1. Jun Liu Department of Statistics Stanford University
Multiple-Try Metropolis
Jun Liu
Department of Statistics
Stanford University
Based on the joint work with F. Liang and W.H. Wong.
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MCMC and Statistics

2. The Basic Problems of Monte Carlo
Draw random variable
Estimate the integral
Sometimes with unknown normalizing constant
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3. 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
c g(x)
p(x) x
u cg(x) c
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4. High Dimensional Problems?
Ising Model a
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.
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5. 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”
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6. Why Does It Work? The detailed balance Actual transition probability
from x to y, where
Transition probability
from y to x.
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7. 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.
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8. 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.
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9. Problems? The moves are very “local”
Tend to be trapped in a local mode.
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10. 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).
Iteration t xc xa
Multiple
chains
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11. 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
A chosen direction
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12. How to sample along a line?
What is the correct conditional distribution?
Random direction:
Directions chosen a priori: the same as above
In ADS?
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13. 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)
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14. Connection with transformation group
WLOG, we let y=0.
The move is now: x  x’=t x
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  .
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15. 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).
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16. Ideas Propose bigger jumps Proposal with mix-sized stepsizes.
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?
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17. Multiple-Try Metropolis
Current is at x
Can be dependent ones
Draw y1,…,yk from the proposal T(x, y) .
Select Y=yj with probability  (yj)T(yj,x).
Draw from T(Y, x). Let
Accept the proposed yj with probability
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18. A Modification
If T(x,y) is symmetric, we can have a different rejection probability:
Ref: Frankel and Smit (1996)
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19. Back to the example y3 y5 y4 x y2 y1 Random Ray Monte Carlo:
Propose random direction
Pick y from y1 ,…, y5
Correct for the MTM bias y5 y4 x y2 y1
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20. An Interesting Twist x Pick y from y1 ,…, y8
One can choose multiple tries semi-deterministically. y7 y8 x y5 y6 y3 y4 y1 y2
Random equal grids y
Pick y from y1 ,…, y8
The correction rule is the same:
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21. 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.
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22. Distribution contour xc xa (anchor point) A gradient or conjugate
gradient direction.
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23. Numerical Examples An easy multimodal problem 9/17/2018
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25. 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.
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26. Fitting a Mixture model
Likelihood:
Prior: uniform in all, but with constraints
And each group has at least one data point.
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28. Bayesian Neural Network Training
Setting: Data =
1-hidden layer feed-forward NN Model
Objective function for optimization:
Nonlinear curve fitting: y
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29. 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
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