  The analysis of photometric large-scale structure data is often complicated
by the need to account for many observational and astrophysical systematics.
The elaborate models needed to describe them often introduce many ``nuisance
parameters'', which can be a major inhibitor of an efficient parameter
inference. In this paper we introduce an approximate method to analytically
marginalise over a large number of nuisance parameters based on the Laplace
approximation. We discuss the mathematics of the method, its relation to
concepts such as volume effects and profile likelihood, and show that it can be
further simplified for calibratable systematics by linearising the dependence
of the theory on the associated parameters. We quantify the accuracy of this
approach by comparing it with traditional sampling methods in the context of
existing data from the Dark Energy Survey, as well as futuristic Stage-IV
photometric data. The linearised version of the method is able to obtain
parameter constraints that are virtually equivalent to those found by exploring
the full parameter space for a large number of calibratable nuisance
parameters, while reducing the computation time by a factor 3-10. Furthermore,
the non-linearised approach is able to analytically marginalise over a large
number of parameters, returning constraints that are virtually
indistinguishable from the brute-force method in most cases, accurately
reproducing both the marginalised uncertainty on cosmological parameters, and
the impact of volume effects associated with this marginalisation. We provide
simple recipes to diagnose when the approximations made by the method fail and
one should thus resort to traditional methods. The gains in sampling efficiency
associated with this method enable the joint analysis of multiple surveys,
typically hindered by the large number of nuisance parameters needed to
describe them.
