Schematic illustration of bucketing. Rejection sampling as per Fig. 1 is inefficient if the upper bound B significantly overestimates the probability distribution function p(x), necessitating a large number of evaluations of p(x) per accepted sample. To increase efficiency, the domain D can be divided into disjoint subdomains D₁,…,D_v (shaded with different colours), henceforth referred to as ‘buckets’. Then, separate upper bounds B₁,…,B_v (solid coloured lines) can be estimated for the individual buckets. To sample a value from p(x), first choose a bucket J ∈ {l, … ,v} with relative probability |D_J|/|D| using discrete inverse transform sampling. Then, sample a value uniformly distributed in bucket D_J. Evaluate and accept the sample with a probability of , and repeat the entire process until a sample has been accepted. If the per-bucket bounds B_J are lower than the global upper bound, fewer samples are rejected, and hence fewer evaluations of p(x) are required.