Fundamentals of nonparametric Bayesian inference / Subhashis Ghosal, North Carolina State University ; Aad van der Vaart, Leiden University
- Author:
- Ghosal, Subhashis
- Published:
- Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2017.
- Copyright Date:
- ©2017
- Physical Description:
- xxiv, 646 pages ; 27 cm.
- Additional Creators:
- Vaart, A. W. van der
- Series:
- Contents:
- Machine generated contents note: 1.1.Motivation -- 1.1.1.Classical versus Bayesian Nonparametrics -- 1.1.2.Parametric versus Nonparametric Bayes -- 1.2.Challenges of Bayesian Nonparametrics -- 1.2.1.Prior Construction -- 1.2.2.Computation -- 1.2.3.Asymptotic Behavior -- 1.3.Priors, Posteriors and Bayes's Rule -- 1.3.1.Absolute Continuity -- 1.4.Historical Notes -- 2.1.Random Basis Expansion -- 2.2.Stochastic Processes -- 2.2.1.Gaussian Processes -- 2.2.2.Increasing Processes -- 2.3.Probability Densities -- 2.3.1.Exponential Link Function -- 2.3.2.Construction through Binning -- 2.3.3.Mixtures -- 2.3.4.Feller Approximation -- 2.4.Nonparametric Normal Regression -- 2.5.Nonparametric Binary Regression -- 2.6.Nonparametric Poisson Regression -- 2.7.Historical Notes -- Problems -- 3.1.Random Measures -- 3.1.1.Other Topologies -- 3.2.Construction through a Stochastic Process -- 3.3.Countable Sample Spaces -- 3.3.1.Construction through Normalization -- 3.3.2.Construction through Stick Breaking -- 3.3.3.Countable Dirichlet Process -- 3.4.Construction through Structural Definitions -- 3.4.1.Construction through a Distribution on a Dense Subset -- 3.4.2.Construction through a Randomly Selected Discrete Set -- 3.4.3.Construction through Random Rectangular Partitions -- 3.4.4.Construction through Moments -- 3.4.5.Construction through Quantiles -- 3.4.6.Construction by Normalization -- 3.5.Construction through a Tree -- 3.6.Tail-Free Processes -- 3.7.Polya Tree Processes -- 3.7.1.Relation with the Polya Urn Scheme -- 3.7.2.Mixtures of Polya Tree Processes -- 3.7.3.Partially Specified Polya Tree -- 3.7.4.Evenly Split Polya Tree -- 3.8.Historical Notes -- Problems -- 4.1.Definition and Basic Properties -- 4.1.1.Expectations, Variances and Co-Variances -- 4.1.2.Self-Similarity -- 4.1.3.Conjugacy -- 4.1.4.Marginal and Conditional Distributions -- 4.1.5.Number of Distinct Values -- 4.2.Constructions -- 4.2.1.Construction via a Stochastic Process -- 4.2.2.Construction through Distribution Function -- 4.2.3.Construction through a Gamma Process -- 4.2.4.Construction through Polya Urn Scheme -- 4.2.5.Stick-Breaking Representation -- 4.3.Further Properties -- 4.3.1.Discreteness and Support -- 4.3.2.Convergence -- 4.3.3.Approximations -- 4.3.4.Mutual Singularity of Dirichlet Processes -- 4.3.5.Tails of a Dirichlet Process -- 4.3.6.Distribution of Median -- 4.3.7.Distribution of Mean -- 4.4.Characterizations -- 4.5.Mixtures of Dirichlet Processes -- 4.6.Modifications -- 4.6.1.Invariant Dirichlet Process -- 4.6.2.Constrained Dirichlet Process -- 4.6.3.Penalized Dirichlet Process -- 4.7.Bayesian Bootstrap -- 4.8.Historical Notes -- Problems -- 5.1.Dirichlet Process Mixtures -- 5.2.MCMC Methods -- 5.3.Variational Algorithm -- 5.4.Predictive Recursion Deconvolution Algorithm -- 5.5.Examples