Intractable posterior - why not use kernel density for the data distribution? In the Bayes rule, it is said that the posterior
$$
   P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}
$$
is intractable, because 
$$
    P(D) = \int P(D,\theta) d\theta
$$
and the latter is often a high-dimensional integral.
See Why is the posterior distribution in Bayesian Inference often intractable?
But this is just one way of computing $P(D)$. There are others? What about instead estimating $P(D)$ using a kernel density
(place a Gaussian or some other lobe at each datapoint, and normalize so it all sums to one). Or simply using delta functions:
$$
 P(x) = \frac{1}{n} \sum_i \delta_{x_i}(x)
$$
This requires touching each bit of data, but that is not intractable.
 A: There are certainly lots of ways to try to numerically estimate high-dimensional definite integrals.  The entire field of high-dimensional numerical integration is devoted to this problem, and it suffers from the dreaded curse of dimensionality.  There are a lot of research papers in this field with a lot of different methods used.  Kernel methods are one method that can be used to obtain approximate integrals (using the delta function would give a terrible approximation for continuous distributions), but I think it is fair to say that the most favoured methods presently used in this field are Monte-Carlo methods (e.g., importance sampling), Markov-Chain Monte-Carlo methods (e.g., Gibbs, Metropolis-Hastings, Hamiltonian MC), and sparse-grid methods.
Most Bayesians make extensive use of Markov-Chain Monte-Carlo (MCMC) methods, and many general pieces of Bayesian software are built on these algorithms.  The Stan package for Bayesian statistics is built on using Hamiltonian Monte-Carlo methods to estimate these integrals.  This is a powerful method that has led to recent improvements in computational power in Bayesian analysis.  I'm not an expert on this stuff myself, but I know it is a very large an complicated field, with lots of methods and lots of literature.
A: Since the marginal density writes as$$m(x)=\int_\Theta f(x|\theta),\pi(\theta)\,\text{d}\theta$$a possible numerical approximation is$$\frac{1}{T}\sum_{t=1}^T f(x|\theta_t)\qquad\theta_1,\ldots,\theta_T\sim\pi(\theta)\tag{1}$$but this Monte Carlo approximation based on simulations does not use kernel estimation. As stressed by Juko Kokkala's comments, the use of a kernel estimator of $m(x)$ would require observations from the marginal, while the classical Bayesian framework only involves observations from $f(x|\theta_0)$ for an unknown $\theta_0$. Plus, (1) is a parametric estimator that converges at the rate $\sqrt{T}$, as opposed to a non-parametric estimator that converges at the rate $
T^{−4/2(d+4)}$ where $d$ is the dimension of $x$.
Note also that an intractable posterior is usually understood as associated with an intractable product 'prior x likelihood' rather than having an unknown normalisation constant $m(x)$, since simulation techniques (like MCMC, importance sampling, &tc.) can bypass this missing term and still deliver. (This issue is also discussed in the post the OP linked to.)
