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Inference V: MCMC MethodsPowerPoint Presentation

Inference V: MCMC Methods

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### Inference V:MCMC Methods

.

Stochastic Sampling

- In previous class, we examined methods that use independent samples to estimate P(X = x |e )
Problem: It is difficult to sample from P(X1, …. Xn |e )

- We had to use likelihood weighting to reweigh our samples
- This introduced bias in estimation
- In some case, such as when the evidence is on leaves, these methods are inefficient

MCMC Methods

- We are going to discuss sampling methods that are based on Markov Chain
- Markov Chain Monte Carlo (MCMC) methods

- Key ideas:
- Sampling process as a Markov Chain
- Next sample depends on the previous one

- These will approximate any posterior distribution

- Sampling process as a Markov Chain
- We start by reviewing key ideas from the theory of Markov chains

...

Xn

X1

X2

X3

Markov Chains- Suppose X1, X2, … take some set of values
- wlog. These values are 1, 2, ...

- A Markov chain is a process that corresponds to the network:
- To quantify the chain, we need to specify
- Initial probability: P(X1)
- Transition probability: P(Xt+1|Xt)

- A Markov chain has stationary transition probability
- P(Xt+1|Xt) is the same for all times t

Irreducible Chains

- A state j is accessible from state i if there is an n such that P(Xn = j | X1 = i) > 0
- There is a positive probability of reaching j from i after some number steps

- A chain is irreducible if every state is accessible from every state

Ergodic Chains

- A state is positively recurrent if there is a finite expected time to get back to state i after being in state i
- If X has finite number of states, then this is suffices that i is accessible from itself

- A chain is ergodic if it is irreducible and every state is positively recurrent

(A)periodic Chains

- A state i is periodic if there is an integer d such thatP(Xn = i | X1 = i ) = 0 when n is not divisible by d
- A chain is aperiodic if it contains no periodic state

Stationary Probabilities

Thm:

- If a chain is ergodic and aperiodic, then the limitexists, and does not depend on i
- Moreover, letthen, P*(X) is the unique probability satisfying

Stationary Probabilities

- The probability P*(X) is the stationary probability of the process
- Regardless of the starting point, the process will converge to this probability
- The rate of convergence depends on properties of the transition probability

Sampling from the stationary probability

- This theory suggests how to sample from the stationary probability:
- Set X1 = i, for some random/arbitrary i
- For t = 1, 2, …, n
- Sample a value xt+1 for Xt+1 from P(Xt+1|Xt=xt)

- return xn

- If n is large enough, then this is a sample from P*(X)

Designing Markov Chains

- How do we construct the right chain to sample from?
- Ensuring aperiodicity and irreducibility is usually easy

- Problem is ensuring the desired stationary probability

Designing Markov Chains

Key tool:

- If the transition probability satisfiesthen, P*(X) = Q(X)
- This gives a local criteria for checking that the chain will have the right stationary distribution

MCMC Methods

- We can use these results to sample from P(X1,…,Xn|e)
Idea:

- Construct an ergodic & aperiodic Markov Chain such that P*(X1,…,Xn) = P(X1,…,Xn|e)
- Simulate the chain n steps to get a sample

MCMC Methods

Notes:

- The Markov chain variable Y takes as value assignments to all variables that are consistent evidence
- For simplicity, we will denote such a state using the vector of variables

Gibbs Sampler

- One of the simplest MCMC method
- At each transition change the state of just on Xi
- We can describe the transition probability as a stochastic procedure:
- Input: a state x1,…,xn
- Choose i at random (using uniform probability)
- Sample x’i from P(Xi|x1, …, xi-1, xi+1 ,…, xn, e)
- let x’j = xj for all j i
- return x’1,…,x’n

Correctness of Gibbs Sampler

- By chain rule
P(x1, …, xi-1, xi, xi+1 ,…, xn|e) =P(x1, …, xi-1, xi+1 ,…, xn|e)P(xi|x1, …, xi-1, xi+1 ,…, xn, e)

- Thus, we get
- Since we choose i from the same distribution at each stage, this procedure satisfies the ratio criteria

Gibbs Sampling for Bayesian Network

- Why is the Gibbs sampler “easy” in BNs?
- Recall that the Markov blanket of a variable separates it from the other variables in the network
- P(Xi | X1,…,Xi-1,Xi+1,…,Xn) = P(Xi | Mbi )

- This property allows us to use local computations to perform sampling in each transition

Gibbs Sampling in Bayesian Networks

- How do we evaluate P(Xi | x1,…,xi-1,xi+1,…,xn) ?
- Let Y1, …, Yk be the children of Xi
- By definition of Mbi, the parents of Yj are in Mbi{Xi}

- It is easy to show that

Sampling Strategy

- How do we collect the samples?
Strategy I:

- Run the chain M times, each run for N steps
- each run starts from a different state points

- Return the last state in each run

M chains

Sampling Strategy

Strategy II:

- Run one chain for a long time
- After some “burn in” period, sample points every some fixed number of steps

“burn in”

M samples from one chain

Comparing Strategies

Strategy I:

- Better chance of “covering” the space of pointsespecially if the chain is slow to reach stationarity
- Have to perform “burn in” steps for each chain
Strategy II:

- Perform “burn in” only once
- Samples might be correlated (although only weakly)
Hybrid strategy:

- run several chains, and sample few samples from each
- Combines benefits of both strategies

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