Markov Chain Monte Carlo

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Markov Chain Monte Carlo

• Auxilliary Variable Methods

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Tools for Improving MCMC Performance

• the slice sampler

• the Swendsen-Wang algorithm

• simulated tempering

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The Slice Sampler
Example Suppose we want to draw from a normal
distribution

• Start at some value x.

• Consider a vertical slice of the density at x.

• Draw a value uniformly from the vertical slice.

• Draw a value uniformly from the corresponding
  horizontal slice.

• Drop down to consider the resulting x value.

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The Slice Sampler
Example Suppose we want to draw from a normal
distribution
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The Slice Sampler
Claim
We are creating a chain of x-values that has, in
this case, the normal distribution as its
stationary distribution!
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The Slice Sampler
Note
Although it would be difficult to implement, this
would also work for densities like this
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The Slice Sampler
In this case, we would be drawing uniformly from
disjoint intervals that look like this
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The Slice Sampler The Theory
Suppose we want to draw from a (possibly
unnormalized) density on
We introduce the auxilliary variable Ygt0 and
the un-normalized joint density
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The Slice Sampler The Theory
Clearly what we are interested in is the marginal
density of X
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The Slice Sampler The Theory
So, if we draw (x,y)s from the bivariate density
and just consider the x values,
we will be drawing from the marginal density
Lets draw from the bivariate density using the
Gibbs sampler
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The Slice Sampler The Theory

• draw x from

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The Slice Sampler The Theory

• draw y from

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The Slice Sampler The Theory
Note By definition of the stationary
distribution of X, if the starting value is drawn
from the resulting x after one iteration (one
vertical and horizontal slice) will have
distribution
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The Swendsen-Wang Algorithm

• possibly the first auxilliary variable technique

• developed for the Ising model where it proves
  to be much more efficient than Metropolis or
  Gibbs

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The Swendsen-Wang Algorithm
Consider the joint density
where 0ltplt1 and
and V and E are vertex and edge sets for an
undirected square lattice graph.
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The Swendsen-Wang Algorithm
The marginal density of X is
This is an Ising model if we let
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The Swendsen-Wang Algorithm
The marginal density of Y is
This is a random cluster model.
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The Swendsen-Wang Algorithm
Gibbs Sampling from

• draw y from

specifies that the bonds ye,
are conditionally independent given x
with
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The Swendsen-Wang Algorithm
Gibbs Sampling from

• draw x from

specifies that the connected
components, with
are conditionally independent given y where
and the common value of xi being
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The Swendsen-Wang Algorithm
Here, C(y) denotes a partition of V into maximal
connected y-components

• A set is a connected y-component
  if any are connected by y.

• C is maximal if there exists no other connected
  y-component D such that

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Simulated Tempering

• Marinari and Parisi

• technique invented to study the Ising model

• phase transition at the critical temperature,
  there are two coexisting states (one high
  energy, one low energy)

• standard Metropolis algorithm tends to get
  stuck in one or the other

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Simulated Tempering

• the tempering part of this algorithm refers to
  a Monte Carlo update of the temperature

• sweeping slightly above and below the critical
  temperature in and out of high and low energy
  states provides a sampling of configurations at
  the critical temperature

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Simulated Tempering
Target density (possibly
un-normalized)
We will make a mixture (linear combination) of
un-normalized densities
with
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Simulated Tempering
Original Temperature Based Physical Applications
where

• is the cold density

• is the hot density

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Simulated Tempering
Consider the un-normalized joint density defined
by , where
are constants.
Let (X,I) be a random vector with un-normalized
density
Then

• X has a mixture distribution with density

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Simulated Tempering
Consider the un-normalized joint density defined
by , where
are constants.
Let (X,I) be a random vector with un-normalized
density
Then

• XIm has the cold distribution

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Simulated Tempering
Consider the un-normalized joint density defined
by , where
are constants.
Let (X,I) be a random vector with un-normalized
density
Then

• I has a distribution given by

(Choosing gives a uniform
mixture.)
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Simulated Tempering

• Let Pi(x,y) denote a transition probability
  that is stationary wrt gi.

ie define a MC with transitions given by Pi(x,y)
where gi is the stationary distribution perhaps
using the MH algorithm to get Pi from a candidate
q.
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Simulated Tempering

• Let Q(x,y) be the transition probabilities for
  the temperature index defined by

for 0ltplt1/2.
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Simulated Tempering The Algorithm
Given a current state (x,i),

• update x to y using Pi(x,y)

• propose an update from i to j by selecting j
  from Q(i,j) and accepting j w.p.

• ultimately, collect the xs for Im