Applications of Extended Ensemble Monte Carlo

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Applications of Extended Ensemble Monte Carlo

• Yukito IBA
• The Institute of Statistical Mathematics, Tokyo,
  Japan

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Extended Ensemble MCMC

• A Generic Name which indicates
• Parallel Tempering,
• Simulated Tempering,
• Multicanonical Sampling,
• Wang-Landau,
• Umbrella Sampling

Valleau and Torrie
1970s
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Contents

• 1. Basic Algorithms
• Parallel Tempering .vs Multicanonical
• 2. Exact Calculation with soft Constraints
• Lattice Protein / Counting Tables
• 3. Rare Events and Large Deviations
• Communication Channels
• Chaotic Dynamical Systems

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Basic Algorithms

• Parallel Tempering
• Multicanonical Monte Carlo

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References in physics

• Iba (2001) Extended Ensemble Monte Carlo
• Int. J. Mod. Phys. C12 p.623.
• A draft version will be found at
• http//arxiv.org/abs/cond-mat/0012323
• Landau and Binder (2005)
• A Guide to Monte Carlo Simulations in
  Statistical Physics (2nd ed. , Cambridge)
• A number of preprints will be found in
• Los Alamos Arxiv on the web.

This slide is added after the talk
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Slow mixing by multimodal dist.
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Bridging
fast mixing high temperature
slow mixing low temperature
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Path Sampling
1.Facilitate Mixing 2.Calculate Normalizing
Constant (free energy)
Path Sampling Gelman and Meng (1998) stress
2. but 1. is also important
In Physics from 2. to 1. 1970s ? 1990s
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Parallel Tempering

• a.k.a. Replica Exchange MC
• Metropolis Coupled MCMC

Geyer (1991), Kimura and Taki (1991) Hukushima
and Nemoto (1996) Iba(1993, in Japanese)
Simulate Many Replicas in Parallel
MCMC in a Product Space
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Examples

• Gibbs Distributions with different temperatures
• Any Family parameterized by
• a hyperparameter

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Exchange of Replicas
K4
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Accept/Reject Exchange

• Calculate Metropolis Ratio
• Generate a Uniform Random Number
• in 0,1) and accept exchange
• iff

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Detailed Balance in Extended Space
Combined Distribution
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Multicanonical Monte Carlo
Berg et al. (1991,1992)

sufficient statistics
Energy not Expectation
Exponential Family
sufficient statistics
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Density of States

The number of which satisfy
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Multicanonical Sampling
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Weight and Marginal Distribution
Original (Gibbs) Multicanonical
Random
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flat marginal distribution
Scanning broad range of E
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Reweighting

• Formally, for arbitrary it holds.
• Practically, is required,
  
• else the variance diverges in a large system.

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Q. How can we do without knowledge on D(E)

• Ans.
• Estimate D(E) in the preliminary runs
• k th simulation

Simplest Method Entropic Sampling
in
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Estimation of Density of States
(Ising Model on a random net)
30000 MCS
2
k1
3
5
4
10
14
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k15
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Estimation of D(E)

• Histogram
• Piecewise Linear
• Fitting, Kernel Density Estimation ..
• Wang-Landau
• Flat Histogram

Entropic Sampling
Original Multicanonical
Continuous Cases D(E)dE Non-trivial Task
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Parallel Tempering / Multicanonical
parallel tempering combined distribution simula
ted tempering mixture distribution to
approximate
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Potts model (2-dim, q10 states)
disordered
ordered
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Phase Coexistence/ 1st order transition

• parameter (Inverse Temperature) changes
• sufficient statistics (Energy) jumps

water and ice coexists
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Potts model (2-dim, q10 states)
disordered
ordered
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Comparison

• _at_ Simple Liquids , Potts Models ..
• Multicanonical seems better than Parallel
  Tempering
• _at_ But, for more difficult cases ?
• ex. Ising Model with three spin Interaction

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Soft Constraints

• Lattice Protein
• Counting Tables

The results on Lattice Protein are taken from
joint works with G Chikenji (Nagoya Univ) and
Macoto Kikuchi (Osaka Univ) Some examples are
also taken from the other works by Kikuchi and
coworkers.
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Lattice Protein Model

• Motivation
• Simplest Models of Protein
• Lattice Protein
• Prototype of Protein-like molecules
• Ising Model
• Prototype of Magnets

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Lattice Protein (2-dim HP)
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sequence of
FIXED
and
corresponds to 2-types of amino acids (H and P)
conformation of chain
STOCHASTIC VARIABLE
SELF AVOIDING (SELF OVERLAP is not
allowed) IMPORTANT!
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Energy (HP model)
the energy of conformation x is defined as
E(X) - the number of
in x
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Examples
E -1
E0
Here we do not count the pairs neighboring on the
chain but it is not essential because the
difference is const.
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MCMC

• Slow Mixing
• Even Non-Ergodicity with local moves

Bastolla et al. (1998) Proteins 32 pp. 52-66
Chikenji et al. (1999) Phys. Rev. Lett. 83
pp.1886-1889
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Multicanonical

• Multicanonical w.r.t. E only
• NOT SUFFUCIENT
• Self-Avoiding condition is essential

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Soft Constraint

• Self-Avoiding condition is essential

Soft Constraint
is the number of monomers that occupy the site i
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Multi Self-Overlap Sampling

• Multi Self-Overlap Ensemble
• Bivariate Density of States
• in the (E,V) plane

V (self-overlap)
E
EXACT !!
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Generation of Paths by softening of constraints
E
V0
large V
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Comparison with multicanonical with hard
self-avoiding constraint
switching between three groups of minimum
energy states of a sequence
conventional (hard constraint)
proposed (soft constraint)
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optimization
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optimization (polymer pairs)
Nakanishi and Kikuchi (2006) J.Phys.Soc.Jpn. 75
pp.064803 / q-bio/0603024
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double peaks
An Advantage of the method is that it can
use for the sampling at any temperature as well
as optimization
3-dim
Yue and Dill (1995) Proc. Nat. Acad. Sci. 92
pp.146-150
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Another Sequence
Chikenji and Kikuchi (2000) Proc. Nat. Acad. Sci
97 pp.14273 - 14277
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Related Works

• Self-Avoiding Walk without interaction /
  Univariate Extension
• Vorontsov-Velyaminov et al.
• J.Phys.Chem.,100,1153-1158 (1996)
• Lattice Protein but not exact / Soft-Constraint
  without control
• Shakhnovich et al.
• Physical Review Letters 67 1665 (1991)
• Continuous homopolymer -- Relax core
• Liu and Berne
• J Chem Phys 99 6071 (1993)
• See References in
• Extended Ensemble Monte Carlo, Int J Phys C
  12 623-656 (2001)
• but esp. for continuous cases,
• there seems more in these five years

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Counting Tables
4 9 2
3 5 7
8 1 6
Pinn et al. (1998) Counting Magic Squares Soft
Constraints Parallel Tempering
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Sampling by MCMC

• Multiple Maxima
• Parallel Tempering

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Normalization Constant

calculated by Path sampling (thermodynamic
integration)
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Latin square (3x3)
For each column, any given number appears once
and only once
For each raw, any given number appears once and
only once
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Latin square (26x26)
This sample is taken from the web.
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Counting Latin Squares

• 6
• 10
• 11

410000 MCS x 27 replicas
510000 MCS x 49 replicas
510000 MCS x 49 replicas
other 3 trials
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Counting Tables

• Soft Constraints Extended Ensemble MC
• Quick and Dirty ways of calculating the number
  of tables that satisfy given constraints.
• It may not be optimal for a special case,
• but no case-by-case tricks, no mathematics,
• and no brain is
  required.

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Rare Events and Large Deviations

• Communication Channels 1
• Chaotic Dynamical Systems 2

1 Part of joint works with Koji
Hukushima (Tokyo Univ).
2 Part of joint works with Tatsuo
Yanagita (Hokkaido Univ). (The result shown
here is mostly due to him )
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Applications of MCMC

• Statistical Physics (1953 )
• Statistical Inference (1970s,1980s, 1990)
• Solution to any problem on
• sampling counting
• estimation of large deviation
• generation of rare events

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Noisy Communication Channel
prior
encoded degraded
decode
distance (bit errors)
by Viterbi, loopy BP, MCMC
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Distribution of Bit Errors
Kronecker delta
tails of the distribution is not easy to estimate
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Introduction of MCMC
NOT sampling from the posterior

• Sampling noise in channels by the MCMC
• Given an error-correcting code
• Some patterns of noise are very harmful
• difficult to correct
• Some patterns of noise are safe
• easy to correct

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Multicanonical Strategy

• MCMC sampling of
• Broad distribution of
• ? Broad distribution of distance

and
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Multicanonical Sampling

• MCMC Sampling and
• with the weight

Estimated by the iteration of preliminary runs
exactly what we want, but can be ..
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flat marginal distribution
Enable efficient calculation of the tails of the
distribution (large deviation)
Scanning broad range of bit errors
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Example

• Convolutional Code

Viterbi decoding
Binary Symmetric Channel Fix the number of
noise (flipped bits)
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Simplification

• In this case

is independent of
Set
Binary Symmetric Channel Fix the number of
noise (flipped bits)
sum over the possible positions of the noise
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Simulation
difficult to calculate by simple sampling
the number of bit errors
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Correlated Channels

• It will be useful for the study of
  error-correcting code in a correlated channel.
• Without assuming models of correlation
• in the channel we can sample relevant
• correlation patterns.

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Rare events in Dynamical Systems

• Deterministic Chaos
• Doll et al. (1994), Kurchan et al. (2005)
• Sasa, Hayashi, Kawasaki .. (2005 )
• (Mostly) Stochastic Dynamics
• Chandler Group
• Frenkel et al.
• and more

Stagger and Step Method Sweet, Nusse, and Yorke
(2001)
Transition Path Sampling
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Sampling Initial Condition

• Sampling initial condition of
• Chaotic dynamical systems
• Rare Events

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Double Pendulum
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Unstable fixed points
control and stop the pendulum one of the three
positions
energy dissipation (friction) is assumed i.e.,
no time reversal sym.
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Definition of artificial energy
stop zero velocity
stopping position
penalty to long time
T is max time
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Metropolis step
Evaluate Energy
Perturb Initial State
Integrate Equation of Motion and Simulate
Trajectory
Reject or Accept
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? Parallel Tempering
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An animation by Yanagita is shown in the talk,
but might not be seen on the web.
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Summary

• Extended Ensemble Soft Constraint strategy
  gives simple solutions to a number of difficult
  problems
• The use of MCMC should not be restricted to the
  standard ones in Physics and Bayesian Statistics.
  
• To explore new applications of MCMC extended
  ensemble MC will play an essential role.
  

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END