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(3) Contents
(5) Preface
(13) Information Theory
(13) 1 Introduciton to Information Theory
(13) About Chapter 1
(15) 1.1 How can we achieve perfect communication over an imperfect, noisy communication channel?
(17) 1.2 Error-correcting codes for the binary symmetric channel
(26) 1.3 What performance can the best codes achieve?
(27) 1.4 Summary
(28) 1.5 Further exercises
(28) 1.6 Solutions
(34) 2 Probability, Entropy, and Inference
(34) 2.1 Probabilities and ensembles
(37) 2.2 The meaning of probability
(39) 2.3 Forward probabilities and inverse probabilities
(44) 2.4 Definition of entropy and related functions
(45) 2.5 Decomposability of the entropy
(46) 2.6 Gibbs inequality
(47) 2.7 Jensens inequality for convex functions
(48) 2.8 Exercises
(50) 2.9 Further exercises
(52) 2.10 Solutions
(59) 3 More about Inference
(59) About Chapter 3
(60) 3.1 A first inference problem
(63) 3.2 The bent coin
(64) 3.3 The bent coin and model comparison
(67) 3.4 An example of legal evidence
(69) 3.5 Exercises
(71) 3.6 Solutions
(77) I- Data Compression
(78) 4 The Source Coding Theorem
(78) About Chapter 4
(79) 4.1 How to measure the information content of a random variable?
(85) 4.2 Data compression
(86) 4.3 Information content defined in terms of lossy compression
(90) 4.4 Typicality
(93) 4.5 Proofs
(95) 4.6 Comments
(96) 4.7 Exercises
(98) 4.8 Solutions
(102) 5 Symbol Codes
(102) About Chapter 5
(103) Introduction
(104) 5.1 Symbol codes
(106) 5.2 What limit is imposed by unique decodeability?
(109) 5.3 What’s the most compression that we can hope for?
(110) 5.4 How much can we compress?
(110) 5.5 Optimal source coding with symbol codes: Huffman coding
(112) 5.6 Disadvantages of the Huffman code
(114) 5.7 Summary
(114) 5.8 Exercises
(116) 5.9 Solutions
(121) 6 Stream Codes
(121) About Chapter 6
(122) 6.1 The guessing game
(123) 6.2 Arithmetic codes
(130) 6.3 Further applications of arithmetic coding
(131) 6.4 Lempel–Ziv coding
(133) 6.5 Demonstration
(134) 6.6 Summary
(135) 6.7 Exercises on stream codes
(136) 6.8 Further exercises on data compression
(139) 6.9 Solutions
(144) 7 Codes for Integers
(149) II- Noisy-Channel Coding
(150) 8 Dependent Random Variables
(150) 8.1 More about entropy
(152) 8.2 Exercises
(153) 8.3 Further exercises
(154) 8.4 Solutions
(157) 9 Communication over a Noisy Channel
(157) About Chapter 9
(158) 9.1 The big picture
(159) 9.2 Review of probability and information
(159) 9.3 Noisy channels
(160) 9.4 Inferring the input given the output
(161) 9.5 Information conveyed by a channel
(163) 9.6 The noisy-channel coding theorem
(165) 9.7 Intuitive preview of proof
(167) 9.8 Further exercises
(169) 9.9 Solutions
(173) 10 The Noisy-Channel Coding Theorem
(173) About Chapter 10
(174) 10.1 The theorem
(174) 10.2 Jointly-typical sequences
(176) 10.3 Proof of the noisy-channel coding theorem
(179) 10.4 Communication (with errors) above capacity
(180) 10.5 The non-achievable region (part 3 of the theorem)
(181) 10.6 Computing capacity
(183) 10.7 Other coding theorems
(184) 10.8 Noisy-channel coding theorems and coding practice
(184) 10.9 Further exercises
(185) 10.10 Solutions
(188) 11 Error-Correcting Codes and Real Channels
(188) About Chapter 11
(189) 11.1 The Gaussian channel
(191) 11.2 Inferring the input to a real channel
(191) 11.3 Capacity of Gaussian channel
(195) 11.4 What are the capabilities of practical error-correcting codes?
