  We derive an effective spin-1/2 exchange model for non-Kramers Tb$^{3+}$
states in the pyrochlore Tb$_2$Ti$_2$O$_7$. The four anisotropic
nearest-neighbour exchange constants, as well as next-neighbour exchange
constants are derived for the effective model. This work goes beyond the
independent tetrahedra model by considering all nearest-neighbour exchange
paths on the pyrochlore lattice. Estimates of the exchange constants reveal
that Tb$_2$Ti$_2$O$_7$ is described by a quantum spin ice Hamiltonian.
