• palaeoclimate;
  • multivariate;
  • spatial;
  • INLA;
  • compositional

We introduce a rich class of models π(Y | C,θ) for multivariate constrained, zero-inflated counts data. We use new methodology for fast Bayesian inference on unobserved θ given matched training data inline image and propose a new algorithm for fast inversion to π(C | Y). Such models arise in palaeoclimate reconstruction, where Y represents vector of counts of a proxy such as pollen and C represents climate. Subsequently, inline image is a vector of counts in a sample reflecting ancient pollen rain and thus ancient climate (the palaeoclimate). The methodology applies in principle to palaeoclimate reconstruction from other proxy types (e.g. chironomids, diatoms, testate amoebae). Furthermore, the generic issue—the statistical inversion of a multivariate relationship—is found in many areas of application (e.g. clustering, supervised classification, medical imaging, oil shale modelling). The contribution of the paper is the application of a Nested-Dirichlet–Multinomial model in conjunction with models for zero inflation and non-parametric response surface modelling. We present the models and the tools to perform fast Bayesian inference on the forward problem and inversion, by using integrated nested Laplace approximations. We demonstrate the increase in precision associated with including more climate proxies and the increase in error associated with doing this naïvely. Copyright © 2012 John Wiley & Sons, Ltd.