(198) 11.5 The state of the art
(199) 11.6 Summary
(199) 11.7 Nonlinear codes
(199) 11.8 Errors other than noise
(200) 11.9 Exercises
(201) 11.10 Solutions
(203) III- Further Topics in Information Theory
(204) 12 Hash Codes: Codes for Efficient Information Retrieval
(204) About Chapter 12
(205) 12.1 The information-retrieval problem
(207) 12.2 Hash codes
(209) 12.3 Collision resolution
(210) 12.4 Planning for collisions: a birthday problem
(210) 12.5 Other roles for hash codes
(213) 12.6 Further exercises
(214) 12.7 Solutions
(217) 13 Binary Codes
(217) About Chapter 13
(218) 13.1 Distance properties of a code
(218) 13.2 Obsession with distance
(220) 13.3 Perfect codes
(222) 13.4 Perfectness is unattainable – first proof
(223) 13.5 Weight enumerator function of random linear codes
(225) 13.6 Berlekamps bats
(226) 13.7 Concatenation of Hamming codes
(227) 13.8 Distance isnt everything
(228) 13.9 The union bound
(228) 13.10 Dual codes
(231) 13.11 Generalizing perfectness to other channels
(232) 13.12 Summary
(232) 12.13 Further exercises
(235) Solutions
(240) 14 Very Good Linear Codes Exist
(240) About Chapter 14
(241) 14.1 A simultaneous proof of the source coding and noisy-channel coding theorems
(243) 14.2 Data compression by linear hash codes
(245) 15 Further Exercises on Information Theory
(251) Solutions
(253) 16 Message Passing
(253) 16.1 Counting
(256) 16.2 Path-counting
(257) 16.3 Finding the lowest-cost path
(258) 16.4 Summary and related ideas
(258) 16.5 Further exercises
(259) 16.6 Solutions
(260) 17 Communication over Constrained Noiseless Channels
(260) 17.1 Three examples of constrained binary channels
(262) 17.2 The capacity of a constrained noiseless channel
(263) 17.3 Counting the number of possible messages
(266) 17.4 Back to our model channels
(266) 17.5 Practical communication over constrained channels
(268) 17.6 Variable symbol durations
(269) 17.7 Solutions
(272) 18 Crosswords and Codebreaking
(272) 18.1 Crosswords
(274) 18.2 Simple language models
(276) 18.3 Units of information content
(277) 18.4 A taste of Banburismus
(280) 18.5 Exercises
(281) 19 Why have Sex? Information Acquisition and Evolution
(281) Introduction
(282) 19.1 The model
(283) 19.2 Rate of increase of fitness
(287) 19.3 The maximal tolerable mutation rate
(288) 19.4 Fitness increase and information acquisition
(289) 19.5 Discussion
(291) 19.6 Further exercises
(292) 19.7 Solutions
(293) IV- Probabilities and Inference
(294) About Part IV
(296) 20 An Example Inference Task: Clustering
(297) 20.1 K-means clustering
(301) 20.2 Soft K-means clustering
(301) 20.3 Conclusion
(302) 20.4 Exercises
(303) 20.5 Solutions
(305) 21 Exact Inference by Complete Enumeration
(305) 21.1 The burglar alarm
(307) 21.2 Exact inference for continuous hypothesis spaces
(312) 22 Maximum Likelihood and Clustering
(312) 22.1 Maximum likelihood for one Gaussian
(314) 22.2 Maximum likelihood for a mixture of Gaussians
(315) 22.3 Enhancements to soft K-means
(317) 22.4 A fatal flaw of maximum likelihood
(318) 22.5 Further exercises
(322) 22.6 Solutions
(323) 23 Useful Probability Distributions
(323) 23.1 Distributions over integers
