Swapped at birth - Faster Monte Carlo methods for point ......“I think you’re begging the question,” said Haydock, “and I can see looming ahead one of those terrible exercises

On probability...

“I think you’re begging the question,” said Haydock, “and I can seelooming ahead one of those terrible exercises in probability where sixmen have white hats and six men have black hats and you have towork it out by mathematics how likely it is that the hats will get mixedup and in what proportion. If you start thinking about things like that,you would go round the bend. Let me assure you of that!”

The Mirror Crack’dAgatha Christie

Mark Huber (Duke University) Swap moves for spatial point processes Graduate/Faculty Seminar 1 / 42

Swapped at birthFaster Monte Carlo methods for point processes

Mark Huber

Department of Mathematics and Institute of Statistics and Decision SciencesDuke University

April 13, 2007


Outline

1 The ProblemSpatial point processesThe Hard Core Gas Model

2 Birth Death ChainsUsing Markov chainsStandard Birth Death ApproachWhy it’s good to swap at birth

3 Perfect SamplingDominated Coupling From the PastdCFTP with standard chainsdCFTP with the swap

4 Results


Spatial point processes

The idea:Begin with grid where each cell on or offProbability proportional to size of cellTake limit as cells get small


Coarse Grid


Better Grid


Even better Grid


Limit: Spatial Poisson process


Number of points is Poisson

Let N = the number of points in area

N ∼ Poisson(µ),µ = λ · area of region

P(N = i) = e−µµi

i!, i ∈ {0, 1, 2, . . .}


Generating from a Poisson point process

Two-step process:

1 Generate N ∼ Poisson(µ)2 Generate X1, X2, . . . , XN independently uniformly from region


The Hard Core Gas Model

Point processes used to model gasesEach point center of hard core of moleculeThat the core is hard means cannot overlap:


Densities for Poisson point processes

Formally, use densities to exact constraintsLet n(x) denote number of points in a configuration

Hard Core:

fhardcore(x) ∝{

1 d(x(i), x(j)) > R for all i , j ∈ {1, . . . , n(x)}0 otherwise

Soft Core (Strauss Point Process)

fsoftcore(x) ∝ γv(x)

v(x) = number of pairs {i , j} with d(x(i), x(j)) ≤ R


Normalizing densities

What makes these problems difficult?Note ∝ in density descriptionsNeed to multiply by constant to make probability densityCalled the normalizing constant

Key fact: finding normalizing constant NP-complete problem


Outline




4 Results


Birth Death Chains

For everything there is a seasonAnd a time for every matter under heaven:A time to be born, and time to die;

Ecclesiastes

Birth Death chains evolve the configuration through timePoints in process are born and then later on, die


Why do this?

For same reason shuffle a deck of cards:Small random changes accumulate over timeResult in the global state being random

Markov chainsNext step only depends on current configurationUnder mild conditions, state eventually “becomes random”


Exponential random variables

Say X ∼ Exp(λ)(read as: X has the exponential distribution with rate parameter λ)if for t ≥ 0,

P(X ∈ [t , t + ∆t ] ≈ λe−λt∆t .

Limit as ∆t → 0:P(X ∈ dt) = λe−λt dt


Birth times

Let X1, X2, . . . be iid mboxExp(λ) with rate µ = λ · area of regionThen Xi is time between birth i and i + 1

X1 X2 X3 X4

time

Called a Poisson process


Points are born...

Run time forwardAt a birth event, add a point uniformly in region


Death: who needs it?

In this world nothing can be said to be certain, except death and taxesBenjamin Franklin

When each point is born, roll Di ∼ mboxExp(1)Then Di is the time until the point dies, removed from state

D1


Why use birth-death chains

The point of birth-death chains:Used for problems with densities with unknown normalizingconstantsPreston [3] worked out detailsDo not always accept births

Hard Core Model:Always accept deathOnly accept birth if does not cause conflict


Example of rejected birth

New point (in blue) is rejectedToo close to existing points


Example of accepted birth

New point (in blue) is accepted an added to configurationToo close to existing points


Example of death

Deaths are always accepted


New move swap

When blocked by exactly one point, “swap” with blocking point:

⇒


Some details

Things to consider:Does swapping give correct distribution?Does it improve performance in a theoretical way?Does the swap move generalize?

Preprint: Huber [2]Can set up probability of swapping to give correct distributionDoes work faster than original chainUse perfect sampling techniques to show faster


Outline




4 Results


Perfect Sampling

“Practice makes perfect, but nobody’s perfect, so why practice?”

