In the preceding sections we have frequently employed the saddle-point approximation (Gaussian approximation) to problems which are nonlinear in the parameters. Two requirements have been more or less tacitly assumed in doing so: the first assumption is that the true posterior resembles relatively closely a multivariate Gaussian. This is frequently the case, but of course not necessarily generally so. The second assumption is that the limits of integration and hence the support of the parameters is (—∞, ∞). This latter assumption can be relaxed to the requirement that the parameter support and position and width of the posterior are such that extending the limits of integration causes negligible approximation error to the integral. This assumption is, however, frequently not justified. Since analytic integration of a multivariate Gaussian within finite limits is not possible, the only solution to the problem is numerical integration. This does not sound like a complicated task, but we shall see in the sequel that things are different in many dimensions and frequently people discussing the topic are misled by a dimension fallacy.

The evaluation of multidimensional integrals of arbitrary posterior distributions, which would of course also include multimodal posteriors, can become rather complicated. For pedagogical reasons, we shall pursue a much simpler problem and limit the discussion to problems where a Gaussian approximation exists and discuss how to account by numerical integration for approximation errors introduced by differences between the true shape of the posterior and its Gaussian approximation and by finite limits of integration.