Monte Carlo Methods1 T-61.182 Special Course In Information Science II Tomas Ukkonen...
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Monte Carlo Methods 1 Monte Carlo Methods T-61.182 Special Course In Information Science II Tomas Ukkonen [email protected]
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Transcript of Monte Carlo Methods1 T-61.182 Special Course In Information Science II Tomas Ukkonen...
Monte Carlo Methods 1
Monte Carlo Methods
T-61.182 Special Course In Information Science II
Tomas Ukkonen
Monte Carlo Methods 2
Problem
1. generate samples from given probability distribution P(x)
2. estimate
The second problem can be solved by using random samples from P(x)
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Monte Carlo Methods 3
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
Monte Carlo Methods 4
Brute force method
why don’t just calculate expected value directly
problem grows exponentiallyas the function of dimension d
number states to check grow exponentially
i
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Monte Carlo Methods 5
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
Monte Carlo Methods 6
Uniform sampling
for small dimensional problems
Just sample uniformly and weight with
Required number of samples for reliable estimators still grows exponentially
x )(xP
HNR 2min
Monte Carlo Methods 7
Importance sampling
idea: approximate complicated distribution with simpler one
only works when correct shape of distribution is known
doesn’t scale to high dimensionseven when approximation is almostright
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Monte Carlo Methods 8
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
Monte Carlo Methods 9
The Metropolis-Hastings method
The previous approaches didn’t scale to high dimensions
In Metropolis algorithm sampling distribution depends on samples sampled so far
Monte Carlo Methods 10
The Metropolis-Hastings, cont.
A new state is drawn from distributionand 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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Monte Carlo Methods 11
Gibbs sampling
a special case of the metropolis method where only single dimension is updated per iteration
useful when only conditional densities are known
one dimensional distributions are easier to work with
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Monte Carlo Methods 12
Gibbs sampling, cont.
Monte Carlo Methods 13
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
Monte Carlo Methods 14
Slice sampling, cont.
faster integer based algorithm has been also developed
Monte Carlo Methods 15
Slice sampling, cont.
Monte Carlo Methods 16
Slice sampling, cont.
Monte Carlo Methods 17
Practical issues
Hard to know for certain when Monte Carlo simulation has converged
Caculating normalization constant
allocation of computational resources:one long simulation or more shorter ones?
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Monte Carlo Methods 18
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
Monte Carlo Methods 19
Exercises & References
exercise 29.4. exercise NN.N.
David J.C. Mackay: Information Theory, Inference, and Learning Algorithms, 2003
