Consider the probability function as follows. We want to generate random samples from p(x) through the rejection method.

x 0 1 2
p(x) 1/3 2/5 4/15

Suppose we have two candidate probability functions:

(1)

x 0 1 2
g1(x) 1/3 1/3 1/3

(2)

x 0 1 2
g2(x) 1/2 1/4 1/4

Which candidate probability function is more efficient in the best-case scenario?

Question

Consider the probability function as follows. We want to generate random samples from p(x) through the rejection method.

x	0	1	2
p(x)	1/3	2/5	4/15

Suppose we have two candidate probability functions:

(1)

x	0	1	2
g1(x)	1/3	1/3	1/3

(2)

x	0	1	2
g2(x)	1/2	1/4	1/4

Which candidate probability function is more efficient in the best-case scenario?

BlackTom AI · Accepted Answer

To compare the efficiency of rejection sampling with two candidate proposal distributions, recall that the acceptance probability in a rejection sampler using a proposal g(x) is approximately 1/M where M = max_x [p(x) / g(x)]. A smaller M yields higher best-case efficiency.

First, examine candidate (1) where g1(x) = 1/3 for x = 0,1,2. Compute the ratios:
p(0)/g1(0) = (1/3) / (1/3) = 1,
p(1)/g1(1) = (2/5) / (1/3) = (2/5) * 3 = 6/5 = 1.2,
p(2)/g1(2) = (4/15) / (1/3) = (4/15) * 3 = 12/15 = 0.8.
The maximum of these ratios is M = 1.2. Therefore, the best-case acceptance probability is 1/1.2 ≈ 0.833.

Next, examine candidate (2) where g2(0) = 1/2, g2(1) = 1/4, g2(2) = 1/4. Compute the ratios:
p(0)/g2(0) = (1/3) / (1/2) = (1/3) * 2 = 2/3 ≈ 0.6667,
p(1)/g2(1) = (2/5) / (1/4) = (2/5) * 4 = 8/5 = 1.6,
p(2)/g2(2) = (4/15) / (1/4) = (4/15) * 4 = 16/15 ≈ 1.0667.
The maximum of these ratios is M = 1.6. Therefore, the best-case acceptance probability is 1/1.6 = 0.625.

Comparing the two maxima, M is smaller for the first candidate (1.2 vs. 1.6), which means the first candidate yields a higher best-case acceptance probability and thus greater best-case efficiency in the rejection sampling context.

In other words, the proposal distribution whose largest p(x)/g(x) ratio is minimized provides the better best-case efficiency, and here that occurs for the first candidate. This stems from its uniformity relative to p, which keeps the ratios more balanced and lowers the worst-case ratio across outcomes.

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