1. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Monte Carlo Sampling methods using Markov
Chains and their Applications
Hastings-University of Toronto
Reading seminar on classics: C.P.Robert
presented by:Donia Skanji
December 3, 2012
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Hastings-University of Toronto Reading Seminar:MCMC
2. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Outline
1 Introduction
2 Monte Carlo Principle
3 Markov Chain Theory
4 MCMC
5 Conclusion
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Hastings-University of Toronto Reading Seminar:MCMC
3. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Introduction to MCMC Methods
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Hastings-University of Toronto Reading Seminar:MCMC
4. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Introduction:
There are several numerical problems such as Integral
computing and Maximum evaluation in large dimensional
spaces
Monte Carlo Methods are often applied to solve integration
and optimisation problems.
Monte Carlo Markov chain (MCMC) is one of the most known
Monte Carlo methods.
MCMC methods involve a large class of sampling algorithms
that have had a greatest influence on science development.
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Hastings-University of Toronto Reading Seminar:MCMC
5. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Study objectif
To expose some relevant theory and techniques of
application related to MCMC methods ♣
To present a generalization of Metropolis sampling method.
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Hastings-University of Toronto Reading Seminar:MCMC
6. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Next Steps
Monte Carlo Principle
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Hastings-University of Toronto Reading Seminar:MCMC
7. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Next Steps
Monte Carlo Principle
Markov Chain
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Hastings-University of Toronto Reading Seminar:MCMC
8. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Next Steps
Monte Carlo Principle
To introduce:
Markov Chain
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Hastings-University of Toronto Reading Seminar:MCMC
9. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Next Steps
Monte Carlo Principle
To introduce:
-MCMC Methods
Markov Chain
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Hastings-University of Toronto Reading Seminar:MCMC
10. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Next Steps
Monte Carlo Principle
To introduce:
-MCMC Methods
-MCMC Algorithms
Markov Chain
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Hastings-University of Toronto Reading Seminar:MCMC
11. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Monte Carlo Methods
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Hastings-University of Toronto Reading Seminar:MCMC
12. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Overview
The idea of Monte Carlo simulation is to draw an i.i.d. set of
samples{x i }N from a target density π.
i=1
These N samples can be used to approximate the target
density with the following empirical point-mass function:
1 N
πN (x) = N i=1 δx (i) (x)
For independent samples, by Law of Large numbers, one can
approximate the integrals I (f ) with tractable sums IN (f ) that
converge as follows:
1 N i
IN (f ) = N i=1 f (x ) → I (f ) = f (x)π(x)dx a.s
see example
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Hastings-University of Toronto Reading Seminar:MCMC
13. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
N sample from π
xN
x3
6 9 1
x x x
7
x x2
8 5
x x
x4
But independent sampling from π may be difficult especially in a
high dimensional space.
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Hastings-University of Toronto Reading Seminar:MCMC
14. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
It turns out that N N f (x i ) → f (x)π(x)dx (N → ∞)
1
i=1
still applies if we generate samples using a Markov
chain(dependent samples).
The idea of MCMC is to use Markov chain convergence
properties to overcome the dimensionality problems met by
regular Monte carlo methods.
But first, some revision of Markov chains in a discrete set χ.
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Hastings-University of Toronto Reading Seminar:MCMC
15. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Markov Chain Theory
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Hastings-University of Toronto Reading Seminar:MCMC
16. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Definition
Finite Markov Chain
A Markov chain is a mathematical system that undergoes
transitions from one state to another, between a finite or countable
number of possible states. It is a random process usually
characterized as memoryless:
P(X (t+1) /X (0) , X (1) , . . . , X (t) ) = P(X (t+1) /X (t) )
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Hastings-University of Toronto Reading Seminar:MCMC
17. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Transition Matrix
Let P = {Pij } the transition Matrix of a markov chain with states
0, 1, 2 . . . , S then, if X (t) denotes the state occupied by the
process at time t, we have:
Pr (X (t+1) = j/X (t) = i) = Pij
X (t+1) = X (t) .P
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Hastings-University of Toronto Reading Seminar:MCMC
18. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Properties
Stationarity/Irreducibility
Stationarity
♣
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Hastings-University of Toronto Reading Seminar:MCMC
19. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Properties
Stationarity/Irreducibility
Stationarity
As t → ∞,the Markov chain converges to its
stationary(invariant) distribution:π = π.P
♣
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Hastings-University of Toronto Reading Seminar:MCMC
20. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Properties
Stationarity/Irreducibility
Stationarity
As t → ∞,the Markov chain converges to its
stationary(invariant) distribution:π = π.P
Irreducibility ♣
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Hastings-University of Toronto Reading Seminar:MCMC
21. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Properties
Stationarity/Irreducibility
Stationarity
As t → ∞,the Markov chain converges to its
stationary(invariant) distribution:π = π.P
Irreducibility ♣
Irreducible means any set of states can be
reached from any other state in a finite number
of moves (p(i, j) > 0 for every i and j).
