Hence, a set $\{x_{i}, \omega^{i}\}_{i=1}^{N}$ would be an estimation of PDF, $\hat{p}(x)$ . At this point we can invoke dirac delta function,

$ \hat{p}(x) = \sum_{i=1}^{N} \omega^{i} \delta(x-x_{i})$

Let's revisit the R code given there, this time let's draw uniform numbers between $[-2, 2]$ to get 100 $x_{i}$ values. Simply these numbers will indicate the locations on the x-axis, a spike train. For simplicity, let's use Gausian distribution for target PDF, $ \mathcal{N}(0, 1)$. Than, for weights we need to draw numbers using the spike locations. This approach is easier to understand compare to my previous double index notation.

**R Example code**

Above explained procedure is trivial to implement in R.

1 2 3 4 5 6 7 8 9 10 | # Generate locations 100 x locations # out out 1000 points in [-2.0, 2.0] set.seed(42) # Domain where Dirac comb operates Xj = seq(-2,2,0.002) Xi = sample(Xj, 100) # Now generate weights from N(0,1) at those given locations Wi = dnorm(Xi) # Now visualise plot(Xi, Wi, type="h",xlim=c(-2.0,2.0),ylim=c(0,0.6),lwd=2,col="blue",ylab="p") |

**Conclusion**

Above notation introduces second abuse of notation while actually there must be a secondary regular grid that pics $x_{i}$ values using dirac delta in practice. Because, the argument of $\hat{p}(x)$, is in the discrete domain. So a little better notation that reflects the above code would be

$ \hat{p}(x_j) = \sum_{i=1}^{N} \omega^{i} \delta(x_{j}-x_{i})$

The set $x_j$ is simply defined in a certain domain, for example regularly. Hence I only recommend not to introduce dirac delta for explaining a particle approximation to PDFs for novice students in the class. It will only confuse them even more.

Figure: Spike trains with weights $\hat{p}(x) = \sum_{i=1}^{N} \omega^{i} \delta(x-x_{i})$ |