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Jun Liu

Department of Statistics

Stanford University

Based on the joint work with F. Liang and W.H. Wong.

MCMC and Statistics

The Basic Problems of Monte Carlo

- Draw random variable
- Estimate the integral

Sometimes with unknown normalizing constant

MCMC and Statistics

c

u cg(x)

p(x)

x

How to Sample from p(x)- The Inversion Method.If U ~ Unif (0,1) then
- The Rejection Method.
- Generate x from g(x);
- Draw u from unif(0,1);
- Accept x if
- The accepted x follows p(x).

The “envelope” distrn

MCMC and Statistics

High Dimensional Problems?

Ising Model

Partition

function

Metropolis Algorithm:

(a) pick a lattice point, say a, at random

(b) change current xa to 1- xa (so X(t) ® X*)

(c) compute r= p(X*)/ p(X(t) )

(d) make the acceptance/rejection decision.

MCMC and Statistics

General Metropolis-Hastings Recipe

- Start with any X(0)=x0, and a “proposal chain” T(x,y)
- Suppose X(t)=xt . At time t+1,
- Draw y~T(xt ,y) (i.e., propose a move for the next step)
- Compute the Metropolis ratio (or “goodness” ratio)
- Acceptance/Rejection decision: Let

“Thinning

down”

MCMC and Statistics

- The detailed balance

Actual transition probability

from x to y, where

Transition probability

from y to x.

MCMC and Statistics

General Markov Chain Simulation

- Question: how to simulate from a target distribution p(X) via Markov chain?
- Key: find a transition function A(X,Y) so that

f0 An ® p

that is, p is an invariant distribution of A.

- Different from traditional Markov Chain theory.

MCMC and Statistics

If the actual transition probability is

I learnt it from Stein

where (x,y) is a symmetric function of x,y,

Then the chain has (x) as its invariant distribution.

MCMC and Statistics

xa

xc

Other Approaches?- Gibbs sampler/Heat Bath:better or worse?
- Random directional search --- should be better if we can do it. “Hit-and-run.”
- Adaptive directional sampling (ADS) (Gilks, Roberts and George, 1994).

Multiple

chains

MCMC and Statistics

Gibbs Sampler/Heat Bath

- Define a “neighborhood” structure N(x)
- can be a line, a subspace, trace of a group, etc.
- Sample from the conditional distribution.
- Conditional Move

MCMC and Statistics

How to sample along a line?

- What is the correct conditional distribution?
- Random direction:
- Directions chosen a priori: the same as above
- In ADS?

MCMC and Statistics

The Snooker Theorem

- Suppose x~and y is any point in the d-dim space. Let r=(x-y)/|x-y|.If t is drawn from

Then

follows the target distribution .

If y is generated from distr’n, the new point x’ is indep. of y.

x

y (anchor)

MCMC and Statistics

Connection with transformation group

- WLOG, we let y=0.
- The move is now:x x’=tx

The set {t: t0} forms a transformation group.

Liu and Wu (1999) show that if t is drawn from

Then the move is invariant with respect to .

MCMC and Statistics

Another Hurdle

- How to draw from something like
- Adaptive rejection? Approximation? Griddy Gibbs?
- M-H Independence Sampler(Hastings, 1970)
- need to draw from something that is close enough to p(x).

MCMC and Statistics

- Propose bigger jumps
- may be rejected too often
- Proposal with mix-sized stepsizes.
- Try multiple times and select good one(s) (“bridging effect”) (Frankel & Smit, 1996)
- Is it still a valid MCMC algorithm?

MCMC and Statistics

Current is at x

Can be dependent ones

- Draw y1,…,yk from the proposal T(x, y) .
- Select Y=yjwith probability (yj)T(yj,x).
- Draw from T(Y, x). Let
- Accept the proposed yj with probability

MCMC and Statistics

A Modification

- If T(x,y) is symmetric, we can have a different rejection probability:

Ref: Frankel and Smit (1996)

MCMC and Statistics

Back to the example

Random Ray Monte Carlo:

y3

- Propose random direction
- Pick y from y1 ,…, y5
- Correct for the MTM bias

y5

y4

x

y2

y1

MCMC and Statistics

y4

y6

y8

y1

y3

y5

y7

An Interesting Twistx

- One can choose multiple tries semi-deterministically.

Random equal grids

y

- Pick y from y1 ,…, y8
- The correction rule is the same:

MCMC and Statistics

Use Local Optimization in MCMC

- The ADS formulation is powerful, but its direction is too “random.”
- How to make use of their framework?
- Population of samples
- Randomly select to be updated.
- Use the rest to determine an “anchor point”
- Here we can use local optimization techniques;
- Use MTM to draw sample along the line, with the help of the Snooker Theorem.

MCMC and Statistics

xc

xa

(anchor point)

A gradient or conjugate

gradient direction.

MCMC and Statistics

A More DifficultTest Example

- Mixture of 2 Gaussians:
- MTM with CG can sample the distribution.
- The Random-Ray also worked well.
- The standard Metropolis cannot get across.

MCMC and Statistics

Fitting a Mixture model

- Likelihood:
- Prior: uniform in all, but with constraints

And each group has at least one data point.

MCMC and Statistics

Bayesian Neural Network Training

Nonlinear curve fitting:

- Setting: Data =
- 1-hidden layer feed-forward NN Model
- Objective function for optimization:

MCMC and Statistics

Liang and Wong (1999) proposed a method that combines the snooker theorem, MTM, exchange MC, and genetic algorithm.

Activation function: tanh(z)

# hidden units M=2

MCMC and Statistics

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