List of Figures

2.1 Directed Acyclic Graph
2.2 Univariate response: forward and inverse
2.3 Results for various model parameters
3.1 Examples of easy, medium and hard for given new data
3.2 Examples of easy, medium and hard; general new data
3.3 Disjoint-Decomposable Model
3.4 % outside 95% HPD region against number of surfaces modelled
3.5 Non-Disjoint-Decomposable Models
3.6 Error as a Function of Interaction
3.7 Inverse Predictive Distributions: Zero-Inflated Counts
3.8 The Simplex Space
3.9 Nested Compositions: Example
3.10 Nested Compositions: Example
3.11 Nested Compositions: Simplest Case
3.12 Prior Nested Dirichlets
3.13 Posterior Nested Dirichlets
3.14 % outside 95% HPD region; repititions and density, high overdispersion
3.15 % outside 95% HPD region; repititions and density, low overdispersion
4.1 Quadratic Approximation to Poisson
4.2 Gaussian Univariate Approximation
4.3 Quadratic Approximation to Zero-Inflated Poisson
4.4 MCMC and GMRF Approximations: Zero-Inflated Poisson Likelihood
4.5 MCMC Chain Trace Plot: Log-rate
4.6 MCMC Chain Trace Plot: Probability of Presence
4.7 Percentage of Points Outside 95% HPD: Zero-Inflation
5.1 % outside 95% HPD region versus replications
5.2 % outside 95% HPD region versus independent surfaces
5.3 Convergence of inverse predictive distribution
5.4 % outside 95% HPD region; repititions and density
6.1 Modern climates; buffer zone
6.2 Localised Histograms of Proportions Data
6.3 Positive Abundances versus Probability of Presence
6.4 Effect of using the modal hyperparameters
6.5 Distribution of the Δ statistic across taxa; Pollen dataset
6.6 Plot of the Δ statistic against number of taxa T
6.7 Plot of the D statistic against number of taxa T
6.8 Distribution of the D statistic; with and without Gaussian Blurring
6.9 Structure of the Nesting of pollen dataset
6.10 Distribution of Dl_j, the expected squared distances to observations
6.11 Plot of the Δ statistic against number of taxa T; within nest
6.12 Example inverse cross-validation predictive distributions
6.13 Inverse Predictive Densities: Extreme Outlier
6.14 Outliers and altitudes; sample densities
6.15 Outliers and AET/PET; sample densities

• Haslett J., Bhattacharya S., Whiley M., Salter-Townshend M., Wilson S., Allen J.R.M., Huntley B., and Mitchell F. (2006). Bayesian Palaeoclimate Reconstruction J. R. Statist. Soc. A. , 169, Part 3, 1–36.

• Salter-Townshend M., and Haslett J. (2006) Zero-Inflation of Compositional Data, Proceedings of the 21st International Workshop on Statistical Modelling, 21, 2006, 448–456.