Example: Accepted candidate
Step 1:\({r(\beta}_{\text{new}},\ \beta_{t-1})=\frac{Posterior(\beta_{\text{new}})}{Posterior(\beta_{t-1})}=\frac{Beta(1,1,0.4)\times Binomial(10,4,0.4)}{Beta(1,1,0.5)\times Binomial(10,4,0.5)}=1.19\)
Step 2: Acceptance probability\(\text{α\ }{(\beta}_{\text{new}},\ \beta_{t-1})=min(1,\ r{(\beta}_{\text{new}},\ \beta_{t-1}))=min(1,\ 1.19)=1\)
Step3: Draw a random number, u , from a Uniform (0, 1), here u=0.345
Step4: If u is less than the acceptance probability, the proposed value of \(\beta_{\text{new}}\) will be accepted. Otherwise, we reject\(\beta_{\text{new}}\) and keep\(,\ \beta_{t-1}\). Here we accept it.