Question

In rejection sampling, if the proposal density multiplied by the bounding constant is significantly larger than the target density across the support, what is the direct mathematical consequence for the algorithm&#x27;s efficiency?

Accepted Answer

In rejection sampling, the efficiency of the algorithm is inversely proportional to the bounding constant, which is the value M that satisfies M times the proposal density q(x) being greater than or equal to the target density p(x) for all x. The probability of any single generated sample being accepted is equal to one divided by M. When the product of the proposal density and the bounding constant is significantly larger than the target density, it implies that the bounding constant M is very large. As M increases, the acceptance rate, which is the ratio of the target density area to the proposal density area multiplied by M, decreases toward zero. Consequently, the algorithm must discard a larger proportion of proposed samples, leading to a direct mathematical consequence of lower computational efficiency because the expected number of iterations required to generate a single valid sample from the target distribution increases linearly with M.

In rejection sampling, if the proposal density multiplied by the bounding constant is significantly larger than the target density across the support, what is the direct mathematical consequence for the algorithm's efficiency?

In rejection sampling, the efficiency of the algorithm is inversely proportional to the bounding constant, which is the value M that satisfies M times the proposal density q(x) being greater than or equal to the target density p(x) for all x. The probability of any single generated samp....

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In rejection sampling, if the proposal density multiplied by the bounding constant is significantly larger than the target density across the support, what is the direct mathematical consequence for the algorithm's efficiency?

In rejection sampling, the efficiency of the algorithm is inversely proportional to the bounding constant, which is the value M that satisfies M times the proposal density q(x) being greater than or equal to the target density p(x) for all x. The probability of any single generated samp....