# Sampling of beta in bayesian regression (variable selection)

I am sampling a beta using a Gibbs sampling. It is variable selection model. So in different iteration of gibbs different covariates are included to the model (denoted by a variable selection indicator $\gamma$). However, this indicator have high variance ( this is because my covariates are correlated). I use: $\hat\beta = 1/n\sum \beta_{samples}$ as my estimate for further analysis. This turn out to be problematic because if I calculate the explained variance it is negative. On the other hand if I just use a single sample of $\beta_{sample}$ to find explained variance it is correct. What's wrong here?

Do I need to re-estimate $\beta$ for nonzero mean $\gamma = 1/n \sum \gamma_{sample}$? This does not sound correct because this estimate does not correspond to mean of posterior of $\beta$. thanks

I found on thing that helps to reduce the variance in $\gamma$ was to use Rao-blackwelization. Its just taking the expected the posterior of $\gamma$ as final extimate i.e. $\hat\gamma = 1/n\sum_i p_{samples}(\gamma| Data, other-params)$. If I take only those $\beta$ which $\hat \gamma >> 0$, it removes the problem of negative explained variance. However, I think still it is not correct samples from posterior of $\beta$.