Probabilistic machine learning: advanced topics
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| Format: | Buch |
| Sprache: | Englisch |
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Cambridge, Massachusetts ; London, England
The MIT Press
[2023]
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| Schriftenreihe: | Adaptive Computation and Machine Learning series
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| Schlagwörter: | |
| Links: | https://probml.github.io/pml-book/book2.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034319734&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xxxi, 1319 Seiten Illustrationen, Diagramme |
| ISBN: | 9780262048439 |
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| 245 | 1 | 0 | |a Probabilistic machine learning |b advanced topics |c Kevin P. Murphy |
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Datensatz im Suchindex
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Brief Contents 1 Introduction Fundamentals I 2 3 4 5 6 Inference 337 Inference algorithms: an overview 339 Gaussian filtering and smoothing 353 Message passing algorithms 395 Variational inference 433 Monte Carlo methods 477 Markov chain Monte Carlo 493 Sequential Monte Carlo 537 III 14 15 16 17 18 19 3 Probability 5 Statistics 63 Graphical models 143 Information theory 217 Optimization 255 II 7 8 9 10 11 12 13 1 Prediction 567 Predictive models: an overview 569 Generalized linear models 583 Deep neural networks 623 Bayesian neural networks 639 Gaussian processes 673 Beyond the iid assumption 727
viii IV 20 21 22 23 24 25 26 V 27 28 29 30 31 32 33 VI BRIEF CONTENTS 763 Generation Generative models: an overview Variational autoencoders 781 Autoregressive models 811 Normalizing flows 819 Energy-based models 839 Diffusion models 857 Generative adversarial networks Discovery 883 915 Discovery methods: an overview Latent factor models 919 State-space models 969 Graph learning 1031 Nonparametric Bayesian models Representation learning 1037 Interpretability 1061 Action 765 917 1035 1091 34 Decision making under uncertainty 35 Reinforcement learning 1133 36 Causality 1171 1093
Contents Preface 1 I 2 xxix 1 Introduction Fundamentals Probability 2.1 2.2 2.3 2.4 2.5 2.6 3 5 Introduction 5 2.1.1 Probability space 5 2.1.2 Discrete random variables 5 2.1.3 Continuous random variables 6 2.1.4 Probability axioms 7 2.1.5 Conditional probability 7 2.1.0 Bayes’ rule 8 Some common probability distributions 8 2.2.1 Discrete distributions 9 2.2.2 Continuous distributions on R 10 2.2.3 Continuous distributions on R+ 13 2.2.4 Continuous distributions on [0,1] 17 2.2.5 Multivariate continuous distributions 17 Gaussian joint distributions 22 2.3.1 The multivariate normal 22 2.3.2 Linear Gaussian systems 28 2.3.3 A general calculus for linear Gaussian systems 30 The exponential family 33 2.4.1 Definition 34 2,4.2 Examples 34 2.4.3 Log partition function is cumulant generating function 39 2.4.4 Canonical (natural) vs mean (moment) parameters 41 2.4.5 MLE for the exponential family 42 2.4.6 Exponential dispersion family 43 2.4.7 Maximum entropy derivation of the exponential family 43 Transformations of random variables 44 2.5.1 Invertible transformations (bijections) 44 2.5.2 Monte Carlo approximation 45 2.5.3 Probability integral transform 45 Markov chains 46 2.6.1 Parameterization 47 2.6.2 Application: language modeling 49
CONTENTS X 2.7 2.6.3 Parameter estimation 49 2.6.4 Stationary distribution of a Markov chain 51 Divergence measures between probability distributions 55 2.7.1 /-divergence 55 2.7.2 Integral probability metrics 57 2.7.3 Maximum mean discrepancy (MMD) 58 2.7.4 Total variation distance 61 2.7.5 Density ratio estimation using binary classifiers 61 3 Statistics 63 3.1 Introduction 63 3.2 Bayesian statistics 63 3.2.1 Tossing coins 64 3.2.2 Modeling more complex data 70 3.2.3 Selecting the prior 71 3.2.4 Computational issues 71 3.2.5 Exchangeability and de Finetti’s theorem 71 3.3 Frequentist statistics 72 3.3.1 Sampling distributions 72 3.3.2 Bootstrap approximation of the sampling distribution 73 3.3.3 Asymptotic normality of the sampling distribution of the MLE 3.3.4 Fisher information matrix 75 3.3.5 Counterintuitive properties of frequentist statistics 79 3.3.6 Why isn’t everyone a Bayesian? 82 3.4 Conjugate priors 83 3.4.1 The binomial model 83 3.4.2 The multinomial model 83 3.4.3 The univariate Gaussian model 85 3.4.4 The multivariate Gaussian model 90 3.4.5 The exponential family model 96 3.4.6 Beyond conjugate priors 98 3.5 Noninformative priors 102 3.5.1 Maximum entropy priors 102 3.5.2 Jeffreys priors 103 3.5.3 Invariant priors 106 3.5.4 Reference priors 107 3.6 Hierarchical priors 107 3.6.1 A hierarchical binomial model 108 3.6.2 A hierarchical Gaussian model 110 3.6.3 Hierarchical conditional models 113 3.7 Empirical Bayes 114 3.7.1 EB for the hierarchical binomial model 114 3.7.2 EB for the hierarchical Gaussian model 115 3.7.3 EB for Markov models (n-gram smoothing)
116 3.7.4 EB for non-conjugate models 118 3.8 Model selection 118 3.8.1 Bayesian model selection 119 3.8.2 Bayes model averaging 121 3.8.3 Estimating the marginal likelihood 121 3.8.4 Connection between cross validation and marginal likelihood 3.8.5 Conditional marginal likelihood 123 3.8.6 Bayesian leave-one-out (LOO) estimate 124 3.8.7 Information criteria 125 3.9 Model checking 127 3.9.1 Posterior predictive checks 128 3.9.2 Bayesian p-values 130 74 122