Problem with Markov chainsHow long should they be run?Perfect sampling algorithms share good properties of Markovchains......but terminate in finite time (with probability 1)


One dimensional Poisson process

time

space

0


Dominated Coupling From the Past

Kendall and Møller [1] ideaSay we don’t know if a point should be in the set or notIf it dies, great!If point born within range before it dies, bad


Dominated Coupling From the Past (part 2)

Start at fixed time in the past with some unknownsRun forward up until time 0If no “?” points, quit, return sampleOtherwise go farther back in time and begin again

TheoremThis actually works!


Example where “?” go away


Example where “?” do not go away


How to update “?”

Notation:

L = { points definitely in process}U = L ∪ { ? points}

Note that L ⊆ U and when L = U there are no ? points.


Pseudocode for updating “?”

Updates for regular birth-death chain

Hard-core bounding process updateInput: move, M, L, U, Output: L,U1) If move = death of point w2) Let L← L− w , let U ← U − w3) Else (move = birth of point v )4) Let NU ← {w ∈ U : ρ(w , v) ≤ R}5) Let NL ← {w ∈ L : ρ(w , v) ≤ R}6) Execute one of the following cases:7) Case I: |NU | = |NL| = 0, let L← L + v , let U ← U + v8) Case II: |NU | ≥ 1,, |NL| = 0, let U ← U + v


When are you guaranteed good performance?

Theorem (Huber [2])Suppose that N events are generated backwards in time and then runforward to get UN(0) and LN(0). Let B(v , R) denote the area withindistance R of v ∈ S, let r = supv∈S λ · B(v , R), and suppose µ ≥ 4.If µr < 1, then for the chain without the swap move

P(UN(0) 6= LN(0)) ≤ 2µ exp(−N(1− µr)/(10µ). (1)

CorollaryRunning time of dCFTP is Θ(µ ln µ) for λ small.


Pseudocode for updating “?” with swap

Updates for regular birth-death chain

Hard-core bounding process updateInput: move, M, L, U, Output: L,U

1) If move = death of point w2) Let L← L− w , let U ← U − w3) Else (move = birth of point v )4) Let NU ← {w ∈ U : ρ(w , v) ≤ R}5) Let NL ← {w ∈ L : ρ(w , v) ≤ R}6) Execute one of the following cases:7) Case I: |NU | = |NL| = 0, let L← L + v , let U ← U + v8) Case II: |NU | = 1, let L← L + v − NL, let U ← U + v − NU9) Case III:|NU | > 1, |NL| = 0 let U ← U + v

10) Case IV: |NU | > 1, |NL| = 1, let L← L− NL, U ← U + v + NL11) Case V: |NU | > 1, |NL| > 1 (do nothing)


When are you guaranteed good performance?

Theorem (Huber [2])Suppose that N events are generated backwards in time and then runforward to get UN(0) and LN(0). Let B(v , R) denote the area withindistance R of v ∈ S, let r = supv∈S λ · B(v , R), and suppose µ ≥ 4.If µr < 2, then for the chain where a swap is executed with probability1/4,

P(UN(0) 6= LN(0)) ≤ 2µ exp(−N(1− .5βr)/(30µ). (2)

CorollaryRunning time of dCFTP is Θ(µ ln µ) for λ twice as large as without theswap.


Running time results

20 30 40 50 60 700

1000

2000

3000

4000

5000

λ

Ave

rage

num

ber

of e

vent

s pe

r sa

mpl

eRunning time of dCFTP for hard core gas model

no swapswap


Outline




4 Results


Conclusions

What is known:Swap move easy to add to point processesAlso can be used in dCFTP to get perfect sampling algorithmResults in about a 4-fold speedup for hard-core gas model

Future work:Running time comparison for Strauss processExperiment better than theory–can theory be improved?


References

W.S. Kendall and J. Møller.Perfect simulation using dominating processes on ordered spaces, with application tolocally stable point processes.Adv. Appl. Prob., 32:844–865, 2000.

M. L. Huber.Spaital Birth-Death-Swap Chainspreprint, 2007

C.J. Preston.Spatial birth-and-death processes.Bull. Inst. Int. Stat., 46(2):371–391, 1977.


The ProblemSpatial point processesThe Hard Core Gas Model

Birth Death ChainsUsing Markov chainsStandard Birth Death ApproachWhy it's good to swap at birth

Perfect SamplingDominated Coupling From the PastdCFTP with standard chainsdCFTP with the swap

Results

Documents

Swapped at birth - Faster Monte Carlo methods for point ......“I think you’re begging the question,” said Haydock, “and I can see looming ahead one of those terrible exercises