I am confused with metropolis hastings. This is a simple question. In the metropolis hastings, it is assumed that we know the un-normalised posterior, $\pi(x)$. We can obtain the density by normalising $\pi(x)$. No doubt, if you have a closed-form expression of $\pi(x)$, this can be difficult because you still evaluate the integral $\int \pi(x)$. If we can evaluate this integral or $\int f(x)\pi(x)$ for some function, $f$, we can obtain inference on the posterior. I am guessing this is what motivates the use of Met-Hastings because we would have difficulty evaluating this integral.
This is where I am confused. Is this the only reason why we apply MCMC using Met Hastings? Is there any way to not know the density up to being able to "evaluate", and still sample from an "unknown" posterior.
More specifically, if we have a state space model (hidden markov model) and we know the process model $x_t=f(x_{t-1})+\epsilon_t$ and $y_t=g(x_{t-1})+w_t$, where $f$ and $g$ are non-linear operators, $x_t$ is the state vector at time $t$ and $\epsilon_t$ and $w_t$ is some sort of stochastic error which can be modelled. Can we construct the un-normalised posterior $\pi(x_{0:t}|y_{0:t})$ from knowing these information and apply met-hastings?
