Mathematical Foundations of Markov Chain Monte Carlo Algorithms

1
Mathematical FoundationsofMarkov Chain Monte
Carlo Algorithms

• Based on lectures given by
• Alistair Sinclair
• Computer Science Division
• U.C. Berkeley

2
Overview

1. Random Sampling
2. The Markov Chain Monte-Carlo Paradigm
3. Mixing Time
4. Coupling
5. Flow
6. Geometry

Techniques for Bounding the Mixing Time
3
Random Sampling
x

• ? - very large sample set.
• ? - probability distribution over ?.

Goal Sample points x?? at random from
distribution ?.
4
The Probability Distribution

• Typically,

w??R is an easily-computed weight function
ZSx w(x) is an unknown normalization factor
5
Application 1 Card Shuffling

• ? - all 52! permutations of a deck of cards.
• ? - uniform distribution ?x w(x)1.

Goal pick a permutation uniformly at random
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Application 2 Counting

• How many ways can we tile some given pattern with
  dominos?

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Application 2 Counting (cont.)

• Sample tilings uniformly at random.
• Let P1 proportion of sample of type 1.
• Compute estimate N1 of N1 recursively.
• output N N1 / P1.

N1
N2
N N1 N2
sample size O(n), levels O(n) ? O(n2)
samples total
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Application 3 Volume Integration
Dyer\Frieze\Kannan

• ? a convex body in Rd (d large)
• Problem estimate vol(?)

sequence of concentric balls B0 ? ? Br
estimate by sampling uniformly from ??Bi
Generalization Integration of log-concave
function over a cube A?Rd
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Application 4 Statistical Physics

• ? - set of configurations of a physical system
• ? - Gibbs distribution
• ?(x)Pr system in config. xw(x)/Z
• where w(x)e-H(x)/KT

temperature
energy
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The Ising Model

• n atomic magnets
• configuration x?-,n
• H(x) -(aligned neighbors)

• - - -
• - - -
• - - -
• - - - -
• -

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Why Sampling?

• statistics of typical configurations.
• mean energy (E?H(x)), specific heat,
• estimate of partition function ZZ(T)?x??w(x)

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Estimating the Partition Function

• Let ?e-1/KT ? ZZ(?)?x?? ?-H(x).
• Define 1?0 lt ?1 lt lt ?r?.

? r ? nlog?O(n2)
can be estimated by random sampling from
??i-1 ?i? ?i-1(11/n) ensures small variance ?
O(n) samples suffice for each ratio
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Application 5 Optimization

• ? - set of feasible solutions to an optimization
  problem
• f(x) - value of solution x.
• Goal maximize f(x).
• Idea sample solutions where w(x)?f(x).

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Application 5 Optimization

• Idea sample solutions where w(x)?f(x).

concentration on good solutions (large values
f(x))
large ?
?
greater mobility (local optima are less high)
small ?
Simulated Annealing heuristic Slowly increase ?
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Application 6 Hypothesis Verification in
Statistical Models

• ? - set of hypotheses
• X - observed data

Let w(?)P(?)P(X/?).
prior
easy
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Application 6 Hypothesis Verification in
Statistical Models (cont.)

• Sampling from ?(?)P(?/X) gives
• Statistical estimate of hypotheses ?.
• Prediction
• Model comparison
• normalization factor P(X)
• Prob model generated X

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Markov Chains

• Sample space ?
• Random variables (r.v) over ?
• X1,X2,,Xt,
• Memoryless ?tgt0, ?x1,,xt1??,

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Sampling Algorithm

• Start at an arbitrary state X0.
• Simulate MC for sufficiently many steps t.
• Output Xt.
• Then, ?x?? Prob Xt x ?(x)

X0
?
Xt
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Transitions Matrix
PrXt1y/Xtx
y

• P is non-negative
• P is stochastic (?x ?xP(x,y)1)
• PrXt1y/X0xPt(x,y)
• PxtPx0 Pt
• Definition ? is a stationary distribution, if
  ?P?.

