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Monte Carlo Methods
T-61.182 Special Course In Information Science II
Tomas Ukkonen
[email protected]
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Problem
1.
2.

generate samples from given probability
distribution P(x)
estimate E[ ( x)]    ( x) P( x)d x
The second problem can be solved by using
random samples from P(x)
1
 (xr )

R r
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Why sampling is hard?

densities may be unscaled:
hard to know how probable a certain point is
when the rest of function is unknown

curse of dimensionality
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Brute force method

why don’t just calculate expected value directly

problem grows exponentially
as the function of dimension d
Z p
*
i
i

number states to
check grow exponentially
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pi  p / Z
*
i
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Brute force method, cont.

going through most of the cases is likely to be
unnecessary

high-dimensional, low entropy densities are
often concentrated to small regions
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Uniform sampling

for small dimensional problems

Just sample x uniformly and weight with

Required number of samples for reliable estimators
still grows exponentially
Rmin  2
P (x )
N H
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Importance sampling


idea: approximate complicated
distribution with simpler one
only works when correct shape of
distribution is known
*
P ( xr )
wr  *
Q ( xr )
w (x )


w
r

doesn’t scale to high dimensions
even when approximation is almost
right
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r
r
r
r
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Rejection sampling

Alternative approximation
based sampling method

sample uniformly from
(x,u) = (x,cQ(x)) and reject
samples where u > P(x)

doesn’t scale to high
dimensions
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The Metropolis-Hastings
method

The previous approaches didn’t scale
to high dimensions

In Metropolis algorithm sampling
distribution depends on samples
sampled so far
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The Metropolis-Hastings, cont.

A new state is drawn from distribution Q( x' ; xt )
and accepted with a certain probability
which guarantees convergence to the
target density

The method doesn’t depend on
dimensionality of a problem, but samples
are correlated and a random walk based
moving is slow
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Gibbs sampling

a special case of the metropolis method where only
single dimension is updated per iteration

useful when only conditional densities
p( xi | x1..xi 1 xi 1..xN ) are known

one dimensional distributions are easier to work
with
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Gibbs sampling, cont.
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Slice sampling

a newer method which is combination of
rejection, Gibbs and Metropolis sampling

still a random walk method but with a self
tuning step length
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Slice sampling, cont.

faster integer based algorithm has been also
developed
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Slice sampling, cont.
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Slice sampling, cont.
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Practical issues



Hard to know for certain when Monte Carlo
simulation has converged
1 *
Caculating normalization constant P( x)  P ( x)
Z
allocation of computational resources:
one long simulation or more shorter ones?
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Practical issues II, cont.
Big Models
 Metropolis method & Gibbs sampling
- update variables in batches

-
How many samples
how much accuracy is needed?
typically 10-1000 samples is enough
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Exercises & References
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

exercise 29.4.
exercise NN.N.
David J.C. Mackay: Information Theory,
Inference, and Learning Algorithms, 2003
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