Through substitution and rearranging terms, we can write the target posterior density as An important restriction on the choice of g ($_{q}$) is that the inequality 0 ≤ F ($_{q}$ | y ) ≤ 1 must hold, at least for any q with a non-negligible posterior density. We discuss this restriction, along with the choice of g ($_{q}$) , in more detail a little later. Next, let u |$_{q}$ , y be an auxiliary variable that is distributed uniformly on ( 0, F ($_{q}$ | y ) $_{p}$($_{q}$ | y )$^{)}$ , so that p ( u |$_{q}$ , y ) = $_{p}$($_{q}$ | y ) F ($_{q}$ | y ) = c$_{1}$ c$_{2}$ L ( y ) g ($_{q}$) . Then construct a joint density of $_{q}$| y and u |$_{q}$ , y , where By integrating Equation 6 over u , the marginal density of $_{q}$| y is Simulating from p ($_{q}$ | y ) is now equivalent to simulating from the target posterior $_{p}$($_{q}$ | y ) . Using Equations 5 and 6, the marginal density of u | y is where q ( u ) = $^{∫}$$_{q}$1 [ u < F ($_{q}$ | y )] g ($_{q}$) d$_{q}$ . This q ( u ) function is the probability that any candidate draw from g ($_{q}$) will satisfy F ($_{q}$ | y ) > u . The sampler comes from recognizing that p ($_{q}$ , u | y ) can be written differently from, but equivalently to, Equation 6. The method involves sampling a u from an approximation to p ( u | y ) , and then sampling from 5 p ($_{q}$ , u | y ) = p ($_{q}$ | u , y ) p ( u | y ) (11) p ( u | y ) = ∫ q $_{p}$($_{q}$ | y ) F ($_{q}$ | y ) 1 [ u < F ($_{q}$ | y )] d$_{q}$ (8) = c$_{1}$ c$_{2}$ L ( y ) ∫ q 1 [ u < F ($_{q}$ | y )] g ($_{q}$) d$_{q}$ (9) = c$_{1}$ c$_{2}$ L ( y ) q ( u ) (10) p ($_{q}$ | y ) = $_{p}$($_{q}$ | y ) F ($_{q}$ | y ) ∫ F ($_{q}$ | y ) 0 du = $_{p}$($_{q}$ | y ) (7) p ($_{q}$ , u | y ) = $_{p}$($_{q}$ | y ) F ($_{q}$ | y ) 1 [ u < F ($_{q}$ | y )] (6) $_{p}$($_{q}$ | y ) = F ($_{q}$ | y ) · g ($_{q}$) · c$_{1}$ c$_{2}$ · L ( y ) (5)