Abstract
Acknowledgements
List of Tables
List of Figures
Publications
1 Introduction
 1.1 Palaeoclimate Reconstruction Project
  1.1.1 The RS10 Pollen Dataset
  1.1.2 Response Surfaces
  1.1.3 The Classical Approach
  1.1.4 The Bayesian Approach
 1.2 Computational Challenges
 1.3 Overview of Chapters
 1.4 Research Contributions
2 Literature Review and Statistical Methodology
 2.1 Palaeoclimate Reconstruction Literature Review
  2.1.1 Classical Approach
  2.1.2 Bayesian Approach
 2.2 Relevant Bayesian Methods
  2.2.1 Bayesian Hierarchical Model
  2.2.2 Markov Chain Monte Carlo
  2.2.3 Directed Acyclic Graphs
  2.2.4 Gaussian Markov Random Fields
 2.3 Integrated Nested Laplace Approximations
 2.4 Spatial Zero-Inflated Models
  2.4.1 Single Process Model for Zero-Inflation
 2.5 Inverse Regression
  2.5.1 Non-parametric Response Surfaces
  2.5.2 Toy Problem Example
 2.6 Model Validation
  2.6.1 Inverse Predictive Power
  2.6.2 Cross-Validation
 2.7 Conclusions
  2.7.1 Advances in this Thesis
3 Models with Known Parameters
 3.1 The Univariate Problem
  3.1.1 Given New Counts Data
  3.1.2 Given Training Data Only
  3.1.3 Percentage Outside Highest Predictive Distribution Region
 3.2 Disjoint-Decomposable Models
  3.2.1 The Marginals Model
  3.2.2 Non-Disjoint-Decomposable Models
  3.2.3 Sources of Interaction
 3.3 Multivariate Normal Model
  3.3.1 General Case Normal Models
  3.3.2 Sensitivity to Dependence
 3.4 Counts Data
  3.4.1 Poisson Model
  3.4.2 Scaled Poisson
  3.4.3 Overdispersion
  3.4.4 Sensitivity to Zero-Inflated Likelihood
 3.5 Compositional Data
  3.5.1 The Simplex Space
  3.5.2 Dirichlet Distribution
  3.5.3 Generalized Dirichlet Distribution
  3.5.4 Logistic-Normal Class of Distributions
  3.5.5 Multivariate, Constrained Likelihood Functions
  3.5.6 Nested Compositional Models
  3.5.7 Disjoint-Decomposing Compositional Models
 3.6 Conclusions
  3.6.1 Disjoint-Decomposition of Models
  3.6.2 Zero-Inflated Models
  3.6.3 Nested Constrained Models
4 INLA Inference and Cross-Validation
 4.1 The Integrated Nested Laplace Approximation
  4.1.1 The Gaussian Markov Random Field Approximation
  4.1.2 Spatial Zero-Inflated Counts Data
  4.1.3 Posterior for the Hyperparameters
  4.1.4 Laplace Approximation for Parameters
  4.1.5 Approximation for Parameters: Inverse Problem
 4.2 Cross Validation
  4.2.1 Importance Resampling
  4.2.2 Cross-Validation in Inverse Problems
  4.2.3 Fast Augmentation of the Multivariate Normal Moments
  4.2.4 More Computational Savings
  4.2.5 Summary Statistics of Model Fit
  4.2.6 Toy Problem Example
 4.3 Conclusions
5 Inference Methodology
 5.1 Reasons for Disjoint-Decomposition
  5.1.1 Parallelisation
  5.1.2 Memory Usage
  5.1.3 Inverse Problem
  5.1.4 Compatibility with the INLA Method
 5.2 Multivariate Normal Model
  5.2.1 Conditions for Perfect Disjoint-Decomposition
  5.2.2 Compositional Independence
 5.3 Sensitivity to Inference via Marginals
  5.3.1 Discrete HPD Regions
  5.3.2 Nested Constrained Models
 5.4 Conclusions
6 Application: the Palaeoclimate Reconstruction Project
 6.1 Bayesian Palaeoclimate Reconstruction Project
  6.1.1 The RS10 Dataset
  6.1.2 Software and Hardware
 6.2 Model Description
  6.2.1 Cross-Validation
  6.2.2 Fast Inversion of the Forward Model
  6.2.3 Buffer Zone for Inverse Problem
 6.3 Zero-Inflation
 6.4 Results
  6.4.1 Treatment of Hyperparameters
  6.4.2 Marginals Model
  6.4.3 Uncertainty in Climate Measurements
  6.4.4 Zero-Inflated Model
  6.4.5 Nested Compositional Model
  6.4.6 Outliers
 6.5 Conclusions
  6.5.1 Advances
  6.5.2 Shortcomings
7 Conclusions and Further Work
 7.1 Conclusions
 7.2 Further Work
  7.2.1 3 Dimensional Climate Space
  7.2.2 Covariates
  7.2.3 Inference Procedures
  7.2.4 Model Validation