I am having a lot of difficulty understanding how to apply the algorithm to a real scenario.
The part that confuses me is that we are looking for a target distribution (the real distribution of our parameters), but we somehow know a kernel distribution that is proportional to the target distribution, since we use it to compute the acceptance ratio. How do we find/define that kernel distribution?
I would like to understand this through a concrete example. Let's say I have real-life data recorded from a system that is modelled by a PDE with 5 parameters. My goal is to create probability distributions of each of those parameters. I understand the idea of the MH-MCMC algorithm, but when it comes to computing the acceptance ratio (which requires the kernel distribution), I don't understand how we define it.
I might be missing something obvious due to my lack of statistics background.