A note on estimating chaotic systems

This is a short note unlike other posts but it is necessary: Recently I came across with some complex systems presentations in Youtube. Most of the scientists, especially complex system scientists, treat chaotic dynamical systems as mysterious objects. The models are beautiful (so you should be careful around them)  and they are not that difficult to deal with in real world. The typical example you would hear that a chaotic system is extremely sensitive to the uncertainty in the initial condition. That's true: If I give you the initial condition of a deterministic chaotic system with a machine epsilon uncertainty along with the exact dynamics, the trajectory you would predict will differ from the real one vastly after a while. This, then they say, can be counted as the evidence of undecidability in chaotic systems.


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.