CONTENTS xi 3.10 Hypothesis testing 131 3.10.1 Frequentist approach 131 3.10.2 Bayesian approach 131 3.10.3 Common statistical tests correspond to inference in linear models 3.11 Missing data 141 4 Graphical models 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5.1 143 Introduction 143 Directed graphical models (Bayes nets) 143 4.2.1 Representing the joint distribution 143 4.2.2 Examples 144 4.2.3 Gaussian Bayes nets 148 4.2.4 Conditional independence properties 149 4.2.5 Generation (sampling) 154 4.2.6 Inference 155 4.2.7 Learning 155 4.2.8 Plate notation 161 Undirected graphical models (Markov random fields) 164 4.3.1 Representing the joint distribution 165 4.3.2 Fully visible MRFs (Ising, Potts, Hopfield, etc.) 166 4.3.3 MRFs with latent variables (Boltzmann machines, etc.) 172 4.3.4 Maximum entropy models 174 4.3.5 Gaussian MRFs 177 4.3.6 Conditional independence properties 179 4.3.7 Generation (sampling) 181 4.3.8 Inference 181 4.3.9 Learning 182 Conditional random fields (CRFs) 185 4.4.1 Id CRFs 186 4.4.2 2d CRFs 189 4.4.3 Parameter estimation 192 4.4.4 Other approaches to structured prediction 193 Comparing directed and undirected PGMs 193 4.5.1 CI properties 193 4.5.2 Converting between a directed and undirected model 195 4.5.3 Conditional directed vs undirected PGMs and the label bias problem 4.5.4 Combining directed and undirected graphs 197 4.5.5 Comparing directed and undirected Gaussian PGMs 199 PGM extensions 201 4.6.1 Factor graphs 201 4.6.2 Probabilistic circuits 204 4.6.3 Directed relational PGMs 205 4.6.4 Undirected relational PGMs 207 4.6.5 Open-universe probability models
210 4.6.6 Programs as probability models 210 Structural causal models 211 4.7.1 Example: causal impact of education on wealth 212 4.7.2 Structural equation models 213 4.7.3 Do operator and augmented DAGs 213 4.7.4 Counterfactuals 214 5 Information theory 136 217 KL divergence 217 5.1.1 Desiderata 218 5.1.2 The KL divergence uniquely satisfies the desiderata 5.1.3 Thinking about KL 222 5.1.4 Minimizing KL 223 219 196
xii CONTENTS 5.2 5.3 5.4 5.5 5.6 6 5.1.5 Properties of KL 226 5.1.6 KL divergence and MLE 228 5.1.7 KL divergence and Bayesian inference 229 5.1.8 KL divergence and exponential families 230 5.1.9 Approximating KL divergence using the Fisher information matrix 5.1.10 Bregman divergence 231 Entropy 232 5.2.1 Deflnition 233 5.2.2 Differential entropy for continuous random variables 233 5.2.3 Typical sets 234 5.2.4 Cross entropy and perplexity 235 Mutual information 236 5.3.1 Definition 236 5.3.2 Interpretation 237 5.3.3 Data processing inequality 237 5.3.4 Sufficient statistics 238 5.3.5 Multivariate mutual information 239 5.3.6 Variational bounds on mutual information 242 5.3.7 Relevance networks 244 Data compression (source coding) 245 5.4.1 Lossless compression 245 5.4.2 Lossy compression and the rate-distortion tradeoff 246 5.4.3 Bits back coding 248 Error-correcting codes (channel coding) 249 The information bottleneck 250 5.6.1 Vanilla IB 250 5.6.2 Variational IB 251 5.6.3 Conditional entropy bottleneck 252 Optimization 6.1 6.2 6.3 6.4 6.5 255 Introduction 255 Automatic differentiation 255 6.2.1 Differentiation in functional form 255 6.2.2 Differentiating chains, circuits, and programs 260 Stochastic optimization 265 6.3.1 Stochastic gradient descent 265 6.3.2 SGD for optimizing a finite-sum objective 267 6.3.3 SGD for optimizing the parameters of a distribution 267 6.3.4 Score function estimator (REINFORCE) 268 6.3.5 Reparameterization trick 269 6.3.6 Gumbel softmax trick 271 6.3.7 Stochastic computation graphs 272 6.3.8 Straight-through estimator 273 Natural gradient
descent 273 6.4.1 Defining the natural gradient 274 6.4.2 Interpretations of NGD 275 6.4.3 Benefits of NGD 276 6.4.4 Approximating the natural gradient 276 6.4.5 Natural gradients for the exponential family 278 Bound optimization (MM) algorithms 281 6.5.1 The general algorithm 281 6.5.2 Example: logistic regression 282 6.5.3 The EM algorithm 283 6.5.4 Example: EM for an MVN with missing data 285 6.5.5 Example: robust linear regression using Student likelihood 6.5.6 Extensions to EM 289 287 231
CONTENTS 6.6 6.7 6.8 6.9 Bayesian optimization 291 6.6.1 Sequential model-based optimization 292 6.6.2 Surrogate functions 292 6.6.3 Acquisition functions 294 6.6.4 Other issues 297 Derivative-free optimization 298 6.7.1 Local search 298 6.7.2 Simulated annealing 301 6.7.3 Evolutionary algorithms 301 6.7.4 Estimation of distribution (EDA) algorithms 304 6.7.5 Cross-entropy method 306 6.7.6 Evolutionary strategies 306 Optimal transport 307 6.8.1 Warm-up: matching optimally two families of points 308 6.8.2 From optimal matchings to Kantorovich and Monge formulations 308 6.8.3 Solving optimal transport 311 Submodular optimization 316 6.9.1 Intuition, examples, and background 316 6.9.2 Submodular basic definitions 318 6.9.3 Example submodular functions 320 6.9.4 Submodular optimization 322 6.9.5 Applications of submodularity in machine learning and AI 327 6.9.6 Sketching, coresets, distillation, and data subset and feature Selection 327 6.9.7 Combinatorial information functions 331 6.9.8 Clustering, data partitioning, and parallel machine learning 332 6.9.9 Active and semi-supervised learning 332 6.9.10 Probabilistic modeling 333 6.9.11 Structured norms and loss functions 335 6.9.12 Conclusions 335 Inference II xiii 337 7 Inference algorithms: an overview 7.1 7.2 7.3 7.4 7.5 8 Gaussian Altering and smoothing 8.1 8.2 339 Introduction 339 Common inference patterns 340 7.2.1 Global latents 340 7.2.2 Local latents 341 7.2.3 Global and local latents 341 Exact inference algorithms 342 Approximate inference algorithms 342 7.4.1 The MAP approximation and its problems 7.4.2 Grid
