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MCMC in practice

- Start collecting samples after the Markov chain has “mixed”.
- How do you know if a chain has mixed or not?
- In general, you can never “proof” a chain has mixed
- But in may cases you can show that it has NOT. (If you fail to do so using several different methods, you probably convinced yourself that it has mixed.)
- Now it becomes: how do you know a chain has not mixed?
- Log likelihood plot
- Marginal plot (for several chains)
- Marginal scatter plot (for two chains)

Initialized from a bad (non-informative) configuration

Initialized from a “good” configuration (found by other methods)

Mixing?

Log likelihood

Probably

iterations

iterations

Log likelihood

NO

iterations

iterations

Each dot is a statistic (e.g., P(X_1 = x_10))

- x-position is its estimation value from chain 1
- y-position is its estimation value from chain 2

Mixing?

NO

Probably

Toy Model for Data Association

- Blue dots: variables, x_i (i=1,2,3,4)
- Red dots: observations (values that we assign to variables)
- What does the distribution look like?

distance

(A)

(B)

(C)

How do we sample from it?

- Add one observation (such that Gibbs would work)
- Two modes:
- Gibbs
- How does it traverse between the two modes?
- Block Gibbs (block size = 2)
- How do we sample?
- Metropolis Hasting
- Take larger steps using a proposal distribution.

(We will come to details of this later.)

Try it yourself

- Connect to clusters with graphics
- Windows

https://itservices.stanford.edu/service/unixcomputing/unix/moreX

- MacOS or Linux

ssh –x [email protected]

- Copy the code to your own directory

cp –r /afs/ir/class/cs228/mcmc ./

cdmcmc

- Run Matlab and execute the following scripts

VisualMCMC1(10000, 0.1, 0.05);

% live animation of sampling

% parameters: num of samples, sigma, pause time after each sample

Plot1;

% the first few lines of Plot1.m contain the parameters you may want to play around with

Proposal distributions for M-H

- Back to the model with 4 observations
- What will Gibbs do on this?
- Proposal distribution 1 (flip two)
- Randomly pick two variables, flip their assignments
- What is the acceptance probability?

Proposal distributions for M-H

- Proposal distribution 2 (augmented path)
- 1. randomly pick one variable
- 2. sample it pretending that all observations are available
- 3. pick the variable whose assignment was taken (conflict), goto step 2
- 4. loop until step 2 creates no conflict
- What is the acceptance probability?

Proposal distributions for M-H

- Proposal distribution 3 (“smart” augmented path)
- Same as the previous one except for the highlighted
- 1. randomly pick one variable
- 2. sample it pretending that all observations are available (excluding the current one)
- 3. pick the variable whose assignment was taken (conflict), goto step 2
- 4. loop until step 2 creates no conflict
- What is the acceptance probability?

Try it yourself

- Run the following Matlab scripts:

VisualMCMC2(10000, 0.7, 0.05);

% live animation of sampling

% parameters: num of samples, sigma, pause time after each sample

Plot2;

% the first few lines of Plot2.m contain the parameters you may want to play around with

Try to fix Gibbs with annealing

- A skewed distribution

The right blue dot is moved up a little bit such that a < b

- How does the distribution look like?
- What would you use for annealing?

(A) Multiple shorter chains.

(B) One longer chain

(suppose that our computation resource cannot afford multiple long chains)

a

b

Try it yourself

- Run the following Matlab scripts:

VisualMCMC3(200, 0.06, 0.05, 50);

% parameters: num of samples, sigma, pause time after each sample, num of chains to run

- What you will see using default parameters:
- Live animation of sampling for 5 chains with annealing, 5 chains without.
- At the end of 200 samples:

Estimate P(x1=1) using Gibbs without annealing: 0.664000

Estimate P(x1=1) using Gibbs with annealing: 0.876000

Estimate P(x1=1) using Metropolis Hasting: 0.909100

(numbers may vary in different runs)

should be close to the true value

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