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Through substitution and rearranging terms, we can write the target posterior density as 

<img src='content_image/323249.jpg'>

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 

<img src='content_image/323248.jpg'>

<img src='content_image/323247.jpg'>

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 

<img src='content_image/323246.jpg'>

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. 

<img src='content_image/323245.jpg'>

The method involves sampling a u from an approximation to p ( u | y ) , and then sampling from 

