Recently I have read a paper, in a part of that the authors try to sample from the distribution $p(\alpha|\tau )=P(\tau | \alpha)\times p(\alpha | \alpha_0)$ where $\tau$ is some evidences in the problem. They propose to generate $N$ samples from the prior, $p(\alpha | \alpha_0)$, call them $\left \{ \alpha^{i} \right \}_{i=1,...,N}$ and then estimate the posterior $\alpha$ using this formula: $$ \hat{\alpha}=\sum_{i=1}^{N}w_i\alpha^{i} $$ where: $$ w_i=\frac{p(\tau|\alpha^{i})P(\alpha^i|\alpha_0)}{\sum_i p(\tau|\alpha^{i})P(\alpha^i|\alpha_0)} $$
I am skeptical of their method. actually I think that their weighting schema is wrong because they double count the weakness of samples by importing $P(\alpha^i|\alpha_0)$ in their weight calculation.Actually I think that the weight calculation should be : $$ w_i=\frac{p(\tau|\alpha^{i})}{\sum_i p(\tau|\alpha^{i})} $$
I would be apperciated if give me some hints of what is wrong with my reasoning?