  We compute a family of scalar loop diagrams in $AdS$. We use the spectral
representation to derive various bulk vertex/propagator identities, and these
identities enable to reduce certain loop bubble diagrams to lower loop
diagrams, and often to tree-level exchange or contact diagrams. An important
example is the computation of the finite coupling 4-point function of the
large-$N$ conformal $O(N)$ model on $AdS_3$. Remarkably, the re-summation of
bubble diagrams is equal to a tree-level contact diagram: the
$\bar{D}_{1,1,\frac{3}{2},\frac{3}{2}} (z,\bar z)$ function. Another example is
a scalar with $\phi^4$ or $\phi^3$ coupling in $AdS_3$: we compute various
4-point (and higher point) loop bubble diagrams with alternating integer and
half-integer scaling dimensions in terms of a finite sum of contact diagrams
and tree-level exchange diagrams. The 4-point function with external scaling
dimensions differences obeying $\Delta_{12}=0$ and $\Delta_{34}=1$ enjoys
significant simplicity which enables us to compute in quite generality. For
integer or half-integer scaling dimensions, we show that the $M$-loop bubble
diagram can be written in terms of Lerch transcendent functions of the
cross-ratios $z$ and $\bar z$. Finally, we compute 2-point bulk bubble diagrams
with endpoints in the bulk, and the result can be written in terms of Lerch
transcendent functions of the AdS chordal distance. We show that the similarity
of the latter two computations is not a coincidence, but arises from a vertex
identity between the bulk 2-point function and the double-discontinuity of the
boundary 4-point function.
