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We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFT d four point functions and expands them in terms of crossing symmetric combinations of AdS d+ 1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d= 4. We reproduce Feynman diagram results for operator dimensions to O (ϵ 3) rather straightforwardly. This approach also yields new analytic predictions for OPE …