approximation 344 7.4.3 Laplace (quadratic) approximation 345 7.4.4 Variational inference 346 7.4.5 Markov chain Monte Carlo (MCMC) 348 7.4.6 Sequential Monte Carlo 349 7.4.7 Challenging posteriors 350 Evaluating approximate inference algorithms 350 353 Introduction 353 8.1.1 Inferential goals 353 8.1.2 Bayesian filtering equations 355 8.1.3 Bayesian smoothing equations 356 8.1.4 The Gaussian ansatz 357 Inference for linear-Gaussian SSMs 357 343
CONTENTS xiv 8.3 8.4 8.5 8.6 8.7 9 8.2.1 Examples 358 8.2.2 The Kalman filter 359 8.2.3 The Kalman (RTS) smoother 363 8.2.4 Information form filtering and smoothing 366 Inference based on local linearization 369 8.3.1 Taylor series expansion 369 8.3.2 The extended Kalman filter (EKF) 370 8.3.3 The extended Kalman smoother (EKS) 373 Inference based on the unscented transform 373 8.4.1 The unscented transform 373 8.4.2 The unscented Kalman filter (UKF) 376 8.4.3 The unscented Kalman smoother (UKS) 376 Other variants of the Kalman filter 376 8.5.1 General Gaussian filtering 376 8.5.2 Conditional moment Gaussian filtering 379 8.5.3 Iterated filters and smoothers 380 8.5.4 Ensemble Kalman filter 382 8.5.5 Robust Kalman filters 383 8.5.6 Dual EKF 383 Assumed density filtering 383 8.6.1 Connection with Gaussian filtering 385 8.6.2 ADF for SLDS (Gaussian sum filter) 386 8.6.3 ADF for online logistic regression 387 8.6.4 ADF for online DNNs 390 Other inference methods for SSMs 390 8.7.1 Grid-based approximations 390 8.7.2 Expectation propagation 391 8.7.3 Variational inference 392 8.7.4 MCMC 392 8.7.5 Particle filtering 392 Message passing algorithms 9.1 9.2 9.3 9.4 395 Introduction 395 Belief propagation on chains 395 9.2.1 Hidden Markov Models 396 9.2.2 The forwards algorithm 397 9.2.3 The forwards-backwards algorithm 398 9.2.4 Forwards filtering backwards smoothing 401 9.2.5 Time and space complexity 402 9.2.6 The Viterbi algorithm 403 9.2.7 Forwards filtering backwards sampling 406 Belief propagation on trees 406 9.3.1 Directed vs undirected trees 406 9.3.2 Sum-product algorithm
408 9.3.3 Max-product algorithm 409 Loopy belief propagation 411 9.4.1 Loopy BP for pairwise undirected graphs 412 9.4.2 Loopy BP for factor graphs 412 9.4.3 Gaussian belief propagation 413 9.4.4 Convergence 415 9.4.5 Accuracy 417 9.4.6 Generalized belief propagation 418 9.4.7 Convex BP 418 9.4.8 Application: error correcting codes 418 9.4.9 Application: affinity propagation 420 9.4.10 Emulating BP with graph neural nets 421
CONTENTS 9.5 9.6 9.7 XV The variable elimination (VE) algorithm 422 9.5.1 Derivation of the algorithm 422 9.5.2 Computational complexity of VE 424 9.5.3 Picking a good elimination order 426 9.5.4 Computational complexity of exact inference 9.5.5 Drawbacks of VE 427 The junction tree algorithm (JTA) 428 Inference as optimization 429 9.7.1 Inference as backpropagation 429 9.7.2 Perturb and MAP 430 10 Variational inference 426 433 10.1 Introduction 433 10.1.1 The variational objective 433 10.1.2 Form of the variational posterior 435 10.1.3 Parameter estimation using variational EM 436 10.1.4 Stochastic VI 438 10.1.5 Amortized VI 438 10.1.6 Semi-amortized inference 439 10.2 Gradient-based VI 439 10.2.1 Reparameterized VI 440 10.2.2 Automatic differentiation VI 446 10.2.3 Blackbox variational inference 448 10.3 Coordinate ascent VI 449 10.3.1 Derivation of CAVI algorithm 450 10.3.2 Example: CAVI for the Ising model 452 10.3.3 Variational Bayes 453 10.3.4 Example: VB for a univariate Gaussian 454 10.3.5 Variational Bayes EM 457 10.3.6 Example: VBEM for a GMM 458 10.3.7 Variational message passing (VMP) 464 10.3.8 Autoconj 465 10.4 More accurate variational posteriors 465 10.4.1 Structured mean field 465 10.4.2 Hierarchical (auxiliary variable) posteriors 465 10.4.3 Normalizing flow posteriors 466 10.4.4 Implicit posteriors 466 10.4.5 Combining VI with MCMC inference 466 10.5 Tighter bounds 467 10.5.1 Multi-sample ELBO (IWAE bound) 467 10.5.2 The thermodynamic variational objective (TVO) 468 10.5.3 Minimizing the evidence upper bound 468 10.6 Wake-sleep algorithm 469 10.6.1 Wake
phase 469 10.6.2 Sleep phase 470 10.6.3 Daydream phase 471 10.6.4 Summary of algorithm 471 10.7 Expectation propagation (EP) 472 10.7.1 Algorithm 472 10.7.2 Example 474 10.7.3 EP as generalized ADF 474 10.7.4 Optimization issues 475 10.7.5 Power EP and a-divergence 475 10.7.6 Stochastic EP 475 11 Monte Carlo methods 11.1 Introduction 477 477
xvi CONTENTS 11.2 Monte Carlo integration 477 11.2.1 Example: estimating π by Monte Carlo integration 478 11.2.2 Accuracy of Monte Carlo integration 478 11.3 Generating random samples from simple distributions 480 11.3.1 Sampling using the inverse cdf 480 11.3.2 Sampling from a Gaussian (Box-Muller method) 481 11.4 Rejection sampling 481 11.4.1 Basic idea 482 11.4.2 Example 483 11.4.3 Adaptive rejection sampling 483 11.4.4 Rejection sampling in high dimensions 484 11.5 Importance sampling 484 11.5.1 Direct importance sampling 485 11.5.2 Self-normalized importance sampling 485 11.5.3 Choosing the proposal 486 11.5.4 Annealed importance sampling (AIS) 486 11.6 Controlling Monte Carlo variance 488 11.6.1 Common random numbers 488 11.6.2 Rao-Blackwellization 11.6.3 11.6.4 11.6.5 Control variates 489 Antithetic sampling 490 Quasi-Monte Carlo (QMC) 12 Markov chain Monte Carlo 488 491 493 12.1 Introduction 493 12.2 Metropolis-Hastings algorithm 494 12.2.1 Basic idea 494 12.2.2 Why MH works 495 12.2.3 Proposal distributions 496 12.2.4 Initialization 498 12.3 Gibbs sampling 499 12.3.1 Basic idea 499 12.3.2 Gibbs sampling is a special case of MH 499 12.3.3 Example: Gibbs sampling for Ising models 500 12.3.4 Example: Gibbs sampling for Potts models 502 12.3.5 Example: Gibbs sampling for GMMs 502 12.3.6 Metropolis within Gibbs 504 12.3.7 Blocked Gibbs sampling 504 12.3.8 Collapsed Gibbs sampling 505 12.4 Auxiliary variable MCMC 507 12.4.1 Slice sampling 507 12.4.2 Swendsen-Wang 509 12.5 Hamiltonian Monte Carlo (HMC) 510 12.5.1 Hamiltonian mechanics 511 12.5.2 Integrating Hamilton’s