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Hastings-University of Toronto Reading Seminar:MCMC
22. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
MCMC
The idea of Markov Monte Carlo Method is to choose P the
transition Matrix so that π(the target density which is very
difficult to sample from) is its unique stationary distribution.
Assume the Markov Chain:
has a stationary distribution π(X )
is irreducible and aperiodic
Then we have an Ergodic Theorem:
Theorem(Ergodic Theorem)
if the Markov chain xt is irriducible, aperiodic and stationary then
for any function h with E |h| ∞
1
N i h(xi ) → h(x)dπ(x) when N → ∞
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Hastings-University of Toronto Reading Seminar:MCMC
23. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Summary
Recall that our goal is to build a markov chain (X t )
using a transition matrix P so that the limiting distri-
bution of (X t ) is the target density π and integrals can
be approximated using the ergodic theorem.
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Hastings-University of Toronto Reading Seminar:MCMC
24. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Question
How do we construct a Markov chain whose stationary
distribution is the target distribution,π
Metropolis et al (1953) showed how.
The method was generalized by Hastings (1970).
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Hastings-University of Toronto Reading Seminar:MCMC
25. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Construction of the transition matrix
in order to construct a markov chain with π as its stationary
distribution, we have to consider a transition matrix P that
satisfy the reversibility condition that for all i and j
πi p(i → j) = πj p(j → i)
πi pij = πj pji
This property ensures that πi pij = πj (definition of a
stationary distribution) and hence that π is a stationary
distribution of P
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Hastings-University of Toronto Reading Seminar:MCMC
26. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Construction of the transition matrix
How to choose the
transition Matrix
P so that the πi Pij = πj Pji
reversibility con-
dition is verified?
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Hastings-University of Toronto Reading Seminar:MCMC
27. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Overview
Suppose that we have a proposal matrix denoted Q where
j qij = 1 .
If it happens that Q itself satisfies the reversibility
condition:πi qij = πj qji for all i and j then our research is
over,but most likely it will not.
We might find for example that for some i and j:πi qij > πj qji
A convenient way to correct this condition is to reduce the
number of moves from i to j by introducing a probability αij
that the move is made.
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Hastings-University of Toronto Reading Seminar:MCMC
28. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
The choice of the transition matrix
we assume that the transition matrix P has this form:
Pij = qij αij if i = j
Pii = 1 − j=i Pij if i = j
where:
Q = qij is the proposal matrix or jumping matrix of an
arbitrary Markov chain on the states 0, 1..S, which suggests a
new sample value j given a sample value i.
αij is the acceptance probability to move from state i to
state j.
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Hastings-University of Toronto Reading Seminar:MCMC
29. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
In order to obtain the reversibility condition, we have to verify :
πi pij = πj pji
πi αij qij = πj αji qji (∗)
The probabilities αij and αji are introduced to ensure that the
two sides of (∗) are in balance.
In his paper, Hastings defined a generic form of the acceptance
probability:
sij
αij = π q
1+ πi qij
j ji
Where:sij is a symetric function of i and j(sij = sji ) chosen so
that 0 αij 1 for all i and j
With this form of Pij and αij suggested by Hastings, it’s readily
verified the reversibility condition.
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Hastings-University of Toronto Reading Seminar:MCMC
30. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
2-The acceptance probability α
The choice of α
Recall that in this paper, Hastings defined the acceptance
probability αij as follows:
sij
αij = π q
1+ πi qij
j ji
For a specific choice of sij , we recognize the acceptance
probabilities suggested by both:
⊕Metropolis et al(1953)
⊕Barker(1965)
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Hastings-University of Toronto Reading Seminar:MCMC
31. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
The acceptance probability α
The choice of Sij
Two choices for Sij are given for all i and j by
πi qij πj qji
(M) 1+ πj qji if πi qij 1
sij = πj qji πj qji
1+ πi qij if πi qij 1
(M)
when qij = qji and Sij = Sij we have the method devised
(M) π
by Metropolis et al with αij = min(1, πji )
(B)
whenqij = qji and Sij = Sij = 1 we have the method
(B) πj
devised by Barker with αij = ( πi +πj )
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Hastings-University of Toronto Reading Seminar:MCMC
32. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Remark
In this paper, Hastings mentionned that little is known about
(M) (B)
the merits of these two choices of Sij and Sij
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Hastings-University of Toronto Reading Seminar:MCMC
33. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
The Proposal Matrix Q
The choice of Q
It has been recognised that the choice of the proposal
matrix/density is crucial to the success(rapid convergence)
of MCMC algorithm.
