Accept-reject algorithm (acceptance-rejection method) or reject sampling is a simple and general simulation method to decide observations with or without a trait from the probability of a distribution. In this way, we can convert a probability into a dichotomous condition (i.e. yes or no). Basically, there are three steps:
- Step 1. Generate Y from density g [Y = f(x), the pdf of f(x) is the target distribution]
- Sample a point (an x-position) from the proposal density distribution (g) and draw a vertical line at this point, get the density (an y-position) [X ~ g(x)]. The density function of Y has a upper, a constant c, and c is >=1.
- Step 2. Generate U from the uniform distribution on the interval (0, cg(x)) [U = cg(x), the pdf of cg(x) is the proposal distribution]
- Sample uniformly along in the range of x-position (i.e. uniformly from 0 to the maximum of the probability density function) [U ~ runif(0, 1)]
- Step 3. If U <= Y, then set Y = X ("accept"), else repeat Steps 1 and 2
Pr(accept|X) = f(x)/cg(x)
Pr(X) = g(x)
Pr(accept) = 1/c
therefore, Pr(X|accept) = f(x)
Example: Stata simulation and define the event
clear
set seed 770488
set obs 1000
gen x = runiform() - .5
gen z = runiform() - .5
gen xb = x + 8*z
gen y = 1 / (1 + exp(xb)) < runiform() // y defined as 0 or 1
logistic y x z
set seed 770488
set obs 1000
gen x = runiform() - .5
gen z = runiform() - .5
gen xb = x + 8*z
gen y = 1 / (1 + exp(xb)) < runiform() // y defined as 0 or 1
logistic y x z
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