equations 511 12.5.3 The HMC algorithm 513 12.5.4 Tuning HMC 514 12.5.5 Riemann manifold HMC 515 12.5.6 Langevin Monte Carlo (MALA) 515 12.5.7 Connection between SGD and Langevin sampling 12.5.8 Applying HMC to constrained parameters 517 12.5.9 Speeding up HMC 518 12.6 MCMC convergence 518 12.6.1 Mixing rates of Markov chains 519 12.6.2 Practical convergence diagnostics 520 12.6.3 Effective sample size 523 516
CONTENTS xvii 12.6.4 Improving speed of convergence 525 12.6.5 Non-centered parameterizations and Neal’s funnel 525 12.7 Stochastic gradient MCMC 526 12.7.1 Stochastic gradient Langevin dynamics (SGLD) 527 12.7.2 Preconditionining 527 12.7.3 Reducing the variance of the gradient estimate 528 12.7.4 SG-HMC 529 12.7.5 Underdamped Langevin dynamics 529 12.8 Reversible jump (transdimensional) MCMC 530 12.8.1 Basic idea 531 12.8.2 Example 531 12.8.3 Discussion 533 12.9 Annealing methods 533 12.9.1 Simulated annealing 533 12.9.2 Parallel tempering 536 13 Sequential Monte Carlo 537 13.1 Introduction 537 13.1.1 Problem statement 537 13.1.2 Particle filtering for state-space models 537 13.1.3 SMC samplers for static parameter estimation 539 13.2 Particle filtering 539 13.2.1 Importance sampling 539 13.2.2 Sequential importance sampling 541 13.2.3 Sequential importance sampling with resampling 542 13.2.4 Resampling methods 545 13.2.5 Adaptive resampling 547 13.3 Proposal distributions 547 13.3.1 Locally optimal proposal 548 13.3.2 Proposals based on the extended and unscented Kalman filter 13.3.3 Proposals based on the Laplace approximation 549 13.3.4 Proposals based on SMC (nested SMC) 551 13.4 Rao-Blackwellized particle filtering (RBPF) 551 13.4.1 Mixture of Kalman filters 551 13.4.2 Example: tracking a maneuvering object 553 13.4.3 Example: FastSLAM 554 13.5 Extensions of the particle filter 557 13.6 SMC samplers 557 13.6.1 Ingredients of an SMC sampler 558 13.6.2 Likelihood tempering (geometric path) 559 13.6.3 Data tempering 561 13.6.4 Sampling rare events and extrema 562 13.6.5
SMC-ABC and likelihood-free inference 563 13.6.6 SMC2 563 13.6.7 Variational filtering SMC 563 13.6.8 Variational smoothing SMC 564 III Prediction 567 14 Predictive models: an overview 569 14.1 Introduction 569 14.1.1 Types of model 569 14.1.2 Model fitting using ERM, MLE, and MAP 570 14.1.3 Model fitting using Bayes, VI, and generalized Bayes 14.2 Evaluating predictive models 572 571 549
xviii CONTENTS 14.2.1 Proper scoring rules 572 14.2.2 Calibration 572 14.2.3 Beyond evaluating marginal probabilities 14.3 Conformal prediction 579 14.3.1 Conformalizing classification 581 14.3.2 Conformalizing regression 581 15 Generalized linear models 576 583 15.1 Introduction 583 15.1.1 Some popular GLMs 583 15.1.2 GLMs with noncanonical link functions 586 15.1.3 Maximum likelihood estimation 587 15.1.4 Bayesian inference 587 15.2 Linear regression 588 15.2.1 Ordinary least squares 588 15.2.2 Conjugate priors 589 15.2.3 Uninformative priors 591 15.2.4 Informative priors 593 15.2.5 Spike and slab prior 595 15.2.6 Laplace prior (Bayesian lasso) 596 15.2.7 Horseshoe prior 597 15.2.8 Automatic relevancy determination 598 15.2.9 Multivariate linear regression 600 15.3 Logistic regression 602 15.3.1 Binary logistic regression 602 15.3.2 Multinomial logistic regression 603 15.3.3 Dealing with class imbalance and the long tail 604 15.3.4 Parameter priors 604 15.3.5 Laplace approximation to the posterior 605 15.3.6 Approximating the posterior predictive distribution 607 15.3.7 MCMC inference 609 15.3.8 Other approximate inference methods 610 15.3.9 Case study: is Berkeley admissions biased against women? 15.4 Probit regression 613 15.4.1 Latent variable interpretation 613 15.4.2 Maximum likelihood estimation 614 15.4.3 Bayesian inference 616 15.4.4 Ordinal probit regression 616 15.4.5 Multinomial probit models 617 15.5 Multilevel (hierarchical) GLMs 617 15.5.1 Generalized linear mixed models (GLMMs) 618 15.5.2 Example: radon regression 618 16 Deep neural networks 623 16.1
Introduction 623 16.2 Building blocks of differentiable circuits 623 16.2.1 Linear layers 624 16.2.2 Nonlinearities 624 16.2.3 Convolutional layers 625 16.2.4 Residual (skip) connections 626 16.2.5 Normalization layers 627 16.2.6 Dropout layers 627 16.2.7 Attention layers 628 16.2.8 Recurrent layers 630 16.2.9 Multiplicative layers 631 16.2.10 Implicit layers 632 16.3 Canonical examples of neural networks 632 611
