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.
Best Answer
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!!