explain to me how computing the computation of $\hat{\boldsymbol{\theta}}$ using the formula above is computationally cheaper than computing the full posterior distribution? Knowing that in both cases we have to resort to numerical computations and not to any analytical formula.