Question

Suppose x is a random variable with density f(x). suppose we wish to sample from the distribution of x|x ≥ a where p(x ≥ a) > 0. show how we can accomplish this with rejection sampling with proposal density f. what does this algorithm simplify to?

Answer 1

To sample from the distribution of x given x ≥ a with rejection sampling, draw samples from f(x) and accept if they're ≥ a. The process simplifies with uniform f(x) as outcomes within [a, b] are equally likely and no adjustment to the proposal density is needed.

Step-by-step explanation:

To accomplish sampling from the distribution of x given x ≥ a using rejection sampling with proposal density f(x), you follow a specific process. First, you draw a sample from the original density f(x). After drawing a sample, you check if this sample is greater than or equal to a. If it is, you accept the sample; if not, you reject it and draw another sample. This process until a sample is accepted is repeated.

The area under the probability density function (pdf) for x ≥ a is given by P(X > x), which can be simplified as 1 − P(X < x) for continuous distributions. Since you're sampling from f(x), which is uniform, the process simplifies because each outcome within a ≤ x ≤ b is equally likely. The density of the truncated distribution, f(x|x ≥ a), becomes the original density f(x), normalized over the interval [a, b].

The rejection sampling method simplifies because there's no need to adjust the proposal density—it is already the desired density over the accepted range. In essence, you're just ignoring any samples less than a.