CONTENTS 16.3.1 16.3.2 16.3.3 16.3.4 16.3.5 16.3.6 xix Multilayer perceptrons (MLPs) 632 Convolutional neural networks (CNNs) 633 Autoencoders 634 Recurrent neural networks (RNNs) 636 Transformers 636 Graph neural networks (GNNs) 637 17 Bayesian neural networks 639 17.1 Introduction 639 17.2 Priors for BNNs 639 17.2.1 Gaussian priors 640 17.2.2 Sparsity-promoting priors 642 17.2.3 Learning the prior 642 17.2.4 Priors in function space 642 17.2.5 Architectural priors 643 17.3 Posteriors for BNNs 643 17.3.1 Monte Carlo dropout 643 17.3.2 Laplace approximation 644 17.3.3 Variational inference 645 17.3.4 Expectation propagation 646 17.3.5 Last layer methods 646 17.3.6 SNGP 647 17.3.7 MCMC methods 647 17.3.8 Methods based on the SGD trajectory 648 17.3.9 Deep ensembles 649 17.3.10 Approximating the posterior predictive distribution 17.3.11 Tempered and cold posteriors 656 17.4 Generalization in Bayesian deep learning 657 17.4.1 Sharp vs flat minima 657 17.4.2 Mode connectivity and the loss landscape 658 17.4.3 Effective dimensionality of a model 658 17.4.4 The hypothesis space of DNNs 660 17.4.5 PAC-Bayes 660 17.4.6 Out-of-distribution generalization for BNNs 661 17.4.7 Model selection for BNNs 663 17.5 Online inference 663 17.5.1 Sequential Laplace for DNNs 664 17.5.2 Extended Kalman filtering for DNNs 665 17.5.3 Assumed density filtering for DNNs 667 17.5.4 Online variational inference for DNNs 668 17.6 Hierarchical Bayesian neural networks 669 17.6.1 Example: multimoons classification 670 18 Gaussian processes 673 18.1 Introduction 673 18.1.1 GPs: what and why? 673 18.2
Mercer kernels 675 18.2.1 Stationary kernels 676 18.2.2 Nonstationary kernels 681 18.2.3 Kernels for nonvectorial (structured) inputs 682 18.2.4 Making new kernels from old 682 18.2.5 Mercer’s theorem 683 18.2.6 Approximating kernels with random features 684 18.3 GPs with Gaussian likelihoods 685 18.3.1 Predictions using noise-free observations 685 18.3.2 Predictions using noisy observations 686 18.3.3 Weight space vs function space 687 653
CONTENTS XX 18.4 18.5 18.6 18.7 18.8 18.3.4 Semiparametric GPs 688 18.3.5 Marginal likelihood 689 18.3.6 Computational and numerical issues 689 18.3.7 Kernel ridge regression 690 GPs with non-Gaussian likelihoods 693 18.4.1 Binary classification 694 18.4.2 Multiclass classification 695 18.4.3 GPs for Poisson regression (Cox process) 696 18.4.4 Other likelihoods 696 Scaling GP inference to large datasets 697 18.5.1 Subset of data 697 18.5.2 Nyström approximation 698 18.5.3 Inducing point methods 699 18.5.4 Sparse variational methods 702 18.5.5 Exploiting parallelization and structure via kernel matrix multiplies 18.5.6 Converting a GP to an SSM 708 Learning the kernel 709 18.6.1 Empirical Bayes for the kernel parameters 709 18.6.2 Bayesian inference for the kernel parameters 712 18.6.3 Multiple kernel learning for additive kernels 713 18.6.4 Automatic search for compositional kernels 714 18.6.5 Spectral mixture kernel learning 717 18.6.6 Deep kernel learning 718 GPs and DNNs 720 18.7.1 Kernels derived from infinitely wide DNNs (NN-GP) 721 18.7.2 Neural tangent kernel (NTK) 723 18.7.3 Deep GPs 723 Gaussian processes for time series forecasting 724 18.8.1 Example: Mauna Loa 724 19 Beyond the iid assumption 727 19.1 Introduction 727 19.2 Distribution shift 727 19.2.1 Motivating examples 727 19.2.2 A causal view of distribution shift 729 19.2.3 The four main types of distribution shift 730 19.2.4 Selection bias 732 19.3 Detecting distribution shifts 732 19.3.1 Detecting shifts using two-sample testing 733 19.3.2 Detecting single out-of-distribution (OOD) inputs 733 19.3.3
Selective prediction 736 19.3.4 Open set and open world recognition 737 19.4 Robustness to distribution shifts 737 19.4.1 Data augmentation 738 19.4.2 Distributionally robust optimization 738 19.5 Adapting to distribution shifts 738 19.5.1 Supervised adaptation using transfer learning 738 19.5.2 Weighted ERM for covariate shift 740 19.5.3 Unsupervised domain adaptation for covariate shift 741 19.5.4 Unsupervised techniques for label shift 742 19.5.5 Test-time adaptation 742 19.6 Learning from multiple distributions 743 19.6.1 Multitask learning 743 19.6.2 Domain generalization 744 19.6.3 Invariant risk minimization 746 19.6.4 Meta learning 747 706
CONTENTS xxi 19.7 Continual learning 750 19.7.1 Domain drift 750 19.7.2 Concept drift 751 19.7.3 Task incremental learning 752 19.7.4 Catastrophic forgetting 753 19.7.5 Online learning 755 19.8 Adversarial examples 756 19.8.1 Whitebox (gradient-based) attacks 758 19.8.2 Blackbox (gradient-free) attacks 759 19.8.3 Real world adversarial attacks 760 19.8.4 Defenses based on robust optimization 760 19.8.5 Why models have adversarial examples 761 IV Generation 763 20 Generative models: an overview 765 20.1 Introduction 765 20.2 Types of generative model 765 20.3 Goals of generative modeling 767 20.3.1 Generating data 767 20.3.2 Density estimation 769 20.3.3 Imputation 770 20.3.4 Structure discovery 771 20.3.5 Latent space interpolation 771 20.3.6 Latent space arithmetic 773 20.3.7 Generative design 774 20.3.8 Model-based reinforcement learning 774 20.3.9 Representation learning 774 20.3.10 Data compression 774 20.4 Evaluating generative models 774 20.4.1 Likelihood-based evaluation 775 20.4.2 Distances and divergences in feature space 776 20.4.3 Precision and recall metrics 777 20.4.4 Statistical tests 778 20.4.5 Challenges with using pretrained classifiers 779 20.4.6 Using model samples to train classifiers 779 20.4.7 Assessing overfitting 779 20.4.8 Human evaluation 780 21 Variational autoencoders 781 21.1 Introduction 781 21.2 VAE basics 781 21.2.1 Modeling assumptions 782 21.2.2 Model fitting 783 21.2.3 Comparison of VAEs and autoencoders 21.2.4 VAEs optimize in an augmented space 21.3 VAE generalizations 786 21.3.1 B-VAE 787 21.3.2 InfoVAE 789 21.3.3 Multimodal VAEs 790
21.3.4 Semisupervised VAEs 793 21.3.5 VAEs with sequential encoders/decoders 21.4 Avoiding posterior collapse 796 21.4.1 KL annealing 797 21.4.2 Lower bounding the rate 798 783 784 794
