Flexible Bayesian Regression Modelling

Contents
Contributors
Preface
1 Bayesian quantile regression with the asymmetric Laplace distribution
	1.1 Introduction
	1.2 The asymmetric Laplace distribution for quantile regression
		1.2.1 A simple and efficient sampler
		1.2.2 Quantile curve fitting
		1.2.3 Additive models
	1.3 On coverage probabilities
	1.4 Postprocessing for multiple fittings
	1.5 Final remarks and conclusion
	References
2 A vignette on model-based quantile regression: analysing excess zero response
	2.1 Introduction
	2.2 Excess zero regression analysis
	2.3 Case study data and objective
	2.4 Fitting single covariate basal area models
		2.4.1 Joint quantile regression call
		2.4.2 MCMC progress and convergence assessment
	2.5 Interpreting quantile regressions
		2.5.1 Coefficient plots
		2.5.2 Quantile line plots
	2.6 Assessing model assumptions and making improvements
		2.6.1 Obtaining estimated quantile levels
		2.6.2 Assessing overall fit
		2.6.3 Assessing linearity
		2.6.4 Model improvement
	2.7 Prediction and interpreting predicted responses
		2.7.1 Quantiles for positive reals
		2.7.2 Probability of zero
	2.8 Fitting multiple regression basal area models
		2.8.1 Model terms, transformations and interactions
		2.8.2 Assessing model assumptions
		2.8.3 Interpreting coefficients
		2.8.4 Understanding marginal and interaction effects
		2.8.5 Understanding effects on probability of zero
		2.8.6 Further model refinement and comparison
	2.9 Conclusions and final remarks
	Acknowledgement
	References
3 Bayesian nonparametric density regression for ordinal responses
	3.1 Introduction
	3.2 Bayesian nonparametric density regression
		3.2.1 Dirichlet process priors and Dirichlet process mixtures
		3.2.2 Dirichlet process mixture modelling for density regression
	3.3 Mixture modelling for ordinal responses
		3.3.1 Modelling approach
		3.3.2 Implementation details
			3.3.2.1 Prior specification
			3.3.2.2 Posterior inference
		3.3.3 Illustration: ozone concentration
			3.3.3.1 Density regression
			3.3.3.2 Ordinal regression
		3.3.4 Related work and extensions
	3.4 Summary
	Acknowledgements
	References
4 Bayesian nonparametric methods for financial and macroeconomic time series analysis
	4.1 Introduction
	4.2 Bayesian nonparametric methods for the innovation distribution in volatility models
	4.3 Bayesian nonparametric methods for long-range dependence in SV models
	4.4 Bayesian nonparametric methods for the analysis of macroeconomic time series
	4.5 Conclusion
	References
5 Bayesian mixed binary-continuous copula regression with an application to childhood undernutrition
	5.1 Introduction
	5.2 Bivariate copula models with mixed binary-continuous marginals
		5.2.1 Copula-based construction of bivariate binary-continuous distributions
		5.2.2 Specifying the marginal distributions
		5.2.3 Specifying the copula
		5.2.4 Embedding copula regression in the distributional regression framework
		5.2.5 Predictor specification
			Linear effects
			Nonlinear effects
			Spatial effects
			Other effect types
	5.3 Bayesian inference
		Data augmentation for the binary response
		Iteratively weighted least squares proposals
		Updating parameters of the marginal distributions
		Updating parameters of the copula
		Updating the smoothing variances
		Implementation
	5.4 Model selection and model evaluation
		Quantile residuals for the marginal fits
		Model choice via information criteria
		Predictive ability
	5.5 Results
		Effects of covariates
		Joint probabilities
	5.6 Summary and discussion
	Acknowledgements
	References
6 Nonstandard flexible regression via variational Bayes
	6.1 Introduction
	6.2 Preparatory modelling components
		6.2.1 Mixed model-based penalised splines
		6.2.2 Semiparametric regression
		6.2.3 Priors
		6.2.4 Mean field variational Bayes
		6.2.5 Tricks when VB is not easy to apply
		6.2.6 Comparisons
	6.3 A standard semiparametric regression model
		6.3.1 Generalised additive model for air quality data
	6.4 Robust nonparametric regression
		6.4.1 Hauser respiratory experiment
	6.5 Generalised additive model with heteroscedastic variance
		6.5.1 Milan air pollution example
	6.6 Generalised additive negative binomial model
		6.6.1 Lung cancer data
	6.7 Logistic regression with missing covariates
		6.7.1 Pima-Indians diabetes
	6.8 Conclusion
	Acknowledgements
	References
7 Scalable Bayesian variable selection regression models for count data
	7.1 Introduction
	7.2 Bayesian variable selection via spike-and-slab priors
	7.3 Negative binomial regression models
		7.3.1 Data augmentation
		7.3.2 MCMC algorithm
		7.3.3 Variational inference algorithm
	7.4 Dirichlet-multinomial regression models
		7.4.1 MCMC algorithm
		7.4.2 Variational inference with reparameterisation
			7.4.2.1 Reparameterisation of the gradient
			7.4.2.2 Concrete relaxation
			7.4.2.3 Hard concrete distribution
			7.4.2.4 Variational inference approximation
			7.4.2.5 Posterior inference using tensorflow
	7.5 Simulation study
		Negative binomial - small p large n example
		Negative binomial - large p small n example
		Dirichlet-multinomial example
	7.6 Benchmark applications
		Boston housing data
		Microbiome data
	7.7 Conclusion
	References
8 Bayesian spectral analysis regression
	8.1 Introduction
	8.2 Smooth operators
	8.3 Bayesian spectral analysis regression
	8.4 Shape constraints
		8.4.1 Monotonic functions
		8.4.2 S- and U-shaped functions
	8.5 Nonnormal distributions
	8.6 R library bsamGP
	8.7 Conclusion
	Acknowledgements
	References
9 Flexible regression modelling under shape constraints
	9.1 Introduction
	9.2 Orthonormal design matrices
	9.3 Monotonic polynomial model
		9.3.1 Model specification and estimation
		9.3.2 Posterior simulation
			9.3.2.1 Proposal mechanisms and acceptance probabilities
				Proposal for γ
				Proposal for σ2y
				Acceptance probability
			9.3.2.2 Summary
		9.3.3 Implementation details
			Implementing the monotonicity constraint and calculating the log posterior
			General MCMC scheme
		9.3.4 A simulated data example
	9.4 Covariate selection
		9.4.1 Model specification
			9.4.1.1 Likelihood expression
			9.4.1.2 Prior specification
				Regression coefficient prior
				Variance prior
				Polynomial degree prior
			9.4.1.3 Posterior expression
		9.4.2 Sampling from the posterior distribution
			9.4.2.1 Dimension proposal
			9.4.2.2 Regression coefficient proposal
			9.4.2.3 Variance proposal
		9.4.3 Acceptance probability
			9.4.3.1 Summary
		9.4.4 Simulated data examples
			9.4.4.1 The relationship between noise and estimated degree
			9.4.4.2 The relationship between the number of data points and estimated degree
				Repeated X-values
				Unique X-values
		9.4.5 Child growth data
	9.5 Conclusion
	References
Index