# MCMC Metropolis-Hastings sampler - Estimation of multiple parameters

First time that I ask a question on this platform! Here I go... I have a dataset with two random variables X1 and X2 and an output Y which comes from a discrete Weibull distribution. I've been trying to build a Metropolis-Hasting sampler to sample the following regression parameters for a discrete Weibull regression (reference: https://www.tandfonline.com/doi/abs/10.1080/02664763.2017.1342782?journalCode=cjas20): $$\theta = (\theta_0, \theta1, \theta2)$$ $$\gamma = (\gamma_0, \gamma1, \gamma2)$$ Both theta and gamma have a Laplace prior with two additional parameters lambda and tau which each also have a prior distribution (hyperprior) being the InvGamma(shape = 2, rate = 1) distribution. $$\lambda, \tau$$

My questions:
1. If I want to use a multivariate normal distribution as my proposal distribution, should I include all 8 parameters in this distribution? Generating random draws with a good acceptance rate seems hard...
2. I've set a covariance matrix with all diagonal elements equal to 0.001 to start off. Once the chain starts running, I update this covariance matrix by the covariance of the parameters drawn from the proposal distribution. Would this be a correct approach to update my stepsize while the MCMC chain is running? Or are there better ways?
3. When looking at the acceptance probability: $$\alpha = min(1, \frac{l(x,y |\pi_{prop}) p(\pi_{prop})g(\pi_{curr}|\pi_{prop}))}{l(x,y |\pi_{curr}) p(\pi_{curr}) g(\pi_{prop}|\pi_{curr})} )$$ (with pi: all parameters) Would it be correct to use the exp(log(...)) trick to have log likelihoods and avoid likelihoods of zero (underflow)? With non-informative priors and for a symmetric proposal distribution, I believe the prior and g() probabilities would cancel out?
Thanks!