I'm currently trying to use the Metropolis-Hastings algorithm to sample from a posterior distribution of the form
$$p(\theta | y ) \propto \prod_{ij} \phi (\theta_{ij}) \times \prod_{i=1}^n \pi_{y_i}(x_i|\theta)$$
Where $\phi()$ is the standard normal density function. This posterior is the result of taking the multinomial logit model with multivariate normal prior $N_p (0, I_p)$.
I have taken the proposal distribution to be
$$q(\theta_{new} | \theta_{old}) \sim N_p (\theta_{old}, I_p)$$
As far as I can tell my implementation of the algorithm is correct, but mixing is incredibly poor. The acceptance ratio is consistently on the order of $1\times 10^{-3}$ or worse.
What are some techniques for improving mixing for sampling from a high dimensional posterior? If the dimension was low, I would plot the distribution to get an idea of what proposal distribution would suit it. But when the posterior is high dimensional, this doesn't seem feasible.