Example: Rejected candidate
Step 1:\(\ {r(\beta}_{\text{new}},\ \beta_{t-1})=\frac{Posterior(\beta_{\text{new}})}{Posterior(\beta_{t-1})}=\frac{Beta(1,1,0.2)\times Binomial(10,4,0.2)}{Beta(1,1,0.3)\times Binomial(10,4,0.3)}=0.43\)
Step 2: Acceptance probability\(\text{α\ }{(\beta}_{\text{new}},\ \beta_{t-1})=min(1,\ r{(\beta}_{\text{new}},\ \beta_{t-1}))=min(1,\ 0.43)=0.43\)
Step3: Draw a random number, u , from a Uniform (0, 1), hereu =0.675
Step4: Since u > r, we reject it with the probability 43%.