(324) 23.2 Distributions over unbounded real numbers
(325) 23.3 Distributions over positive real numbers
(327) 23.4 Distributions over periodic variables
(328) 23.5 Distributions over probabilities
(330) 23.6 Further exercises
(331) 24 Exact Marginalization
(331) 24.1 Inferring the mean and variance of a Gaussian distribution
(335) 24.2 Exercises
(335) 24.3 Solutions
(336) 25 Exact Marginalization in Trellises
(336) 25.1 Decoding problems
(338) 25.2 Codes and trellises
(339) 25.3 Solving the decoding problems on a trellis
(343) 25.4 More on trellises
(345) 25.5 Solutions
(346) 26 Exact Marginalization in Graphs
(346) 26.1 The general problem
(348) 26.2 The sum–product algorithm
(351) 26.3 The min–sum algorithm
(352) 26.4 The junction tree algorithm
(352) 26.5 Exercises
(353) 27 Laplaces Method
(354) 27.1 Exercises
(355) 28 Model Comparison and Occams Razor
(355) 28.1 Occams razor
(363) 28.2 Example
(364) 28.3 Minimum description length (MDL)
(366) 28.4 Exercises
(368) 29 Monte Carlo Methods
(368) About Chapter 29
(369) 29.1 The problems to be solved
(373) 29.2 Importance sampling
(376) 29.3 Rejection sampling
(377) 29.4 The Metropolis–Hastings method
(382) 29.5 Gibbs sampling
(384) 29.6 Terminology for Markov chain Monte Carlo methods
(386) 29.7 Slice sampling
(391) 29.8 Practicalities
(391) 29.9 Further practical issues
(393) 29.10 Summary
(394) 29.11 Exercises
(396) 29.12 Solutions
(399) 30 Efficient Monte Carlo Methods
(399) 30.1 Hamiltonian Monte Carlo
(402) 30.2 Overrelaxation
(404) 30.3 Simulated annealing
(404) 30.4 Skillings multi-state leapfrog method
(406) 30.5 Monte Carlo algorithms as communication channels
(407) 30.6 Multi-state methods
(409) 30.7 Methods that do not necessarily help
(409) 30.8 Further exercises
(410) 30.9 Solutions
(411) 31 Ising Models
(411) About Chapter 31
(412) Introduction
(414) 31.1 Ising models – Monte Carlo simulation
(419) 31.2 Direct computation of partition function of Ising models
(424) 31.3 Exercises
(425) 32 Exact Monte Carlo Sampling
(425) 32.1 The problem with Monte Carlo methods
(425) 32.2 Exact sampling concepts
(430) 32.3 Exact sampling from interesting distributions
(432) 32.4 Exercises
(433) 32.5 Solutions
(434) 33 Variational Methods
(434) 33.1 Variational free energy minimization
(436) 33.2 Variational free energy minimization for spin systems
(438) 33.3 Example: mean field theory for the ferromagnetic Ising model
(439) 33.4 Variational methods in inference and data modelling
(441) 33.5 The case of an unknown Gaussian: approximating the posterior distrubution of mu and sigma
(443) 33.6 Interlude
(444) 33.7 K-means clustering and the expectation–maximization algorithm as a variational method
(445) 33.8 Variational methods other than free energy minimization
(446) 33.9 Further exercises
(447) 33.10 Solutions
(449) 34 Independent Component Analysis and Latent Variable Modelling
(449) 34.1 Latent variable models
(449) 34.2 The generative model for independent component analysis
(452) 34.3 A covariant, simpler, and faster learning algorithm
(456) 34.4 Exercises
(457) 35 Random Inference Topics
(457) 35.1 What do you know if you are ignorant?