CONTENTS xxii 21.4.3 Free bits 798 21.4.4 Adding skip connections 798 21.4.5 Improved variational inference 798 21.4.6 Alternative objectives 799 21.5 VAEs with hierarchical structure 799 21.5.1 Bottom-up vs top-down inference 800 21.5.2 Example: very deep VAE 801 21.5.3 Connection with autoregressive models 802 21.5.4 Variational pruning 804 21.5.5 Other optimization difficulties 804 21.6 Vector quantization VAE 805 21.6.1 Autoencoder with binary code 805 21.6.2 VQ-VAE model 805 21.6.3 Learning the prior 807 21.6.4 Hierarchical extension (VQ-VAE-2) 807 21.6.5 Discrete VAE 808 21.6.6 VQ-GAN 809 22 Autoregressive models 811 22.1 Introduction 811 22.2 Neural autoregressive density estimators (NADE) 812 22.3 Causal CNNs 812 22.3.1 Id causal CNN (convolutional Markov models) 22.3.2 2d causal CNN (PixelCNN) 813 22.4 Transformers 814 22.4.1 Text generation (GPT, etc.) 815 22.4.2 Image generation (DALL-E, etc.) 816 22.4.3 Other applications 818 23 Normalizing flows 819 23.1 Introduction 819 23.1.1 Preliminaries 819 23.1.2 How to train a flow model 821 23.2 Constructing flows 822 23.2.1 Affine flows 822 23.2.2 Elementwise flows 822 23.2.3 Coupling flows 825 23.2.4 Autoregressive flows 826 23.2.5 Residual flows 832 23.2.6 Continuous-time flows 834 23.3 Applications 836 23.3.1 Density estimation 836 23.3.2 Generative modeling 836 23.3.3 Inference 837 24 Energy-based models 839 24.1 Introduction 839 24.1.1 Example: products of experts (PoE) 840 24.1.2 Computational difficulties 840 24.2 Maximum likelihood training 841 24.2.1 Gradient-based MCMC methods 842 24.2.2 Contrastive
divergence 842 24.3 Score matching (SM) 846 24.3.1 Basic score matching 846 24.3.2 Denoising score matching (DSM) 847 24.3.3 Sliced score matching (SSM) 848 24.3.4 Connection to contrastive divergence 849 813
CONTENTS xxiii 24.3.5 Score-based generative models 850 24.4 Noise contrastive estimation 850 24.4.1 Connection to score matching 852 24.5 Other methods 852 24.5.1 Minimizing Differences/Derivatives of KL Divergences 24.5.2 Minimizing the Stein discrepancy 853 24.5.3 Adversarial training 854 25 Diffusion models 853 857 25.1 Introduction 857 25.2 Denoising diffusion probabilistic models (DDPMs) 857 25.2.1 Encoder (forwards diffusion) 858 25.2.2 Decoder (reverse diffusion) 859 25.2.3 Model fitting 860 25.2.4 Learning the noise schedule 861 25.2.5 Example: image generation 863 25.3 Score-based generative models (SGMs) 864 25.3.1 Example 864 25.3.2 Adding noise at multiple scales 864 25.3.3 Equivalence to DDPM 866 25.4 Continuous time models using differential equations 867 25.4.1 Forwards diffusion SDE 867 25.4.2 Forwards diffusion ODE 868 25.4.3 Reverse diffusion SDE 869 25.4.4 Reverse diffusion ODE 870 25.4.5 Comparison of the SDE and ODE approach 871 25.4.6 Example 871 25.5 Speeding up diffusion models 871 25.5.1 DDIM sampler 872 25.5.2 Non-Gaussian decoder networks 872 25.5.3 Distillation 873 25.5.4 Latent space diffusion 874 25.6 Conditional generation 875 25.6.1 Conditional diffusion model 875 25.6.2 Classifier guidance 875 25.6.3 Classifier-free guidance 876 25.6.4 Generating high resolution images 876 25.7 Diffusion for discrete state spaces 877 25.7.1 Discrete Denoising Diffusion Probabilistic Models 877 25.7.2 Choice of Markov transition matrices for the forward processes 25.7.3 Parameterization of the reverse process 879 25.7.4 Noise schedules 880 25.7.5 Connections
to other probabilistic models for discrete sequences 26 Generative adversarial networks 883 26.1 Introduction 883 26.2 Learning by comparison 884 26.2.1 Guiding principles 885 26.2.2 Density ratio estimation using binary classifiers 26.2.3 Bounds on /-divergences 888 26.2.4 Integral probability metrics 890 26.2.5 Moment matching 892 26.2.6 On density ratios and differences 892 26.3 Generative adversarial networks 894 26.3.1 From learning principles to loss functions 894 26.3.2 Gradient descent 895 26.3.3 Challenges with GAN training 897 886 878 880
CONTENTS xxiv 26.4 26.5 26.6 26.7 V 26.3.4 Improving GAN optimization 898 26.3.5 Convergence of GAN training 898 Conditional GANs 902 Inference with GANs 903 Neural architectures in GANs 904 26.6.1 The importance of discriminator architectures 26.6.2 Architectural inductive biases 905 26.6.3 Attention in GANs 905 26.6.4 Progressive generation 906 26.6.5 Regularization 907 26.6.6 Scaling up GAN models 908 Applications 908 26.7.1 GANs for image generation 908 26.7.2 Video generation 911 26.7.3 Audio generation 912 26.7.4 Text generation 912 26.7.5 Imitation learning 913 26.7.6 Domain adaptation 914 26.7.7 Design, art and creativity 914 Discovery 915 27 Discovery methods: an overview 27.1 Introduction 917 27.2 Overview of Part V 28 Latent factor models 904 917 918 919 28.1 Introduction 919 28.2 Mixture models 919 28.2.1 Gaussian mixture models (GMMs) 920 28.2.2 Bernoulli mixture models 922 28.2.3 Gaussian scale mixtures (GSMs) 922 28.2.4 Using GMMs as a prior for inverse imaging problems 924 28.2.5 Using mixture models for classification problems 927 28.3 Factor analysis 929 28.3.1 Factor analysis: the basics 929 28.3.2 Probabilistic PCA 934 28.3.3 Mixture of factor analyzers 936 28.3.4 Factor analysis models for paired data 943 28.3.5 Factor analysis with exponential family likelihoods 945 28.3.6 Factor analysis with DNN likelihoods (VAEs) 948 28.3.7 Factor analysis with GP likelihoods (GP-LVM) 948 28.4 LFMs with non-Gaussian priors 949 28.4.1 Non-negative matrix factorization (NMF) 949 28.4.2 Multinomial PCA 950 28.5 Topic models 953 28.5.1 Latent Dirichlet allocation (LDA)
