This is a very quick and dirty introduction to Bayesian quadrature. For a more concrete introduction, see these nice slides

Bayesian Quadrature is a probabilistic numerics methods to compute integrals. The idea is to compute the integral

\[I = \int f(x) p(x)dx,\]

where $p(x)\sim \mathcal{N}$, by representing the function $f(x)$ by a Gaussian Process.

However, to know what the posterior of $f$ is like, we need samples $\{x_i, y_i\}$ where $y_i = f(x_i) + \epsilon$, with $epsilon$ being some noise. In the end, instead of looking for an approximation of $I$, we obtain a posterior distribution for it values:

\[ p(I|\{x_i,y_i\}) = \mathcal{N}(\mu, \sigma^2)\]

Under some conditions, the mean $\mu$ and the variance $\sigma^2$ can be found analytically.