Perfect Simulation

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Perfect Simulation

• (Yes, were finally here!)

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Perfect Simulation

• MCMC without statistical error

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The Propp-Wilson Algorithm
Consider an irreducible, aperiodic Markov chain
on a three point state space S0,1,2 with
transition law P(i,j).
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The Propp-Wilson Algorithm
At time 1, start sample paths from every point
in the state space
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The Propp-Wilson Algorithm
At time -2, start sample paths from every point
in the state space, and append the paths onto the
previous paths
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The Propp-Wilson Algorithm
As an alternative to appending, it will be
helpful to think about the procedure as
reusing the same random numbers.
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The Propp-Wilson Algorithm
Back to time 3
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The Propp-Wilson Algorithm
WHOA!!!! COOL!!!
No matter where we start at time 3, we always
end up in state 1 at time 0!
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The Propp-Wilson Algorithm
Lets start a stationary path at time 4 (or
before)
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The Propp-Wilson Algorithm
Running this path forward, using the previously
used random numbers (and one more to go from time
4 to time 3)
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The Propp-Wilson Algorithm
Then by stationarity, this is a draw from
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The Propp-Wilson Algorithm
Then by stationarity, this is a draw from
The problem is we dont know how to draw from
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The Propp-Wilson Algorithm
Then by stationarity, this is a draw from
The good news is that all roads lead to We
can start anywhere!
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The Propp-Wilson Algorithm
Then by stationarity, this is a draw from
Also, we can start anytime before -3!
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The Propp-Wilson Algorithm
We are seeing the tail end of a path that has
come in from
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Perfect Simulation

• The next time, we use new random numbers to get
  possibly a different value at time zero.

• The next time, the backward coupling time
  might not be at -3.

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Perfect Simulation

• Reusing (appending) the random numbers is
  critical to this algorithm.

• Otherwise, there is nothing backward about it.

• This brings up the real question

Why doesnt forward coupling work?
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Perfect Simulation
One could imagine starting from all states and
running forward until all paths have coupled
(coalesced, met, etc)
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Perfect Simulation
This doesnt work!

• Intuitively, something is wrong since there are
  many MCs where coupling can only happen at the
  top or bottom of the state space.

• Notice that in the perfect sampling (coupling
  from the past) algorithm, we dont stop once
  all paths have met we still follow through to
  time 0.

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Perfect Simulation
Forward coupling doesnt work!

• Mathematically, the problem is

If we start a MC according to a draw from the
stationary distribution and run it according to
its transition law P, it maintains that
distribution at all fixed time points.
That point of coalescence is a random time
point.
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The Propp-Wilson Algorithm
Surely, we cant expect that this PW algorithm
will work for any interesting Markov chains.

• a lot of states

• a continuous state space

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Stochastic Monotonicity
Coupling-from-the-past (CFTP) is particularly
efficient if the chain is stochastically
monotone
For simulational purposes, this means that the
paths wont cross.
(We are assuming that we are using the same
random number stream for each path.)
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Stochastic Monotonicity
Example
Note We might want to reorder the states in
order to have this monotonicity.
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Stochastic Monotonicity
Okay, that was a rather contrived example, but
stochastic monotonicity occurs naturally in a lot
of models.
For example

• queueing models

• storage models

• Ising models

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Stochastic Monotonicity
For a stochastically monotone MC on a continuous
state space, our challenge will be to get the
sample paths to meet
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Unbounded Spaces
What about a stochastically monotone MC on an
unbounded state space?
All hope is not lost
We will try to stochastically bound the space.