Hi.

I have two questions about the random walk metropolis-hasting algorithm.

We have a proposal distribution in this algorithm such as q(x^{\star} / x^{0}). q is a proposal distribution. My first question is that how this proposal distribution for example Normal distribution has relation to the previous sample or x^{0}.

For example we can select a new sample from a normal distribution as a proposal distribution but why in the literature this new sample or x^{\star} is subject to the previous sample x^{0}?

The acceptance ratio is :

α(x^{\star}/x^{0})=min{\frac{π(x^{\star})\times q(x^{\star}/x^{0})}{π(x^{0})\times q(x^{0}/x^{\star})} ,1}

The numerator is posterior density in x^{\star} and denominator is posterior density in x^{0} ??