(458) 35.2 The Luria–Delbruck distribution
(459) 35.3 Inferring causation
(460) 35.4 Further exercises
(461) 35.5 Solutions
(463) 36 Decision Theory
(463) 36.1 Rational prospecting
(465) 36.2 Further reading
(466) 36.3 Further exercises
(469) 37 Bayesian Inference and Sampling Theory
(469) Introduction
(470) 37.1 A medical example
(474) 37.2 Dependence of p-values on irrelevant information
(476) 37.3 Confidence intervals
(477) 37.4 Some compromise positions
(478) 37.5 Further exercises
(479) V- Neural Networks
(480) 38 Introduction to Neural Networks
(480) 38.1 Memories
(482) 38.2 Terminology
(483) 39 The Single Neuron as a Classifier
(483) 39.1 The single neuron
(484) 39.2 Basic neural network concepts
(487) 39.3 Training the single neuron as a binary classifier
(491) 39.4 Beyond descent on the error function: regularization
(492) 39.5 Further exercises
(494) 40 Capacity of a Single Neuron
(494) Problems to look at before Chapter 40
(495) 40.1 Neural network learning as communication
(495) 40.2 The capacity of a single neuron
(497) 40.3 Counting threshold functions
(502) 40.4 Further exercises
(503) 40.5 Solutions
(504) 41 Learning as Inference
(504) 41.1 Neural network learning as inference
(505) 41.2 Illustration for a neuron with two weights
(505) 41.3 Beyond optimization: making predictions
(508) 41.4 Monte Carlo implementation of a single neuron
(513) 41.5 Implementing inference with Gaussian approximations
(516) Postscript on Supervised Neural Networks
(517) 42 Hopfield Networks
(517) 42.1 Hebbian learning
(518) 42.2 Definition of the binary Hopfield network
(519) 42.3 Definition of the continuous Hopfield network
(519) 42.4 Convergence of the Hopfield network
(522) 42.5 The associative memory in action
(522) 42.6 The continuous-time continuous Hopfield network
(524) 42.7 The capacity of the Hopfield network
(527) 42.8 Improving on the capacity of the Hebb rule
(528) 42.9 Hopfield networks for optimization problems
(531) 42.10 Further exercises
(532) 42.11 Solutions
(534) 43 Boltzmann Machines
(534) 43.1 From Hopfield networks to Boltzmann machines
(537) 43.2 Boltzmann machine with hidden units
(538) 43.3 Exercise
(539) 44 Supervised Learning in Multilayer Networks
(539) 44.1 Multilayer perceptrons
(540) 44.2 How a regression network is traditionally trained
(541) 44.3 Neural network learning as inference
(542) 44.4 Benefits of the Bayesian approach to supervised feedforward neural networks
(544) 44.5 Exercises
(546) 45 Gaussian Processes
(546) About Chapter 45
(547) Introduction
(548) 45.1 Standard methods for nonlinear regression
(552) 45.2 From parametric models to Gaussian processes
(554) 45.3 Using a given Gaussian process model in regression
(555) 45.4 Examples of covariance functions
(557) 45.5 Adaptation of Gaussian process models
(559) 45.6 Classification
(559) 45.7 Discussion
(561) 46 Deconvolution
(561) 46.1 Traditional image reconstruction methods
(564) 46.2 Supervised neural networks for image deconvolution
(564) 46.3 Deconvolution in humans
(566) 46.4 Exercises
(567) VI- Sparse Graph Codes
(568) About Part VI
(569) 47 Low-Density Parity-Check Codes
(569) 47.1 Theoretical properties
(569) 47.2 Practical decoding
(571) 47.3 Decoding with the sum–product algorithm
(574) 47.4 Pictorial demonstration of Gallager codes
(578) 47.5 Density evolution
(579) 47.6 Improving Gallager codes
(581) 47.7 Fast encoding of low-density parity-check codes
(583) 47.8 Further reading
(584) 47.9 Exercises
(584) 47.10 Solutions
(586) 48 Convolutional Codes and Turbo Codes
(586) 48.1 Introduction to convolutional codes
(586) 48.2 Linear-feedback shift-registers
(590) 48.3 Decoding convolutional codes
(591) 48.4 Turbo codes
(593) 48.5 Parity-check matrices of convolutional codes and turbo codes
(593) 48.6 Solutions
(594) 49 Repeat-Accumulate Codes
(594) 49.1 The encoder
(594) 49.2 Graph
(595) 49.3 Decoding
(596) 49.4 Empirical distribution of decoding times
(596) 49.5 Generalized parity-check matrices
(600) 50 Digital Fountain Codes
(600) About Chapter 50
(601) Introduction
(602) 50.1 A digital fountains encoder
(603) 50.2 The decoder
(603) 50.3 Designing the degree distribution
(605) 50.4 Applications
(606) 50.5 Further exercises
(608) 50.6 Summary of sparse-graph codes
(608) 50.7 Conclusion
(609) VII- Appendices
(610) A- Notation
(613) B- Some Physics
(617) C- Some Mathematics
(617) C.1 Finite field theory
(618) C.2 Eigenvectors and eigenvalues
(620) C.3 Perturbation theory
(624) C.4 Some numbers
(625) Bibliography
(632) Index