I am confused by the dirac delta function in the formulation of monte carlo sampling. http://www.cs.ubc.ca/~arnaud/doucet_johansen_tutorialPF.pdf, for instance, defines in section 3.1 page 8 the marginal as $\pi(x_k)=\frac{1}{N}\sum_{i=1}^N\delta_{X^i_k}(x_k)$.
What does it mean here? That the density of $x_k$ is the average of the densities of the samples equal to $x_k$?
Thanks in advance.
This is much simpler than you think. You need to understand that what underlies probability is integration. So, if you are interested in the marginal of the $k$th coordinate, what you really are looking for is knowing the value of $\int_A \pi(x_k) dx_k$ for all (measurable, whatever) $A$. How on earth would you estimate it? The solution is simple: you have a bunch of samples $X^{i} \in \mathbb{R}^d$. You estimate the above integral by counting how many times you have $X^{i}_k \in A$ and then divide by the number of samples you have - $N$. That is it.
In the notation you use, $\delta_{X_k^{i}}( \cdot)$ is a "function" s.t. if you integrate it over a set $A$, the result is 1if $X_k^i$ is in $A$ and zero otherwise: $\int_A \delta_{X_k^i}(x) dx = 1$ iff $X_k^i \in A$ and $0$ otherwise.
$$ \int_A \frac{1}{N}\sum_i \delta_{X_k^i}(x)dx = \frac{1}{N} \sum \int_A \delta_{X_k^i}(x )dx\\ =\frac{1}{N} \cdot \text{number of times $X_k^i$ is in $A$ } $$
just like our intuition would expect!!
Dirac delta function, is not exactly a function, but with a little abuse of notation you can consider $\delta_{x_0}(x)$ to be a function getting a non-zero value only at point $x=x_0$ and being a probability distribution, i.e. $\int_{-\infty}^{+\infty}=1$. Now you can extend this definition to multi-dimensional case with the same requirements. This is the case for Doucet et. al. tutorial. They are estimating the main distribution by a union of multi-dimensional dirac functions.
