
Contents
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14.1 Two examples 14.1 Two examples
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14.1.1 Example 1: Ising chain 14.1.1 Example 1: Ising chain
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14.1.2 Example 2: A tree-parity-check code 14.1.2 Example 2: A tree-parity-check code
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14.2 Belief propagation on tree graphs 14.2 Belief propagation on tree graphs
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14.2.1 Three problems 14.2.1 Three problems
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14.2.2 The BP equations 14.2.2 The BP equations
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14.2.3 Correlations and energy 14.2.3 Correlations and energy
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14.2.4 Entropy 14.2.4 Entropy
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14.2.5 Pairwise models 14.2.5 Pairwise models
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14.3 Optimization: Max-product and min-sum 14.3 Optimization: Max-product and min-sum
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14.3.1 Max-marginals 14.3.1 Max-marginals
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14.3.2 Message passing 14.3.2 Message passing
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14.3.3 Warning propagation 14.3.3 Warning propagation
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14.4 Loopy BP 14.4 Loopy BP
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14.4.1 The Bethe free entropy 14.4.1 The Bethe free entropy
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14.4.2 Correlations 14.4.2 Correlations
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14.5 General message−passing algorithms 14.5 General message−passing algorithms
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14.6 Probabilistic analysis 14.6 Probabilistic analysis
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14.6.1 Assumptions 14.6.1 Assumptions
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14.6.2 Density evolution equations 14.6.2 Density evolution equations
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14.6.3 The replica-symmetric cavity method 14.6.3 The replica-symmetric cavity method
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14.6.4 Numerical methods 14.6.4 Numerical methods
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Notes Notes
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Cite
Abstract
This chapter discusses a general method for approximating marginals of large graphical models. This powerful technique has been discovered independently in various fields: statistical physics (under the name ‘Bethe Peierls approximation’), coding theory (‘sum-product’ and ‘min-sum’ algorithms), and artificial intelligence (‘belief propagation’). It is based on an exchange of messages between variables and factors, along the edges of the factor graph. These messages are interpreted as probability distributions for the variable in a graph where a cavity has been dug. The chapter also discusses the statistical analysis of these messages in large random graphical models: density evolution and the replica symmetric cavity method.
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