Einstein-Hilbert action as an effective field theory I have read the following statement: 

"The Einstein-Hilbert action can be seen as an effective field theory: setting $g_{\mu\nu}= \eta_{\mu\nu}+h_{\mu\nu}$, one gets a free Lagrangian for the field $h_{\mu\nu}$ plus interactions."

Now, how is it possible to define all the interaction terms?
EDIT: I ask apologize if my statement was appeared unclear: I have created a similar question which is more precise; please don't continue to consider seriously this one. 
 A: The Einstein-Hilbert action for an $n$-dimensional space-time, $\mathcal M$, is given by,
$$S = \frac{2}{\kappa^2}\int_{\mathcal M} d^n x \, \sqrt{|g|} \, \mathcal R$$
where $\mathcal R$ is the Ricci scalar associated to the metric, $g_{\mu\nu}$ and $\kappa = \sqrt{32 G_N}$. We can linearise the theory, as you have noted, via $g_{\mu\nu} \to \eta_{\mu\nu} + \kappa h_{\mu\nu}$. Since $\mathcal R$ also depends upon the inverse metric, and the inverse in terms of $h_{\mu\nu}$ is an infinite series, the expansion of $S$ consists of an infinite number of terms, including self-interactions of the gauge field:
$$\mathcal L = \mathcal L^0 + \kappa \mathcal L^1 + \kappa^2 \mathcal L^2 + \dots$$
where the first few terms are,
$$\mathcal L^0 = -\frac14 \partial_\mu h \partial^\mu h + \frac12 \partial_\mu h^{\sigma\nu} \partial^\mu h_{\sigma\nu},$$
$$\mathcal L^1 = \frac12 h^{\alpha}_\beta \partial^\mu h^\beta_\alpha \partial_\mu h - \frac12 h^\alpha_\beta \partial_\alpha h^\mu_\nu \partial^\beta h^\nu_\mu - h^\alpha_\beta \partial_\mu h^\nu_\alpha \partial^\mu h^\beta_\nu + \frac14 h \partial^\beta h^\mu_\nu \partial_\beta h^\nu_\mu + h^\beta_\mu \partial_\nu h^\alpha_\beta \partial^\mu h^\nu_\alpha - \frac18 h \partial^\nu h \partial_\nu h$$
where $h := h^\mu_\mu.$ As you can see, we acquire terms of the form, $\mathcal L \sim h^m (\partial h)^n$ which correspond, in terms of Feynman diagrams, to self-interactions of the graviton. We can also identify a kinetic term for the theory, to extract the propagator. This theory can be coupled to others, such as a scalar, fermion, massive vector boson and the electromagnetic field.
In all such cases it is a non-renormalisable theory, but nevertheless as you have stated can be viewed as an effective field theory.

Take the Lagrangian to zeroth order in $h$ for simplicity, that is,
$$\mathcal L^0 = -\frac14 \partial_\mu h \partial^\mu h + \frac12 \partial_\mu h^{\sigma\nu} \partial^\mu h_{\sigma\nu}.$$
Under a diffeomorphism, $x^\mu \to \xi^\mu(x)$, infinitesimally the metric changes as $h_{\mu\nu} \to h_{\mu\nu} + \mathcal L_\xi h_{\mu\nu}$ which is explicitly, $\delta h_{\mu\nu} = \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu$. I now encourage you to apply this to the Lagrangian to check its behaviour under diffeomorphisms.