953 28.5.2 Correlated topic model 957 28.5.3 Dynamic topic model 957 28.5.4 LDA-HMM 958 28.6 Independent components analysis (ICA) 962 28.6.1 Noiseless ICA model 962 28.6.2 The need for non-Gaussian priors 963 28.6.3 Maximum likelihood estimation 964 28.6.4 Alternatives to MLE 965
CONTENTS 28.6.5 28.6.6 XXV Sparse coding Nonlinear ICA 966 967 29 State-space models 969 29.1 Introduction 969 29.2 Hidden Markov models (HMMs) 970 29.2.1 Conditional independence properties 970 29.2.2 State transition model 970 29.2.3 Discrete likelihoods 971 29.2.4 Gaussian likelihoods 972 29.2.5 Autoregressive likelihoods 972 29.2.6 Neural network likelihoods 973 29.3 HMMs: applications 974 29.3.1 Time series segmentation 974 29.3.2 Protein sequence alignment 976 29.3.3 Spelling correction 978 29.4 HMMs: parameter learning 980 29.4.1 The Baum-Welch (EM) algorithm 980 29.4.2 Parameter estimation using SGD 983 29.4.3 Parameter estimation using spectral methods 984 29.4.4 Bayesian HMMs 985 29.5 HMMs: generalizations 987 29.5.1 Hidden semi-Markov model (HSMM) 987 29.5.2 Hierarchical HMMs 989 29.5.3 Factorial HMMs 991 29.5.4 Coupled HMMs 992 29.5.5 Dynamic Bayes nets (DBN) 992 29.5.6 Changepoint detection 993 29.6 Linear dynamical systems (LDSs) 996 29.6.1 Conditional independence properties 996 29.6.2 Parameterization 996 29.7 LDS: applications 997 29.7.1 Object tracking and state estimation 997 29.7.2 Online Bayesian linear regression (recursive least squares) 29.7.3 Adaptive filtering 1000 29.7.4 Time series forecasting 1000 29.8 LDS: parameter learning 1001 29.8.1 EM for LDS 1001 29.8.2 Subspace identification methods 1003 29.8.3 Ensuring stability of the dynamical system 1003 29.8.4 Bayesian LDS 1004 29.9 Switching linear dynamical systems (SLDSs) 1005 29.9.1 Parameterization 1005 29.9.2 Posterior inference 1006 29.9.3 Application: Multitarget tracking 1006 29.10
Nonlinear SSMs 1009 29.10.1 Example: object tracking and state estimation 1010 29.10.2 Posterior inference 1010 29.11 Non-Gaussian SSMs 1010 29.11.1 Example: spike train modeling 1011 29.11.2 Example: stochastic volatility models 1012 29.11.3 Posterior inference 1012 29.12 Structural time series models 1012 29.12.1 Introduction 1013 29.12.2 Structural building blocks 1013 29.12.3 Model fitting 1016 998
CONTENTS xxvi 29.12.4 Forecasting 1016 29.12.5 Examples 1016 29.12.6 Causal impact of a time series intervention 29.12.7 Prophet 1024 29.12.8 Neural forecasting methods 1024 29.13 Deep SSMs 1025 29.13.1 Deep Markov models 1026 29.13.2 Recurrent SSM 1027 29.13.3 Improving multistep predictions 1027 29.13.4 Variational RNNs 1028 30 Graph learning 1031 30.1 Introduction 1031 30.2 Latent variable models for graphs 30.3 Graphical model structure learning 31 Nonparametric Bayesian models 31.1 Introduction 1020 1031 1031 1035 1035 32 Representation learning 1037 32.1 Introduction 1037 32.2 Evaluating and comparing learned representations 1037 32.2.1 Downstream performance 1038 32.2.2 Representational similarity 1040 32.3 Approaches for learning representations 1044 32.3.1 Supervised representation learning and transfer 1045 32.3.2 Generative representation learning 1047 32.3.3 Self-supervised representation learning 1049 32.3.4 Multiview representation learning 1052 32.4 Theory of representation learning 1057 32.4.1 Identifiability 1057 32.4.2 Information maximization 1058 33 Interpretability 1061 33.1 Introduction 1061 33.1.1 The role of interpretability: unknowns and under-specifications 1062 33.1.2 Terminology and framework 1063 33.2 Methods for interpretable machine learning 1066 33.2.1 Inherently interpretable models: the model is its explanation 1067 33.2.2 Semi-inherently interpretable models: example-based methods 1069 33.2.3 Post-hoc or joint training: the explanation gives a partial view of the model 1069 33.2.4 Transparency and visualization 1073 33.3 Properties: the
abstraction between context and method 1074 33.3.1 Properties of explanations from interpretable machine learning 1074 33.3.2 Properties of explanations from cognitive science 1076 33.4 Evaluation of interpretable machine learning models 1077 33.4.1 Computational evaluation: does the method have desired properties? 1078 33.4.2 User study-based evaluation: does the method help a user perform a target task? 1082 33.5 Discussion: how to think about interpretable machine learning 1086 VI Action 1091 34 Decision making under uncertainty 34.1 Statistical decision theory 1093 34.1.1 Basics 1093 34.1.2 Frequentist decision theory 1093 1093
CONTENTS 34.2 34.3 34.4 34.5 34.6 34.7 xxvii 34.1.3 Bayesian decision theory 1094 34.1.4 Frequentist optimality of the Bayesian approach 1095 34.1.5 Examples of one-shot decision making problems 1095 Decision (influence) diagrams 1099 34.2.1 Example: oil wildcatter 1100 34.2.2 Information arcs 1101 34.2.3 Value of information 1101 34.2.4 Computing the optimal policy 1102 A/В testing 1103 34.3.1 A Bayesian approach 1103 34.3.2 Example 1106 Contextual bandits 1107 34.4.1 Types of bandit 1108 34.4.2 Applications 1109 34.4.3 Exploration-exploitation tradeoff 1109 34.4.4 The optimal solution 1110 34.4.5 Upper confidence bounds (UCBs) 1111 34.4.6 Thompson sampling 1113 34.4.7 Regret 1114 Markov decision problems 1116 34.5.1 Basics 1116 34.5.2 Partially observed MDPs 1117 34.5.3 Episodes and returns 1117 34.5.4 Value functions 1119 34.5.5 Optimal value functions and policies 1119 Planning in an MDP 1120 34.6.1 Value iteration 1121 34.6.2 Policy iteration 1122 34.6.3 Linear programming 1123 Active learning 1124 34.7.1Active learning scenarios 1124 34.7.2 Relationship to other forms of sequential decision making 34.7.3 Acquisition strategies 1126 34.7.4 Batch active learning 1128 35 Reinforcement learning 1133 35.1 Introduction 1133 35.1.1 Overview of methods 1133 35.1.2 Value-based methods 1135 35.1.3 Policy search methods 1135 35.1.4 Model-based RL 1135 35.1.5 Exploration-exploitation tradeoff 1136 35.2 Value-based RL 1138 35.2.1 Monte Carlo RL 1138 35.2.2 Temporal difference (TD) learning 1138 35.2.3 TD learning with eligibility traces 1139 35.2.4 SARSA: on-policy TD control 1140