The proposal matrix can be almost arbitrary which allows to
reach all states frequently and assure a high acceptance rate
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Hastings-University of Toronto Reading Seminar:MCMC
34. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
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Hastings-University of Toronto Reading Seminar:MCMC
35. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
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Hastings-University of Toronto Reading Seminar:MCMC
36. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
3 Then, to return a new sample j given the most recent
sample i, we proceed as follows:
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Hastings-University of Toronto Reading Seminar:MCMC
37. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
3 Then, to return a new sample j given the most recent
sample i, we proceed as follows:
4 Generate a proposed new sample value j from the jumping
distribution Q(i → j).
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Hastings-University of Toronto Reading Seminar:MCMC
38. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
3 Then, to return a new sample j given the most recent
sample i, we proceed as follows:
4 Generate a proposed new sample value j from the jumping
distribution Q(i → j).
5 Accept proposal with probability α(i → j)
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Hastings-University of Toronto Reading Seminar:MCMC
39. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
3 Then, to return a new sample j given the most recent
sample i, we proceed as follows:
4 Generate a proposed new sample value j from the jumping
distribution Q(i → j).
5 Accept proposal with probability α(i → j)
-if proposal accepted then move to j/step4
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Hastings-University of Toronto Reading Seminar:MCMC
40. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
1 First, pick a proposal matrix Q(i, j) of an arbitrary Markov
chain on the states 0, 1..S, which suggests a new sample
value j given a sample value i.
2 Also, start with some arbitrary point i0 as the first sample.
3 Then, to return a new sample j given the most recent
sample i, we proceed as follows:
4 Generate a proposed new sample value j from the jumping
distribution Q(i → j).
5 Accept proposal with probability α(i → j)
-if proposal accepted then move to j/step4
-repeat until a sample from the desired size is
obtained
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Hastings-University of Toronto Reading Seminar:MCMC
41. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Remarks
An empirical way for checking convergence is to let two or
more different chains run in parallel and see if they are
concentrating on the some place.
The calculation of α does not require knowledge of the
normalizing constant of π because it appears both in the
numerator and denominator.
Although the Markov chain eventually converges to the
desired distribution, the initial samples may follow a very
different distribution, especially if the starting point is in a
region of low density.
As a result a burn in period is typically necessary.
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Hastings-University of Toronto Reading Seminar:MCMC
42. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Example:Poisson Distribution as the Target Distribution
Consider π as the Poisson distribution with intensity λ > 0
i
πi = e −λ λ where i = 0, 1, 2, · · ·
i!
Hastings(1970)suggests the following proposal transition matrix
1 1
2 2 0 0 ···
1 1 0 1 0 ···
q00 = q01 = 2 if i = 0 2 2
1 1
if j = i − 1 Q = 0 2 0 2 · · ·
1
qij = 2
0 0 1 0 ···
1
2 if j = i + 1 2
0 otherwise . . . .
. . . . ···
. . . .
Q is in fact symmetric, and the algorithm reduces to that of
Metropolis
skip
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Hastings-University of Toronto Reading Seminar:MCMC
43. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
1 i
2 min(1, λ) if j = i − 1
1 λ
2 min(1, i+1 ) if j = i + 1
(M)
pij = qij αij =
1 − pi,i−1 − pi,i+1
j =i
0 otherwise
For i = 0
1
2 min(1, λ) if j = 1
p0j = 1 − 1 min(1, λ)
2 if j = 0
0 otherwise
this transition probability is aperiodic and irreducible
In practice, if λ is small, this choice of Q seems to work fairly
well and fast to approximate π
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Hastings-University of Toronto Reading Seminar:MCMC
44. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
♣
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Hastings-University of Toronto Reading Seminar:MCMC
45. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
♣
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Hastings-University of Toronto Reading Seminar:MCMC
46. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
♣
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Hastings-University of Toronto Reading Seminar:MCMC
47. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
♣
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Hastings-University of Toronto Reading Seminar:MCMC
48. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
We calculate Metropolis and Hastings ratio: ♣
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Hastings-University of Toronto Reading Seminar:MCMC
49. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
We calculate Metropolis and Hastings ratio: ♣
π(j) i!
αij = min{1, π(i) } = min{1, λ(j−i) × j! }
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Hastings-University of Toronto Reading Seminar:MCMC
50. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
We calculate Metropolis and Hastings ratio: ♣
π(j) i!
αij = min{1, π(i) } = min{1, λ(j−i) × j! }
let u ∼ U[0, 1]
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Hastings-University of Toronto Reading Seminar:MCMC
51. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
We calculate Metropolis and Hastings ratio: ♣
π(j) i!
