A primer on filtering

Here, I discuss the core of the filtering idea in a relatively simple language. I will not introduce particle filters here but at the end you should have a really solid idea about what they are aiming at. In the following, I assume some familiarity with probability densities and fundamental rules (e.g. marginalisation or conditional independence or difference between a random variable and its realisation).


A simple bound for optimisation using a grid

If I give you a function on $[0,1]$ and a computer and want you to find the minimum, what would you do? Since you have the computer, you can be lazy: Just compute a grid on $[0,1]$, evaluate the grid points and take the minimum. This will give you something close to the true minimum. But how much?


An $L_2$ bound for Perfect Monte Carlo

Monte Carlo methods are widely used for estimating expectations of complicated probability distributions. Here I provide the well-known $L_2$ bound.