Px
x
P
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Irreducibility

• Definition P is irreducible if

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Aperiodicity

• Definition P is aperiodic if

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Note on Irreducibility and Aperiodicity

• If P is irreducible, we can always make it
  aperiodic, by adding self-loops
• P ½(PI)
• P has same stationary distribution as P.
• Call P a lazy MC.

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Fundamental Theorem

• Theorem If P is irreducible and aperiodic,
• then it is ergodic, i.e
• where ? is the (unique) stationary distribution
  of P i.e ? P?.

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Main Idea (The MCMC Paradigm)

• An ergodic MC provides an effective algorithm for
  sampling from ?.

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Examples

• Random Walks on Graphs
• Ehrenfest Urn
• Card Shuffling
• Coloring of a Graph
• The Ising Model

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1. Random Walk on Undirected Graphs
At each node, choose a neighbor u.a.r and jump to
it
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Random Walk on Undirected Graph G(V,E)
?V
degree

• Irreducible ? G is connected
• Aperiodic ? G is not bipartite

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Random Walk The Stationary Distribution
not essential

• Claim If G is connected and not bipartite, then
  the probability distribution induced by a random
  walk on it converges to ?(x)d(x)/Sxd(x).
• Proof

2E
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2. Ehrenfest Urn
j balls
(n-j) balls

• Pick a ball u.a.r
• Move the ball to the other urn

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2. Ehrenfest Urn

• Xt number of balls in first urn.
• MC is a non-uniform random walk on ?0,1,,n.

j/n
1-j/n

• Irreducible Periodic
• Stationary distribution

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3. Card Shuffling

1. Top-in-at-random

• Irreducible
• Aperiodic
• P is doubly stochastic ?y SxP(x,y)1
• ? ? is uniform ?x ?(x)1/n!

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3. Card Shuffling

1. Random Transpositions

• Irreducible
• Aperiodic
• P is symmetric ?x,y P(x,y)P(y,x)
• ? ? is uniform

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3. Card Shuffling

1. Riffle shuffle Gilbert/Shannon/Reeds

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3. Card Shuffling

1. Riffle shuffle Gilbert/Shannon/Reeds

• Irreducible
• Aperiodic
• P is doubly stochastic
• ? ? is uniform

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4. Colorings of a graph

• G(V,E) connected, undirected
• q number of colors
• ? set of proper q-colorings of G
• ? uniform

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Colorings Markov Chain

• pick v?V and c?1,,q u.a.r.
• recolor v with c if possible.

Gs max degree

• Irreducible if q??2
• Aperiodic
• P is symmetric
• ? ? is uniform

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5. The Ising Model

• Markov chain (Heat bath)
• pick a site i u.a.r
• replace spin x(i) by random spin x(i) s.t

• n sites
• ?-,n
• w(x)?aligned neighbors (x)

• - - -
• - - -
• - - -
• - - - -
• -

neighbors of i
Irreducible, aperiodic, reversible w.r.t ? ?
converges to ?
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Designing Markov Chains

• What do we want?
• Given ?, ?
• MC over ? which converges to ?

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The Metropolis Rule

• Define any connected undirected graph on ?
  (neighborhood structure/(local) moves)

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The Metropolis Rule

• Transitions from state x??
• pick a neighbor y of x w.p ?(x,y)
• move to y w.p minw(y)/w(x),1
• (else stay at x)

?(x,y)?(y,x), ?(x,x)1-Sy-x?(x,y)

• Irreducible
• Aperiodic (make lazy if nec.)
• reversible w.r.t w
• ? converges to ?.

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The Mixing Time

• Key Question How long until Pxt looks like ??
• We will use the variation distance

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The Mixing Time

• Define
• ?x(t) pxt-?
• ?(t) maxx ?x(t)
• The mixing time is
• ?mixmin t ?(t)?1/2e

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Toy Example Top-In-At-Random

• Let T time after initial bottom card reaches
  top
• T is a strong stationary time, i.e
• PrXtx/tT?(x)
• Claim ?(t)?PrTgtt
• Thus, it remains to estimate T.

n
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The Coupon Collector Problem

• Each pack contains one coupon.
• The goal is to complete the series.
• How many packs would we buy?!

