# Sampling with Metropolis-Hastings

In Metropolis-Hastings sampling, if every draw of my proposal distribution (Q) is independent from the previous draw, is the convergence to the stationary distribution still guaranteed? To be more precise, every draw is a fresh draw from the proposal (Q) and the acceptance rate is calculated as follows:

 a = min{1,(P(X')Q(X))/(P(X)Q(X'))}


Where X' is the new drawn sample, and X is the current sample, and P is the actual distribution. Also, can we still call this an MCMC sampling technique?

Nothing forbids setting the proposal distribution $Q(x_{\mathrm{new}} \mid x_{\mathrm{old}})$ to be independent of $x_{\mathrm{old}}$ so the proof for detailed balance condition in Metropolis-Hastings directly holds for this proposal distribution.
Note that to guarantee convergence to the correct stationary distribution, we must further assume that the proposal $Q$ has positive probability for all regions where $P$ has positive probability.