In a linearized DSGE model, after reaching a StateSpace model, we’ll usually have some matrices where each element/component is a function of several structural parameters.
Also, in most economics papers, when doing bayesian estimation, namely using a MetropolisHastings algorithm, I see many authors put priors on the parameters (this procedure seems to help in getting an economic interpretation) instead of on the matrices in the StateSpace model and the proposal chosen for the parameters is most of the times a multivariate Tdistribution, where the location parameter of this distribution follows \theta[t]=\theta[t1]+\frac{\partial}{\partial \theta} p(\mathbf{y}\theta[t1]).
I have the following questions:

The likelihood function maybe very ‘complex’, and increasing with the number of parameters and sample size. How does one compute the above derivative for the proposal? Is it feasible to just define p(\mathbf{y}\theta)=\text{Likelihood}(\theta) and then trying to numerically maximise it?

The chosen proposal does not guarantees that the drawn parameters from it will obey the linearized DSGE, let alone the original DSGE, i.e., when drawing from the proposal I may get several inconsistent values. How does one make sure to get the right draws?