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Multiple-Try Metropolis. Jun Liu Department of Statistics Stanford University. Based on the joint work with F. Liang and W.H. Wong. The Basic Problems of Monte Carlo. Draw random variable Estimate the integral . Sometimes with unknown normalizing constant. c g( x ). c. u cg ( x ).

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Jun Liu Department of Statistics Stanford University

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Multiple-Try Metropolis

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 g(x)

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


a

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


Why Does It Work?

  • 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


Generally

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


Problems?

  • The moves are very “local”

  • Tend to be trapped in a local mode.

MCMC and Statistics


Iteration t

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


A chosen direction

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: t0} 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


Ideas

  • 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


Multiple-Try Metropolis

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


y2

y4

y6

y8

y1

y3

y5

y7

An Interesting Twist

x

  • 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


Distribution contour

xc

xa

(anchor point)

A gradient or conjugate

gradient direction.

MCMC and Statistics


Numerical Examples

  • An easy multimodal problem

MCMC and Statistics


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


MCMC and Statistics


y

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


MCMC and Statistics


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