In a physically based BRDF, what vector should be used to compute the Fresnel coefficient? The well known Schlick approximation of the Fresnel coefficient gives the equation:

$F=F_0+(1 - F_0)(1 - cos(\theta))^5$

And $cos(\theta)$ is equal to the dot product of the surface normal vector and the view vector.

It is still unclear to me though if we should use the actual surface normal $N$ or the half vector $H$. Which should be used in a physically based BRDF and why?

Moreover, as far as I understand the Fresnel coefficient gives the probability of a given ray to either get reflected or refracted. So I have trouble seeing why we can still use that formula in a BRDF, which is supposed to approximate the integral over all the hemisphere.

This observation would tend to make me think this is where $H$ would come, but it is not obvious to me that the Fresnel of a representative normal is equivalent to integrating the Fresnel of all the actual normals.
In Schlick's 1994 paper, &quot;An Inexpensive Model for Physically-Based Rendering&quot;, where they derive the approximation, the formula is:
$$F_{\lambda}(u) = f_{\lambda} + (1 - f_{\lambda})(1 - u)^{5}$$
Where

So, to answer your first question, $\theta$ refers to the angle between the view vector and the half vector. Consider for a minute that the surface is a perfect mirror. So:
$$V \equiv reflect(V')$$
In this case:
$$N \equiv H$$
For microfacet-base BRDFs, the $D(h_{r})$ term refers to the statistical percentage of microfacet normals that are oriented towards $H$. Aka, what percentage of the incoming light will bounce in the outgoing direction.
As for why we use Fresnel in a BRDF, it has to do with the fact that a BRDF by itself is only a portion of the full BSDF. A BRDF attenuates the reflected portion of light and a BTDF attenuates the refracted. We use the Fresnel to calculate the amount of reflected vs. refracted light, so we can properly attenuate it with the BRDF and BTDF.
$$BSDF = BRDF + BTDF\\
$$
$$\begin{align*}
L_{\text{o}}(p, \omega_{\text{o}}) &amp;=  L_{e}(p, \omega_{\text{o}}) \ + \  \int_{\Omega} BSDF * L_{\text{i}}(p, \omega_{\text{i}}) \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}} \\
 &amp;= L_{e}(p, \omega_{\text{o}}) \ + \  \int_{\Omega} BRDF * L_{\text{i, reflected}}(p, \omega_{\text{i}}) \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}}  \ + \  \int_{\Omega} BTDF * L_{\text{i, refracted}}(p, \omega_{\text{i}}) * \left | \cos \theta_{\text{i}} \right | d\omega_{\text{i}}
\end{align*}$$
So, in summary, we use $D$ to get the percentage of light that will bounce in the outgoing direction, and $F$, to find out what percentage of the remaining light will reflect/refract. Both these use $H$, because that is the surface orientation that allows a mirror reflection between $V$ and $V'$