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nielsen f geometry statistics 2022

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Textbook in PDF format Preface Foundations in classical geometry and analysis Geometry, information, and complex bundles Introduction Complex planes Important implications of Liouville's theorem Geometric analysis and Jordan curves Summary References Geometric methods for sampling, optimization, inference, and adaptive agents Introduction Accelerated optimization Principle of geometric integration Conservative flows and symplectic integrators Rate-matching integrators for smooth optimization Manifold and constrained optimization Gradient flow as a high friction limit Optimization on the space of probability measures Hamiltonian-based accelerated sampling Optimizing diffusion processes for sampling Hamiltonian Monte Carlo Statistical inference with kernel-based discrepancies Topological methods for MMDs Smooth measures and KSDs The canonical Stein operator and Poincaré duality Kernel Stein discrepancies and score matching Information geometry of MMDs and natural gradient descent Minimum Stein discrepancy estimators Likelihood-free inference with generative models Adaptive agents through active inference Modeling adaptive decision-making Behavior, agents, and environments Decision-making in precise agents The information geometry of decision-making Realizing adaptive agents The basic active inference algorithm Sequential decision-making under uncertainty World model learning as inference Scaling active inference Acknowledgments References Equivalence relations and inference for sparse Markov models Introduction Improved modeling capabilities of sparse Markov models (SMMs) Fitting SMMs and example applications Model fitting based on a collapsed Gibbs sampler Modeling wind speeds Modeling a DNA sequence Fitting SMM through regularization Application to classifying viruses Equivalence relations and the computation of distributions of pattern statistics for SMMs Notation Computing distributions in higher-order Markovian sequences Specializing the computation to SMM Application to spaced seed coverage Summary Acknowledgments References Information geometry Symplectic theory of heat and information geometry Preamble Life and seminal work of Souriau on lie groups thermodynamics From information geometry to lie groups thermodynamics Symplectic structure of fisher metric and entropy as Casimir function in coadjoint representation Symplectic Fisher Metric structures given by Souriau model Entropy characterization as generalized Casimir invariant function in coadjoint representation and Poisson Cohomology Koszul Poisson Cohomology and entropy characterization Covariant maximum entropy density by Souriau model Gauss density on Poincaré unit disk covariant with respect to SU(1,1) Lie group Gauss density on Siegel unit disk covariant with respect to SU(N,N) Lie group Gauss density on Siegel upper half plane Conclusion References Further reading A unifying framework for some directed distances in statistics Divergences, statistical motivations, and connections to geometry Basic requirements on divergences (directed distances) Some statistical motivations Incorporating density function zeros Some motivations from probability theory Divergences and geometry Some incentives for extensions phi-Divergences between other statistical objects Some non-phi-divergences between probability distributions Some non-phi-divergences between other statistical objects The framework Statistical functionals S and their dissimilarity The divergences (directed distances) D The reference measure λ The divergence generator phi The scaling and the aggregation functions m1, m2, and m3 m1(x) = m2(x) =: m(x), m3(x) = r(x)m(x) [0, ] for some (measurable) function r:XR satisfying r(x)]-,00,[ for λ- ... m1(x) = m2(x):= 1, m3(x) = r(x) for some (measurable) function r:X[0,] satisfying r(x) ]0, [ for λ-a.a. xX m1(x) = m2(x):= Sx(Q), m3(x) = r(x)Sx(Q) [0, ] for some (measurable) function r:XR satisfying r(x)]-,00,[ for ... m1(x) = m2(x):= w(Sx(P), Sx(Q)), m3(x) = r(x)w(Sx(P), Sx(Q)) [0, [ for some (measurable) functions w:R(S(P))xR( ... m1(x)=Sx(P) and m2(x)=Sx(Q) with statistical functional SS, m3(x) 0 Auto-divergences Connections with optimal transport and coupling Aggregated/integrated divergences Dependence expressing divergences Bayesian contexts Variational representations Some further variants Acknowledgments References The analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions Introduction and motivation Differential-geometric structures induced by smooth convex functions Hessian manifolds and Bregman manifolds Bregman manifolds: Dually flat spaces Some illustrating examples Exponential family manifolds Natural exponential family Fisher–Rao manifold of the categorical distributions Regular cone manifolds Mixture family manifolds Definition The categorical distributions: A discrete mixture family Information geometry of the mixture family of two distinct Cauchy distributions Cauchy mixture family of order 1 An analytic example with closed-form dual potentials Conclusion Appendix. Symbolic computing notebook in MAXIMA References Local measurements of nonlinear embeddings with information geometry Introduction α-Divergence and autonormalizing α-Discrepancy of an embedding Empirical α-discrepancy Connections to existing methods Neighborhood embeddings Autoencoders Conclusion and extensions Conclusion and extensions Appendices Appendix A. Proof of Lemma 1 Appendix B. Proof of Proposition 1 Appendix C. Proof of Proposition 2 Appendix D. Proof of Theorem 1 References Advanced geometrical intuition Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac m ... Introduction Diffeomorphometry Longitudinal models Parallel transport for intersubject normalization Chapter organization Parallel transport with ladder methods Numerical accuracy of Schild's and pole ladders Elementary construction of Schild's ladder Taylor expansion Numerical scheme and convergence Pole ladder Infinitesimal schemes A short overview of the LDDMM framework Ladder methods with LDDMM Validation Application to cardiac motion modeling The right ventricle and its diseases Motion normalization with parallel transport Interaction between shape and deformations: A scale problem An intuitive rescaling of LDDMM parallel transport Hypothesis Criterion and estimation of λ Results Relationship between λ and VolED Changing the metric to preserve relative volume changes Model Implementation Geodesics Results Analysis of the normalized deformations Geodesic and spline regression Results Hotelling tests on velocities Conclusion Acknowledgments Abbreviations References Geometry and mixture models Introduction Fundamentals of modeling with mixtures Mixtures and the fundamentals of geometry Structure of article Identification, singularities, and boundaries Mixtures of finite distributions Likelihood geometry General geometric structures Singular learning theory Bayesian methods Singularities and algebraic geometry Singular learning and model selection Nonstandard testing problems Discussion References Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature Introduction Gaussian distributions and RMT From Gauss to Shannon The ``right´´ Gaussian The normalizing factor Z(σ) MLE and maximum entropy Barycenter and covariance Z(σ) from RMT The asymptotic distribution Duality: The Θ distributions Gaussian distributions and Bayesian inference MAP versus MMS Bounding the distance Computing the MMS Metropolis-Hastings algorithm The empirical barycenter Proof of Proposition 13 Appendix A. Riemannian symmetric spaces The noncompact case The compact case Example of Propositions A.1 and A.2 Appendix B. Convex optimization Convex sets and functions Second-order Taylor formula Taylor with retractions Riemannian gradient descent Strictly convex case Strongly convex case Appendix C. Proofs for Section B References Multilevel contours on bundles of complex planes* Introduction Infinitely many bundles of complex planes Multilevel contours in a random environment Behavior of X (zl(t), l) at (l 0) Loss of spaces in bundle B Islands and holes in B Consequences of B\l on multilevel contours PDEs for the dynamics of lost space Concluding remarks Acknowledgments References Index Back Cover

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