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The Coupon Collector Problem

• N total number of different coupons.
• Xi time to get the i-th coupon.

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Toy Example Top-In-At-Random

• By the coupon collector,
• the i-th coupon is a ticket to advance from the
  (n-i1) level to the next one.
• Pr T gt nlnn cn ? e-c
• ? ?mixnlnn cn

n
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Example Riffle Shuffle
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Example Riffle Shuffle

• Inverse shuffle (same mixing time)

0 0 0 1 1 1 1 1
1 0 1 1 1 0 1 0
0/1 u.a.r
sorted stably
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Inverse Shuffle

• After t steps, each card is labeled with t
  digits.
• Cards are sorted by their labels.
• Cards with different labels are in random order
• Cards with same label are in original order

0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1
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Riffle Shuffle (Cont.)

• Let T time until all cards have distinct labels
• T is a strong stationary time.
• Again we need to estimate T.

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B i rthday Paradox

• With which probability two of them have the same
  birthday?

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B I rthday Paradox (Cont.)

• k people, n days (ngtkgt1)
• The probability all birthdays are distinct

arithmetic sum
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Riffle Shuffle (Cont.)

• By the birthday paradox,
• each card (1..n) picks a random label
• there are 2t possible labels
• we want all labels to be distinct
• ?mixO(logn)

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General Techniques for Mixing Time

• Probabilistic Coupling
• Combinatorial Flows
• Geometric - Conductance

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Coupling
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Mixing Time Via Coupling

• Let P be an ergodic MC. A coupling for P is a
  pair process (Xt,Yt) s.t
• Xt,Yt are each copies of P
• XtYt ? Xt1Yt1
• Define Txymint XtYt X0x, Y0Y

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Coupling Theorem

• Theorem Aldous et al.
• ?(t) ? maxx,yPrTx,y gt t

Design a coupling that brings X and Y together
fast
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1. Random Walk On Cube
?0,1n ? is uniform

• Markov Chain
• pick coordinate i?R1,,n
• pick value b?R0,1
• set x(i)b

1/2
1/6
1/6
1/6
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Coupling For Random Walk

• pick same i,b for both X and Y
• Txy ? time to hit all n coordinates
• By coupon collecting,
• Pr Txy gt nlnn cn lt e-c
• ? ?mix ? nlnn cn

( 0 , 0 , 1 , 0 , 1 , 1 )
( 1 , 1 , 0 , 0 , 1 , 0 )
( 0 , 0 , 1 , 0 , 1 , 1 )
( 1 , 1 , 0 , 0 , 1 , 1 )
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Flow
capacity of e(z,z) C(e)?(z)P(z,z)
flow along e denoted f(e)
flow routes ?(x)?(y) units from x to y, for every
x,y
l(f)
Diameter
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Flow Theorem

• Theorem Diaconis/Stroak, Jerrum/Sinclair
• For a lazy ergodic MC and any flow f,
• ?x(?) ? 2p(f)l(f) ln?(x)-1 2ln?-1

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1. Random Walk On Cube

• Flow f Route (x,y) flow evenly along all
  shortest paths xy
• ? ?mix ?
• constp(f)l(f)log?-1 O(n3)

?0,1n ?2nN ?x ?(x)1/N
1/2
1/2n
1/2n
1/2n
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Conductance
bottleneck
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Conductance
S
?-S
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Conductance Theorem

• Theorem Jerrum/Sinclair, Lawler/Sokal, Alon,
  Cheeger For a lazy reversible MC,
• ?x(?) ? 2/?2 ln?(x)-1 ln?-1

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1. Random Walk On Cube

• The sketched S is (essentially) the worst S.
• ? ?mix O(?-2 log?min-1) O(n3)