35.2.5 Q-learning: off-policy TD control 1141 35.2.6 Deep Q-network (DQN) 1142 35.3 Policy-based RL 1144 35.3.1 The policy gradient theorem 1145 35.3.2 REINFORCE 1146 35.3.3 Actor-critic methods 1146 35.3.4 Bound optimization methods 1148 35.3.5 Deterministic policy gradient methods 1150 35.3.6 Gradient-free methods 1151 1125
CONTENTS xxviii 35.4 Model-based RL 1151 35.4.1 Model predictive control (MPC) 1151 35.4.2 Combining model-based and model-free 1153 35.4.3 MBRL using Gaussian processes 1154 35.4.4 MBRL using DNNs 1155 35.4.5 MBRL using latent-variable models 1156 35.4.6 Robustness to model errors 1158 35.5 Off-policy learning 1158 35.5.1 Basic techniques 1159 35.5.2 The curse of horizon 1162 35.5.3 The deadly triad 1163 35.6 Control as inference 1165 35.6.1 Maximum entropy reinforcement learning 1165 35.6.2 Other approaches 1167 35.6.3 Imitation learning 1168 36 Causality 1171 36.1 Introduction 1171 36.2 Causal formalism 1173 36.2.1 Structural causal models 1173 36.2.2 Causal DAGs 1174 36.2.3 Identification 1176 36.2.4 Counterfactuals and the causal hierarchy 1178 36.3 Randomized control trials 1180 36.4 Confounder adjustment 1181 36.4.1 Causal estimand, statistical estimand, and identification 1181 36.4.2 ATE estimation with observed confounders 1184 36.4.3 Uncertainty quantification 1189 36.4.4 Matching 1189 36.4.5 Practical considerations and procedures 1190 36.4.6 Summary and practical advice 1193 36.5 Instrumental variable strategies 1195 36.5.1 Additive unobserved confounding 1196 36.5.2 Instrument monotonicity and local average treatment effect 1198 36.5.3 Two stage least squares 1201 36.6 Difference in differences 1202 36.6.1 Estimation 1205 36.7 Credibility checks 1206 36.7.1 Placebo checks 1206 36.7.2 Sensitivity analysis to unobserved confounding 1207 36.8 The do-calculus 1215 36.8.1 The three rules 1215 36.8.2 Revisiting backdoor adjustment 1216 36.8.3 Frontdoor adjustment
1217 36.9 Further reading 1218 Index 1221 Bibliography 1239 |
| any_adam_object | 1 |
| author | Murphy, Kevin P. 1970- |
| author_GND | (DE-588)1026956137 |
| author_facet | Murphy, Kevin P. 1970- |
| author_role | aut |
| author_sort | Murphy, Kevin P. 1970- |
| author_variant | k p m kp kpm |
| building | Verbundindex |
| bvnumber | BV049057529 |
| classification_rvk | ST 300 |
| ctrlnum | (OCoLC)1397517603 (DE-599)KXP182326283X |
| discipline | Informatik |
| format | Book |
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| id | DE-604.BV049057529 |
| illustrated | Illustrated |
| indexdate | 2025-02-24T09:01:24Z |
| institution | BVB |
| isbn | 9780262048439 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034319734 |
| oclc_num | 1397517603 |
| open_access_boolean | 1 |
| owner | DE-384 DE-83 DE-20 DE-M347 DE-898 DE-BY-UBR DE-573 DE-473 DE-BY-UBG DE-29T DE-Aug4 DE-19 DE-BY-UBM DE-4325 |
| owner_facet | DE-384 DE-83 DE-20 DE-M347 DE-898 DE-BY-UBR DE-573 DE-473 DE-BY-UBG DE-29T DE-Aug4 DE-19 DE-BY-UBM DE-4325 |
| physical | xxxi, 1319 Seiten Illustrationen, Diagramme |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | The MIT Press |
| record_format | marc |
| series2 | Adaptive Computation and Machine Learning series |
| spelling | Murphy, Kevin P. 1970- Verfasser (DE-588)1026956137 aut Probabilistic machine learning advanced topics Kevin P. Murphy Cambridge, Massachusetts ; London, England The MIT Press [2023] © 2023 xxxi, 1319 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Adaptive Computation and Machine Learning series Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Erscheint auch als Online-Ausgabe, EPUB 978-0-262-37600-6 Erscheint auch als Online-Ausgabe, PDF 978-0-262-37599-3 https://probml.github.io/pml-book/book2.html Verlag kostenfrei Volltext Digitalisierung UB Bamberg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034319734&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murphy, Kevin P. 1970- Probabilistic machine learning advanced topics Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maschinelles Lernen (DE-588)4193754-5 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4193754-5 |
| title | Probabilistic machine learning advanced topics |
| title_auth | Probabilistic machine learning advanced topics |
| title_exact_search | Probabilistic machine learning advanced topics |
| title_full | Probabilistic machine learning advanced topics Kevin P. Murphy |
| title_fullStr | Probabilistic machine learning advanced topics Kevin P. Murphy |
| title_full_unstemmed | Probabilistic machine learning advanced topics Kevin P. Murphy |
| title_short | Probabilistic machine learning |
| title_sort | probabilistic machine learning advanced topics |
| title_sub | advanced topics |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maschinelles Lernen (DE-588)4193754-5 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Maschinelles Lernen |
| url | https://probml.github.io/pml-book/book2.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034319734&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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