of Kernels -- 5.6.Historical Notes -- Problems -- 6.1.Consistency and Its Implications -- 6.2.Doob's Theorem -- 6.3.Inconsistency -- 6.4.Schwartz's Theorem -- 6.5.Tail-Free Priors -- 6.6.Permanence of the Kullback-Leibler Property -- 6.7.General Observations -- 6.7.1.Independent Observations -- 6.7.2.Markov Processes -- 6.8.Alternative Approaches -- 6.8.1.Separation -- 6.8.2.Le Cam's Inequality -- 6.8.3.Predictive Consistency -- 6.8.4.Martingale Approach -- 6.8.5.alpha-Posterior -- 6.9.Historical Notes -- Problems -- 7.1.Priors with the Kullback-Leibler Property -- 7.1.1.P61ya Trees -- 7.1.2.Kernel Mixtures -- 7.1.3.Exponential Densities -- 7.2.Density Estimation -- 7.2.1.Normal Mixtures -- 7.2.2.Dirichlet Process Mixtures of a General Kernel -- 7.2.3.Polya Tree Process -- 7.2.4.Exponential Densities -- 7.3.Other Nonparametric Models -- 7.3.1.Nonparametric Binary Regression -- 7.3.2.Nonparametric Regression with Normal Errors -- 7.3.3.Spectral Density Estimation -- 7.4.Semiparametric Models -- 7.4.1.Location Problem -- 7.4.2.Linear Regression with Unknown Error Density -- 7.4.3.Binary Nonparametric Monotone Regression -- 7.5.Historical Notes -- Problems -- 8.1.Introduction -- 8.2.Independent Identically Distributed Observations -- 8.2.1.Further Refinements -- 8.2.2.Priors Based on Finite Approximating Sets -- 8.3.General Observations -- 8.3.1.Independent Observations -- 8.3.2.Gaussian Regression with Fixed Design -- 8.3.3.Markov Chains -- 8.3.4.White Noise Model -- 8.3.5.Gaussian Time Series -- 8.4.Lower Bounds -- 8.5.Misspecification -- 8.5.1.Convex Models -- 8.5.2.Nonparametric Regression -- 8.6.a-Posterior -- 8.7.Historical Notes -- Problems -- 9.1.Log-Spline Priors -- 9.2.Priors Based on Dirichlet Processes -- 9.3.Bernstein Polynomials -- 9.4.Dirichlet Process Mixtures of Normal Kernel -- 9.4.1.Approximation -- 9.4.2.Prior Concentration -- 9.4.3.Entropy Estimate and Controlling Complexity -- 9.4.4.Proof of Theorem 9.9 -- 9.4.5.Wishart Prior -- 9.5.Non-i.i.d. Models -- 9.5.1.Finite Sieves -- 9.5.2.Whittle Estimation of a Spectral Density -- 9.5.3.Nonlinear Autoregression -- 9.5.4.White Noise with Conjugate Priors -- 9.5.5.Nonparametric Regression Using Splines -- 9.5.6.Binary Nonparametric Regression with a Dirichlet Process Prior -- 9.5.7.Interval Censoring Using a Dirichlet Process Prior -- 9.6.Historical Notes -- Problems -- 10.1.Introduction -- 10.2.Independent Identically Distributed Observations -- 10.2.1.Universal Weights -- 10.2.2.Parametric Rate -- 10.2.3.Two Models -- 10.3.Examples -- 10.3.1.Priors Based on Finite Approximating Sets -- 10.3.2.White Noise Model -- 10.3.3.Finite-Dimensional Approximations -- 10.3.4.Log-Spline Models -- 10.4.Finite Random Series -- 10.4.1.Density Estimation -- 10.4.2.Nonparametric Normal Regression -- 10.4.3.Nonparametric Binary Regression -- 10.4.4.Nonparametric Poisson Regression -- 10.4.5.Functional Regression -- 10.4.6.Whittle Estimation of a Spectral Density -- 10.5.Model Selection Consistency -- 10.5.1.Testing a Point Null -- 10.5.2.General Case -- 10.5.3.Testing Parametric versus Nonparametric Models -- 10.6.Historical Notes -- Problems -- 11.1.Definition and Examples -- 11.2.Reproducing Kernel