Abstract In this chapter, we extend the basic exploratory techniques of Chapter 9, aiming to answer the perennial questions ‘What ‘s there, and how much? ‘. As so often in algorithm design, there are some nice tricks to be found. Here are some of them. We now consider problems in which the discrete states of a system are assigned different prior probabilities, which cannot usefully be transformed to uniformity. It is possible to choose potential transitions at random, as before. Indeed, many writers recommend just that, using rejections to control the desired non-uniformity, as in Metropolis exploration described in Section 9.4.4. Our view, though, is that most priors are sufficiently simple to be explored efficiently without rejection, and that the insight required to accomplish this is worthwhile.