αij = min{1, π(i) } = min{1, λ(j−i) × j! }
let u ∼ U[0, 1]
if u ≤ αij then Xk+1 = j
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Hastings-University of Toronto Reading Seminar:MCMC
52. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Algorithm
Given a starting point i we take:
1
j=i+1 with probability 2
1
or j=i-1 with probability 2
qij = 2 δi−1 (j) + 1 δi+1 (j)
1
2
We calculate Metropolis and Hastings ratio: ♣
π(j) i!
αij = min{1, π(i) } = min{1, λ(j−i) × j! }
let u ∼ U[0, 1]
if u ≤ αij then Xk+1 = j
else Xk+1 = Xk = i
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Hastings-University of Toronto Reading Seminar:MCMC
53. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
R implementation
> l i b r a r y ( mcsm )
> f a c t=f u n c t i o n ( n ) {gamma( n+1)}
> p o i s s o n f=f u n c t i o n ( n , lambda , x0 ) {
x=x0
xn=x0
f o r ( i i n 1 : n ){
i f ( xn != 0 )
y=xn +(2∗ rbinom ( 1 , 1 , 0 . 5 ) − 1 )
e l s e { y=rbinom ( 1 , 1 , 0 . 5 ) }
a l p h a=min ( 1 , lambda ˆ ( y−xn ) ∗ f a c t ( xn ) / f a c t ( y ) )
i f ( r u n i f ( 1 ) < a l p h a ) { xn=y }
x=c ( x , xn ) }
x}
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Hastings-University of Toronto Reading Seminar:MCMC
54. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
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Hastings-University of Toronto Reading Seminar:MCMC
55. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Multivariate Target
if the distribution π is d-dimensional and the simulated
process X (t) = {X1 (t), · · · Xd (t)}, we may use the following
techniques to construct the transition matrix P
1 In the transition from t to t + 1 all co-ordinates of X (t) may
be changed
2 In the transition from t to t + 1 only one co-ordinates of X (t)
may be changed, that selection may be made at random
among the d co-ordinates
3 In the transition from time t to t + 1 only one co-ordinate may
change in each transition, and the co-ordinate being selected
in a fixed rather than a random sequence.
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Hastings-University of Toronto Reading Seminar:MCMC
56. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Hastings’justification
Hastings transformed the d dimensional problem to one
dimensional problem
The approach is based on updating one component at each
time
♣
The transition matrix is defined as follow:P = P1 .P2 · · · Pd
For each (k = 1 · · · d), Pk is constructed so that πPk = π
π will be a stationary distribution of P since
πP = πP1 · · · Pd = πP2 · · · Pd
Orthogonal Matrices
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Hastings-University of Toronto Reading Seminar:MCMC
57. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Conclusion
+In this paper, Hastings gives a generalization of Metropolis
et al (1953) approach.
+He also introduiced gibbs sampling strategy when he
presented the multivariate target.
+Hastings treated the continuous case using a discretization
analogy.
-little information about the merits of Metropolis and Barker
acceptance forms.
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Hastings-University of Toronto Reading Seminar:MCMC
58. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Thank You For Your Attention
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Hastings-University of Toronto Reading Seminar:MCMC
59. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Bibliography
[1]:W.K.Hastings(1970).Monte Carlo Sampling Methods Using
Markov chain and their Applications
[2]:Christian P Roberts (2010).Introduicing Monte Carlo Methods
with R
[3]:Kenneth Lange(2010).Numerical Analysis for statisticians
[4]:Siddhartha Chib(1995).Understanding the metropolis Hastings
algorithm
[5]:Robert Gray(2001).Advanced statistical computing
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Hastings-University of Toronto Reading Seminar:MCMC
60. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Random orthogonal Matrices
Hastings suggests an interesting chain on the space n × n
orthogonal matrices(H H = I , det(H) = 1)
The proposal stage of Hasting’s algorithm consists of choosing
at random 2 indices i and j and an angle θ ∈ [0, 2π]
The proposed replacement for the current rotation matrix H is
then H = Eij (θ).H
Eij (θ) coincides with the identity matrix expect for some
entries
since Eij (θ)−1 = Eij (−θ) the transition density is symmetric
and the markov chain induced is reversible
back
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Hastings-University of Toronto Reading Seminar:MCMC
61. Outline
Introduction
Monte Carlo Principle
Markov Chain Theory
MCMC
Conclusion
Estimating Pi using Monte Carlo methods (SAS output)
Problem :Estimate PI using Monte Carlo
Integration
Strategy:Equation of a circle with radius= 1 :
x 2 + y 2 = 1 which can be written y = 1 − x 2
Area of this circle =pi
Area of this circle in the first quadrant = pi 4
Generate Ux Uniform(0, 1) and Uy Uniform(0, 1)
Check to see if Uy ≤ 2
1 − Ux
The proportion of generated points when this
Condition is true is an estimate of pi 4.
Based on 10,000 simulated points using SAS:
PI (SE ) = 3.1056(0.016)
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Hastings-University of Toronto Reading Seminar:MCMC