Hilbert Space -- 11.3.Posterior Contraction Rates -- 11.3.1.Density Estimation -- 11.3.2.Nonparametric Binary Regression -- 11.3.3.Nonparametric Normal Regression -- 11.3.4.White Noise Model -- 11.4.Specific Gaussian Processes as Priors -- 11.4.1.Brownian Motion and Its Primitives -- 11.4.2.Riemann-Liouville Process -- 11.4.3.Fractional Brownian Motion -- 11.4.4.Stationary Processes -- 11.4.5.Series Priors -- 11.5.Rescaled Gaussian Processes -- 11.5.1.Self-Similar Processes -- 11.5.2.Stationary Gaussian Processes -- 11.6.Adaptation -- 11.7.Computation -- 11.7.1.Kernel Methods and the Posterior Mode -- 11.7.2.Density Estimation -- 11.7.3.Nonparametric Binary Regression -- 11.7.4.Expectation Propagation -- 11.7.5.Laplace Approximation -- 11.8.Historical Notes -- Problems -- 12.1.Introduction -- 12.2.Dirichlet Process -- 12.2.1.Strong Approximation -- 12.3.Semiparametric Models -- 12.3.1.Functionals -- 12.3.2.Strict Semiparametric Model -- 12.3.3.Cox Proportional Hazard Model -- 12.4.White Noise Model -- 12.4.1.Full Parameter -- 12.4.2.Linear Functionals -- 12.5.Historical Notes -- Problems -- 13.1.Introduction -- 13.2.Dirichlet Process Prior -- 13.3.Beta Process Prior -- 13.3.1.Discrete Time -- 13.3.2.Continuous Time -- 13.3.3.Sample Path Generation -- 13.3.4.Mixtures of Beta Processes -- 13.4.Neutral to the Right and Independent Increment Processes -- 13.4.1.Consistency -- 13.4.2.Bernstein-von Mises Theorem -- 13.5.Smooth Hazard Processes -- 13.6.Proportional Hazard Model -- 13.6.1.Posterior Distribution -- 13.6.2.Bernstein-von Mises Theorem -- 13.7.The Bayesian Bootstrap for Censored Data -- 13.7.1.Survival Data without Covariates -- 13.7.2.Cox Proportional Hazard Model -- 13.8.Historical Notes -- Problems -- 14.1.Exchangeable Partitions -- 14.1.1.The Chinese Restaurant Process -- 14.1.2.The Chinese Restaurant Franchise Process -- 14.2.Species Sampling Processes -- 14.2.1.Posterior Distribution -- 14.2.2.Species Sampling Process Mixtures -- 14.3.Gibbs Processes -- 14.4.Pitman-Yor Process -- 14.5.Poisson-Kingman Processes -- 14.6.Normalized Inverse-Gaussian Process -- 14.7.Normalized Completely Random Measures -- 14.8.Relations between Classes of Discrete Random Probability Measures -- 14.9.Dependent Random Discrete Distributions -- 14.9.1.Kernel Stick-Breaking Process -- 14.9.2.Local Dirichlet Process -- 14.9.3.Probit Stick-Breaking Process -- 14.9.4.Ordering Dependent Stick-Breaking Processes -- 14.9.5.Nested Dirichlet Processes -- 14.10.The Indian Buffet Process -- 14.11.Historical Notes -- Problems -- A.Space of Probability Measures -- B.Space of Probability Densities -- C.Packing, Covering, Bracketing and Entropy Numbers -- D.Hypothesis Tests -- E.Polynomials, Splines and Wavelets -- F.Elements of Empirical Processes -- G.Finite-Dimensional Dirichlet Distribution -- H.Inverse-Gaussian Distribution -- I.Gaussian Processes -- J.Completely Random Measures -- K.Inequalities and Estimates -- L.Miscellaneous Results -- M.Elements of Markov Chain Monte Carlo.
- Subject(s):
- ISBN:
- 9780521878265 hardcover
0521878268 hardcover - Bibliography Note:
- Includes bibliographical references and indexes.
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