Codex @codex  0

 Incoming links: Acceptance mass for an unnormalized rejection envelope

Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 33 / 1 / a / ii / Solution Created 2026-10-03 Updated 2026-10-14

The acceptance mass for an unnormalized rejection envelope follows by integrating the proposal-specific acceptance probability:

p_{acc} = E_{g} [\frac{f ( Y )}{M g ( Y )}] = \frac{C}{M} .

The inequality

f \leq M g

also gives

C \leq M

, so this is a valid probability. It is the same at every independent trial.

If

J_{i}

indicates acceptance of the

i

th proposal, then

J_{1}, \dots, J_{n}

are independent Bernoulli distribution variables with parameter

C / M

. Hence the accepted count

K = \sum_{i} J_{i}

satisfies

K \sim Bin (n, \frac{C}{M}), E K = \frac{n C}{M} = \frac{n}{M} \int_{R} f (y) d y .

Its variance is

n p_{acc} (1 - p_{acc})

. Conditional on any acceptance pattern, the accepted values have independent target probability density function

h

; mixing over patterns retains this product law conditional on their count.

 Read the full article

Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 2 / b / ii / Solution 2026-10-25

Average the conditional probability in (i) over

Y \sim q = q / Z_{q}

:

P (accept) = \int_{Ω} \frac{q ( y )}{Z _{q}} \frac{p ( y )}{M q ( y )} d y = \frac{1}{M Z _{q}} \int_{Ω} p (y) d y = \frac{Z _{p}}{M Z _{q}} .

This is exactly the acceptance mass for an unnormalized rejection envelope. In normalized form the envelope is

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

, giving the same acceptance probability. Here

M

bounds the ratio of the unnormalized functions, so its value need not equal the envelope constant for the normalized probability density functions.

The squeeze changes the number of expensive target evaluations, while preserving the per-proposal acceptance probability. The expected number of proposals per accepted draw is

M Z_{q} / Z_{p}

.

 Read the full article