Which of the following descriptions is correct about the rejection method for sampling from a target density
𝑓
(
𝑥
)
?

Question

Which of the following descriptions is correct about the rejection method for sampling from a target density 
𝑓
(
𝑥
)
?

BlackTom AI · Accepted Answer

The question asks: Which of the following descriptions is correct about the rejection method for sampling from a target density f(x)?

First, I will summarize what rejection sampling is supposed to do in general terms, since the available content focuses on describing the method rather than comparing multiple options:
- We select a candidate density g(x) from which we can sample easily.
- We must find a constant M ≥ 1 such that f(x) ≤ M g(x) for all x in the domain. This ensures that the target density is bounded above by a multiple of the proposal density.
- The sampling procedure then involves drawing X from g(x) and accepting X with probability f(X) / (M g(X)); otherwise, we reject and repeat.
- The choice of g(x) should make sampling feasible and the computation of f(x) and g(x) (and the ratio f(x)/(M g(x))) should be tractable.

Now, assessing the given statement:
- The statement says: 'We need to provide a candidate density g(x) that we can sample from and find the constant M such that f(x) ≤ M g(x). Here g(x) can be any density.'
- The core idea is correct that we need a tractable g(x) and a constant M so that f(x) ≤ M g(x) holds for all x. This is the standard setup for rejection sampling.
- However, the second part—'Here g(x) can be any density'—requires nuance. Not every density g(x) will satisfy the inequality with a finite M, and not every choice of g(x) will yield an efficient algorithm. In practice, we choose g(x) to majorize f(x) up to a finite multiple M and to ensure efficient sampling and computation of the ratio f(x)/(M g(x)). So while the statement captures the essential steps, the caveats about selecting an appropriate g(x) and an appropriate M are important.

To break down why this statement aligns with the rejection sampling framework and where it could be misleading:
- The necessity of sampling from g(x) is accurate; you must be able to generate samples from g(x).
- The requirement to find an M such that f(x) ≤ M g(x) for all x is also accurate, because this guarantees that the acceptance probability f(x)/(M g(x)) lies in [0,1].
- The potential subtlety is the claim that g(x) can be 'any density.' If g(x) cannot majorize f(x) by any finite M, or if M is impractically large, the rejection sampling method becomes inefficient or infeasible. Thus, while many densities g(x) are candidates, the method implicitly imposes the condition that such an M exists and is reasonably small for practical efficiency.

In summary, the described description captures the essential mechanics of rejection sampling (select g, determine M so that f ≤ M g, and use the acceptance-rejection step). The main caveat is that not every density g will satisfy the needed inequality with a usable M, so the choice of g is not arbitrary in practice—the selection must ensure the bound holds with a finite, preferably small, M.

25F-STATS-102C-LEC-1 Quiz3- Requires Respondus LockDown Browser

Which of the following descriptions is correct about the rejection method for sampling from a target density 𝑓 ( 𝑥 ) ?

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Let 𝑓 ( 𝑥 ) = 𝑐 𝑜 𝑠 ( 𝑥 ) , for 𝑥 ∈ [ 0 , 𝜋 / 2 ] . Suppose we want to sample 𝑓 ( 𝑥 ) through the rejection method, with 𝑈 𝑛 𝑖 𝑓 ( 0 , 𝜋 / 2 ) as the candidate density. What is the multiplicative constant M?

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