Bayesian Machine Learning in Geotechnical Site Characterization (Challenges in Geotechnical and Rock Engineering)

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Table of Contents
Series Preface
	Challenges in Geotechnical and Rock Engineering
Preface
Chapter 1: Bayesian approach
	1.1 Boolean logic
	1.2 Plausibility logic
		1.2.1 Negation function and theorem
			1.2.1.1 Negation theorem
			1.2.1.2 Proof of the negation theorem
		1.2.2 Conjunction function and theorem
			1.2.2.1 Conjunction theorem (stated without proof)
	1.3 Probability logic
	1.4 Axioms for Bayesian probability
	1.5 Continuous variable
Chapter 2: Review of probability and models
	2.1 Definitions and concepts related to continuous random variables
		2.1.1 Probability density function
		2.1.2 Cumulative distribution function
		2.1.3 Quantile function and Monte Carlo simulation
			2.1.3.1 Proof for the procedure of MCS
		2.1.4 Expected value
			2.1.4.1 Mean value of X
			2.1.4.2 Variance of X
			2.1.4.3 Proof of Equation (2.14)
			2.1.4.4 Covariance between X and Y
			2.1.4.5 Proof of Equation (2.19)
			2.1.4.6 Identities for expected values
			2.1.4.7 Law of large numbers
		2.1.5 Independence
			2.1.5.1 Proof for X⊥Y ➔ COV(X,Y) = 0
	2.2 Some univariate PDF models
		2.2.1 Uniform PDF model
		2.2.2 Normal (Gaussian) PDF model
		2.2.3 Lognormal PDF model
			2.2.3.1 Proof for the lognormal PDF in Equation (2.46)
		2.2.4 Johnson system of distributions
		2.2.5 Chi-squared PDF model
		2.2.6 Inverse-Gamma PDF model
		2.2.7 Student’s t-distribution model
	2.3 Bivariate PDF models
		2.3.1 Bivariate normal PDF model
		2.3.2 Bivariate Nataf model
	2.4 Multivariate PDF models
		2.4.1 Multivariate normal PDF model
		2.4.2 Multivariate Nataf model
		2.4.3 Inverse-Wishart PDF model
		2.4.4 Wishart PDF model
	2.5 Conjugate prior PDFs
		2.5.1 Normal PDF as conjugate prior for normal likelihood with unknown μ
		2.5.2 Inverse-Gamma as conjugate prior for normal likelihood with unknown σ 2
		2.5.3 Multivariate normal as conjugate prior for multivariate normal likelihood with unknown μ
		2.5.4 Inverse-Wishart as conjugate prior for multivariate normal likelihood with unknown C
		2.5.5 Wishart as conjugate prior for an inverse-Wishart likelihood with unknown Σ
	2.6 Models for soil spatial variability
Chapter 3: Bayesian parameter estimation and prediction
	3.1 Maximum-likelihood method (frequentist)
	3.2 Bayesian problems with univariate normal data
		3.2.1 Bayesian analysis with conjugate prior
		3.2.2 Bayesian analysis with non-conjugate prior
		3.2.3 Numerical methods for Bayesian analysis
			3.2.3.1 Metropolis–Hastings (MH) algorithm
			3.2.3.2 Gibbs sampler (GS) algorithm
	3.3 Bayesian problems with multivariate normal data
	3.4 Bayesian problems with spatial variability
		3.4.1 Transitional Markov chain Monte Carlo algorithm
		3.4.2 Bayesian model selection
Chapter 4: Geotechnical data and Bayesian modeling
	4.1 Cross-correlated data
		4.1.1 Generic model
			4.1.1.1 Conversion to standard normal variables
			4.1.1.2 Bayesian parameter estimation
			4.1.1.3 Bayesian prediction for a new case
			4.1.1.4 Bayesian prediction for a Taipei site
		4.1.2 Site-specific model
			4.1.2.1 Bayesian parameter estimation
			4.1.2.2 Bayesian prediction
		4.1.3 Quasi-site-specific model – hierarchical Bayesian modeling
			4.1.3.1 Hierarchical Bayesian modeling
			4.1.3.2 Estimation of the hyper-parameters – construction of an informative prior model
			4.1.3.3 Behavior of the prior model constructed by the hyper-parameter samples
			4.1.3.4 Bayesian parameter estimation for the Taipei site
			4.1.3.5 Bayesian prediction for the Taipei site
	4.2 Spatially variable data
		4.2.1 One-dimensional (1D) data
			4.2.1.1 Bayesian parameter estimation
			4.2.1.2 Bayesian prediction
		4.2.2 Three-dimensional (3D) data
			4.2.2.1 Bayesian parameter estimation
			4.2.2.2 Bayesian prediction (conditional random field simulation)
		4.2.3 Three-dimensional (3D) data – GPR method for trend modeling
			4.2.3.1 Karhunen–Loève expansion for the trend
			4.2.3.2 Evaluation of the likelihood function
			4.2.3.3 Bayesian parameter estimation
			4.2.3.4 Bayesian prediction (conditional random field simulation)
	Appendix 4.1: Derivations for Equation (4.70)
	Appendix 4.2: Drawing {t, t′} Samples from f(t,t′ | θ,D)
Chapter 5: Full-scale real case study
	5.1 Real case study – Baytown site
	5.2 Modeling of the Baytown-site data
	5.3 Construction of informative prior using HBM – dealing with missing data in the database
	5.4 Estimation of auto-correlations – dealing with non-lattice CPT data
	5.5 Bayesian parameter estimation for the Baytown site – accommodate spatial correlation and missing data
	5.6 Simulation of conditional random field – dealing with independent simulation domain
	Appendix 5.1: Derivations for the Mean Vector and Covariance Matrix for f(ξ~lat|θ, ξlat)
	Appendix 5.2: Derivations for f(μS | CS,Θ,XS) and f(μS | CS,Θ,XS) under Spatial Correlation
	Appendix 5.3: Derivations for f(Xhz′|μS, CS, Xhz) and f(Xh′(Z∪Z′)|μS, CS, Xh(Z∪Z′))
References
Index