  Laplace approximation is a very useful tool in Bayesian inference and it
claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022}
established some rather accurate finite sample results about the quality of
Laplace approximation in terms of the so called effective dimension $p$ under
the critical dimension constraint $p^{3} \ll n$. However, this condition can be
too restrictive for many applications like error-in-operator problem or Deep
Neuronal Networks. This paper addresses the question whether the dimensionality
condition can be relaxed and the accuracy of approximation can be improved if
the target of estimation is low dimensional while the nuisance parameter is
high or infinite dimensional. Under mild conditions, the marginal posterior can
be approximated by a Gaussian mixture and the accuracy of the approximation
only depends on the target dimension. Under the condition $p^{2} \ll n$ or in
some special situation like semi-orthogonality, the Gaussian mixture can be
replaced by one Gaussian distribution leading to a classical Laplace result.
The second result greatly benefits from the recent advances in Gaussian
comparison from \cite{GNSUl2017}. The results are illustrated and specified for
the case of error-in-operator model.
