 Solution

ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2023/2/b/iii/solution

C

, (i) gives the joint acceptance probability

P (Y \in C, accept) = \int_{C} \frac{q ( y )}{Z _{q}} \frac{p ( y )}{M q ( y )} d y = \frac{1}{M Z _{q}} \int_{C} p (y) d y .

Divide by

Z_{p} / (M Z_{q})

from (ii). The resulting conditional probability is

P (Y \in C ∣ accept) = \frac{1}{Z _{p}} \int_{C} p (y) d y = \int_{C} p (y) d y .

Thus every accepted value has target probability density function

p

. Repeating rejected trials with fresh independent random variables does not change this law: the contribution from acceptance on trial

j

is multiplied by

(1 - Z_{p} / (M Z_{q}))^{j - 1}

, and summing that geometric series gives the same conditional law. Independent repetitions produce independent target draws.

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