MCMC Metropolis-Hastings initial values my posterior values that I obtained via Metroplis-Hasting are always around my initial values. For instance if I chose $\theta_0 =(1,2)$ my posterior values, after either taking mean or median, are around (1,2)
Any idea how to get the MH to explore more values?
For the proposal transition I am using T-distribution with  c$\Sigma$ covariance. constant c as suggested by Gelman. I tried to get the empirical bayes estimator and used them as initial values but it does not work.
CODE: USING R
     Trying to maximized log-likelihood function. ββ is of dimenssion 2
ββ[i,]   # Current value
log_post_curr= lik.b(ββ[i,])   # current log-likelihood values 
ββ[i+1,] =rmvt(n=1,delta=ββ[i+1,],ΣΣ=sig,df=3) : # Generating new value from      T-distribution centralized at the current value with known covariance matrix. df: degree of freedom 

log_post_prop= lik.b(ββ[i+1,]) # log-likelihood value for the proposed ββ

log_prop_curr= log(dmvt(ββ[i,],delta=ββ[i,],ΣΣ=sig,df=3))   # log T-value for the current ββ

log_prop_prop= log(dmvt(ββ[i+1,],delta=ββ[i+1,],ΣΣ=sig,df=3)) # log T-value for the proposed ββ

log_αα = log_prop_curr+log_post_prop - log_prop_prop-log_post_curr # log of the acceptance ratio

ϵϵ[i] = min(1,exp(log_αα))  # Acceptance ratio

u = runif(1)      # Generating a uniform value

if ((u <= ϵϵ[i]))

{
ββ[i,] <- ββ[i+1,]

counter=counter+1     # counting the number of accepted ββ

}

 A: There are several coding issues:
1.The line of code
ββ[i+1,] =rmvt(n=1,delta=ββ[i+1,],ΣΣ=sig,df=3)

seems to show that the proposed value ββ[i+1,] is centred at itself: delta=ββ[i+1,] instead of being centred at the previous value delta=ββ[i,]
Similar corrections apply to the lines
log_prop_curr= log(dmvt(ββ[i,],delta=ββ[i,],ΣΣ=sig,df=3))   
# log T-value for the current ββ

log_prop_prop= log(dmvt(ββ[i+1,],delta=ββ[i+1,],ΣΣ=sig,df=3)) 
# log T-value for the proposed ββ

2.In addition, if the condition u <= ϵ[i]is not satisfied the previous value should be allocated to the new value
ββ[i+1,] <- ββ[i,]

while, if the condition u <= ϵ[i]is satisfied, the previous value should not be replaced by the new value:
β[i,] <- β[i+1,]

3.As an aside, although there is nothing wrong theoretically, you do not need to exponentiate the log Metropolis ratio
ϵϵ[i] = min(1,exp(log_αα))  # Acceptance ratio

as it is more stable and overflow-robust to compare a log uniform to the log-ratio:
if ((log(u) < log_αα))

