I'm using the Bayesian approach to determine a vector $\mathbf y$ of parameters. I will use the Metropolis-Hastings algorithm to numerically determine the probability distribution of each of the sought parameters. As proposal transition kernel $Q(\mathbf y';\mathbf y)$ I would just use a multivariate normal distribution. The simplest thing would be to use a multivariate normal distribution with no correlation at all among the variables, so that the sampling reduces to the independent sampling from many one-dimensional normal distributions.
Are there any disadvantages in such an approach ? Why should I use instead a multivariate normal distribution with covariances $\neq 0$ ? What would be the advantages ?
This is just a quick sketch, but I think it's all accurate.
Suppose you had a model $Y_n = \beta X_n + \epsilon$ where $Y_n \in \mathbb{R}^2$, $X_n \in \mathbb{R}^{2}$, $\epsilon \sim \textrm{N}(0,\sigma I_2)$, and the prior on $\beta$ is $\beta \sim N(0, I_2)$.
Looking at only the covariance, the posterior distribution of $\beta$ is going to be normally distributed with precision matrix $XX^T + I_2$. Suppose that you saw an observation with $X = [.01, 100]$. Then $XX^T = \left[ {\begin{array}{cc} .0001 & 1 \\ 1 & 10000 \\ \end{array} }\right]$ which means that $(XX^T + I_2)^{-1} \approx \left[ {\begin{array}{cc} 1 & 0 \\ 0 & .00001 \\ \end{array}}\right]$.
This means that the posterior distribution of $\beta$ is going to be highly elliptical instead of circular. If you just made standard proposals from a $N(0,I_2)$ distribution, you would have a high probability of going in the direction of the second dimension, which would likely be rejected because of the elliptical shape of the posterior.
The appropriate way to deal with this is to standardize your data, but this is just an example of how using a different proposal could offer some benefit. For example, if you proposed from $N(0,XX^T+I_2)$, this would essentially be equivalent to standardizing your data.