For non-linear posterior distributions :math:`\mathcal{P}(\xi|d)` an approximation with a Gaussian :math:`\mathcal{Q}(\xi)` in the coordinates :math:`\xi` is sub-optimal, as higher order interactions are ignored.
For non-linear posterior distributions :math:`\mathcal{P}(\xi|d)` an approximation with a Gaussian :math:`\mathcal{Q}(\xi)` in the coordinates :math:`\xi` is sub-optimal, as higher order interactions are ignored.
A better approximation can be achieved by constructing a coordinate system :math:`y = g\left(\xi\right)` in which the posterior is close to a Gaussian, and perform VI with a Gaussian :math:`\mathcal{Q}(y)` in these coordinates.
A better approximation can be achieved by constructing a coordinate system :math:`y = g\left(\xi\right)` in which the posterior is close to a Gaussian, and perform VI with a Gaussian :math:`\mathcal{Q}(y)` in these coordinates.
This approach is called Geometric Variation Inference (geoVI).
This approach is called Geometric Variational Inference (geoVI).
It is discussed in detail in [2]_.
It is discussed in detail in [2]_.
One useful coordinate system is obtained in case the metric :math:`M` of the posterior can be expressed as the pullback of the Euclidean metric by :math:`g`:
One useful coordinate system is obtained in case the metric :math:`M` of the posterior can be expressed as the pullback of the Euclidean metric by :math:`g`:
