Markov Chain Monte Carlo

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Markov Chain Monte Carlo
(MCMC)
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The Metropolis Algorithm
Procedure
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The Metropolis Algorithm

• Accept a move from x to y with probability

• Otherwise, stay at state x.

• Rinse and repeat

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Claim This procedure produces a Markov chain
whose stationary distribution is
Proof
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Yuck! Dont go this way to show that
Once we have shown that, we are done since
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So, it remains to show that the detailed balance
condition holds.
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The Metropolis Algorithm
Example
Use the Metropolis algorithm to draw values from
the distribution
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The Metropolis Algorithm
Example (continued)
Find a symmetric candidate density to draw from
how?
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The Metropolis Algorithm
Example (continued)
For this example, lets try
ie if we are at state x, we will draw a
candidate state y from the N(x,1/2) distribution.
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The Metropolis Algorithm
Example (continued)
Specifically,

• start with some value x

• draw y from N(x,1/2) I used Box-Muller

• let yy

• draw Uunif(0,1)

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The Metropolis Algorithm
Simulation Results
Output of 10,000 chains, each of length 1,000.
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The Metropolis Algorithm
The Ising Model The Quintessential Example
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The Metropolis Algorithm
The Ising Model The Quintessential Example

• Nnxn sites on a square lattice

• the spin at site i takes on one of two values

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The Metropolis Algorithm
The Ising Model The Quintessential Example
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The Metropolis Algorithm
The Ising Model The Quintessential Example
Assuming N is large and the system is in thermal
equlibrium, the probability that the system is in
state S is given by the Boltzmann probability
distribution
k Boltzmanns constant
T temperature
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The Metropolis Algorithm
The Ising Model Simulation
The standard practice is to use the Metropolis
algorithm

• pick a spin either at random or in sequence

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The Metropolis Algorithm
The Ising Model Simulation
Note that
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The Metropolis Algorithm
The Ising Model Simulation
This standard method can be very slow

• people run simulations like this for several
  weeks even for small grids (lt20x20)

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The Metropolis Algorithm
The Random Cluster Model

• Let G be a square lattice graph with undirected
  edges.

• Let V be the set of vertices.

• Let E be the set of edges.

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The Metropolis Algorithm
The Random Cluster Model

• A configuration of the random cluster model
  (state of a Markov chain) is a random
  configuration of edges.

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The Metropolis Algorithm
The Random Cluster Model
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The Metropolis Algorithm
The Random Cluster Model

• The random cluster distribution is defined as

where C(S) is the number of connected regions in
configuration S.
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The Metropolis Algorithm
The Random Cluster Model

• Alternatively,

where B(S) is the number of bonds in
configuration S.
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The Metropolis Algorithm
The Random Cluster Model Simulation

• pick an edge either at random or in sequence

• flip a fair coin to propose including or not
  including the edge

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The Metropolis Algorithm
The Random Cluster Model

• Clearly a good algorithm for determining
  connectivity is needed.

• Well talk about a super-fast one later.

Be clever!
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The Metropolis Algorithm
The Random Cluster Model and Ising
For example, if you sampled this configuration
from the random cluster model with appropriate p
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The Metropolis Algorithm
The Random Cluster Model and Ising
Then you would flip a coin for each connected
component to determine up/down
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The Metropolis Algorithm
The Random Cluster Model and Ising
Then you would flip a coin for each connected
component to determine up/down
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