Introduction to Sampling Methods

1
Introduction to Sampling Methods

• Qi Zhao
• Oct.27,2004

2
Introduction to Sampling Methods

• Background
• Seven sampling methods
• Conclusion

3
Monte Carlo Methods
Aim to solve one or both of the following
problems
Problem 1 to generate samples from
a given probability distribution
Problem 2 to estimate expectations of functions
under this distribution, for example
4
Monte Carlo Methods

• Hard to sample from

Huge!

• Write in the following form

known
unknown
what is the cost to evaluate it?
5
Monte Carlo Methods

• To compute , every point in the
  space should be visited

1000
50
Back
6
Monte Carlo Methods

• Difficult to estimate the expectation of
  
• by drawing random samples
  uniformly from the state space and evaluating
  .

Back
7
Sampling Methods
References
link

• Importance sampling
• Rejection sampling
• Metropolis sampling
• Gibbs sampling
• Factored sampling
• Condensation sampling
• Icondensation sampling

Back
8
Importance Sampling

• Not a method for generating samples from
  , just a method for estimating the
  expectation of a function .

• is complex while is of simpler
  density .

9
Rejection Sampling

• Further assumption of importance samplingwe know
  the constant such that for all ,

• Evaluation of the probability
  density of the x-coordinates of the accepted
  points must be proportional to

Back
10
Metropolis Sampling

• is not necessarily look similar to at all

• has a shape that changes as changes

If ,then the new state is
accepted. Otherwise, the new state is accepted
with probability
Back
11
Metropolis Sampling

• Proof

Back
12
Gibbs Sampling

• A special case of Metropolis sampling is
  defined in terms of the conditional distribution
  of the joint distribution

• Sample from distributions over at least two
  dimensions

13
Gibbs Sampling

• An example with two variables

Back
14
Markov chain Monte Carlo

• Comparison

• Rejection sampling
  Accepted points are independent samples
  from the desired distribution

• Markov chain Monte Carlo
• Involve a Markov process in which a sequence
  of states is generated, each sample
  having a probability distribution that
  depends on the previous value .

Back
15
Factored Sampling

• Deal with non-Gaussian observations in single
  image

• Essential idea here is to transform the uniform
  distribution into weighted distribution. So that
  non-Gaussian forms can also use uniform
  distribution (random bits) to generate sample

16
Factored Sampling

• An sample set is
  generated from the prior density

• II. An index is chosen with
  probability

• the value chosen (with probability
  ) in this fashion has a distribution
  ,as .

17
Condensation Sampling

• Based on factored sampling

• Extended to apply iteratively to successive
  images in a sequence

18
Condensation Sampling
19
Condensation Sampling

• Select a sample
• a. Generate a random number
  , uniformly distributed
• b. Find the smallest for which
• II. Predict using dynamic model
• e.g.
• Measure and weight
• a. Calculate the new position in terms of
  the measured features ,
  
• b. Normalize so that
• c. Store together
  ,where

20
ICondensation Sampling

• It is a technique developed to improve the
  efficiency of factored sampling and condensation
  sampling.

• Premise auxiliary knowledge is available in the
  form of an importance function that describes
  which areas of state-space contain most
  information about the posterior .

• Idea to concentrate samples in those areas of
  state-space by generating new samples from the
  importance function instead of from the prior .

Back
21
Summary
is easy to get
Importance Sampling
Rejection Sampling

• Easy to
• get the
• function?

yes
no
Gibbs Sampling
Metropolis Sampling
a special case
MCMC

• To use uniform distribution for non-Gaussian
  distribution

Condensation Sampling
Factored Sampling
successive images
improvement
Icondensation Sampling
22
References

• D.J.C.Mackay, Introduction to Monte Carlo Methods
• Michael Isard and Andrew Blake, Condensation
  conditional density propagation for visual
  tracking
• Michael Isard and Andrew Blake, Icondensation
  Unifying low-level and high-level tracking in a
  stochastic framework

Back
23
Thank you all!