The document discusses the role of simulation as a fundamental tool in statistical computation, particularly focusing on Monte Carlo methods and the bootstrap technique. It highlights the practical applications of simulation in various fields such as finance, epidemiology, and computer gaming, emphasizing the importance of producing randomness deterministically through pseudo-random generators. The content includes references and resources to enhance understanding and skill development in using R programming for simulations.

1. Simulation: a ubiquitous tool for statistical computation
Statistique exploratoire (NOISE)
introduction to computer language R
rudiments of simulation (aka Monte Carlo methods)
illustration by the boostrap method

2. Simulation: a ubiquitous tool for statistical computation
Organisation
weekly computer labs using R and Rstudio
one computer account per student
midterm and final exams on anonymous account
exam made of programming problems (e.g., 3 out of 6)
exam evaluation based on executable R code
handouts and on-line help available on anonymous account
no other document

3. Simulation: a ubiquitous tool for statistical computation
Reference
documents on
https://www.ceremade.dauphine.fr/~xian/Noise.html
R. Drouihlet, P. Lafaye de Micheaux & B. Liquet (2010) Le
logiciel R Springer, Paris
C. Robert & G. Casella (2007) Introduction to Monte Carlo
Methods with R Springer, New York
Manuals on http://cran.r-project.org/manuals.html
(CRAN is the Comprehensive R Archive Network)

4. Simulation: a ubiquitous tool for statistical computation
Simulation: a ubiquitous tool for statistical
computation
Christian P. Robert
Universit´e Paris-Dauphine
http://www.ceremade.dauphine.fr/~xian
September 26, 2013

5. Simulation: a ubiquitous tool for statistical computation
Outline
Simulation, what’s that for?!
Producing randomness by deterministic means
Monte Carlo principles
Simulated annealing

6. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Evaluation of the behaviour of a complex system (network,
computer program, queue, particle system, atmosphere,
epidemics, economic actions, &tc)
[ c Office of Oceanic and Atmospheric Research]

7. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Production of changing landscapes, characters, behaviours in
computer games and flight simulators
[ c guides.ign.com]

8. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Determine probabilistic properties of a new statistical
procedure or under an unknown distribution [bootstrap]
(left) Estimation of the cdf F from a normal sample of 100 points;
(right) variation of this estimation over 200 normal samples

9. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Validation of a probabilistic model
Histogram of 103
variates from a distribution and fit by this distribution density

10. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Approximation of a integral
[ c my daughter’s math book]

11. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Maximisation of a weakly regular function/likelihood
[ c Dan Rice Sudoku blog]

12. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Pricing of a complex financial product (exotic options)
Simulation of a Garch(1,1) process and of its volatility (103
time units)

13. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Handling complex statistical problems by approximate
Bayesian computation (ABC)
core principle
Simulate a parameter value (at random) and pseudo-data from the
likelihood until the pseudo-data is “close enough” to the observed
data, then
keep the corresponding parameter value
[Griffith & al., 1997; Tavar´e & al., 1999]

14. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Handling complex statistical problems by approximate
Bayesian computation (ABC)
demo-genetic inference
Genetic model of evolution from a
common ancestor (MRCA)
characterized by a set of parameters
that cover historical, demographic, and
genetic factors
Dataset of polymorphism (DNA sample)
observed at the present time
!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03!
1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+
Différents scénarios possibles, choix de scenario par ABC
Le scenario 1a est largement soutenu par rapport aux

15. Simulation: a ubiquitous tool for statistical computation
Simulation, what’s that for?!
Illustrations
Necessity to “(re)produce chance” on a computer
Handling complex statistical problems by approximate
Bayesian computation (ABC)
Pygmies population demo-genetics
Pygmies populations: do they
have a common origin? when
and how did they split from
non-pygmies populations? were
there more recent interactions
between pygmies and
non-pygmies populations?
94
!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03!
1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+

16. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
[ c MMP World]

17. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
0.1333139
0.3026299
0.4342966
0.2395357
0.3223723
0.8531162
0.3921457
0.7625259
0.1701947
0.2816627
...

18. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
Definition (Pseudo-random generator)
A pseudo-random generator is a deterministic function f from ]0, 1[
to ]0, 1[ such that, for any starting value u0 and any n, the
sequence
{u0, f(u0), f(f(u0)), . . . , fn
(u0)}
behaves (statistically) like an iid U (0, 1) sequence

19. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Pseudo-random generator
Pivotal element/building block of simulation: always requires
availability of uniform U (0, 1) random variables
q
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q
0.0 0.2 0.4 0.6 0.8
0.00.20.40.60.8
xt+1
10 steps (ut , ut+1) of a uniform generator

20. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Philosophical foray
¡Paradox!
While avoiding randomness, the deterministic sequence
(u0, u1 = f(u0), . . . , un = f(un−1))
must resemble a random sequence!
Debate on whether or not true
randomness does exist (Laplace’s
demon versus Schroedinger’s
cat), in which case pseudo
random generators are not
random (von Neuman’s state of
sin)

21. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Philosophical foray
¡Paradox!
While avoiding randomness, the deterministic sequence
(u0, u1 = f(u0), . . . , un = f(un−1))
must resemble a random sequence!
Debate on whether or not true
randomness does exist (Laplace’s
demon versus Schroedinger’s
cat), in which case pseudo
random generators are not
random (von Neuman’s state of
sin)

22. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Philosophical foray
¡Paradox!
While avoiding randomness, the deterministic sequence
(u0, u1 = f(u0), . . . , un = f(un−1))
must resemble a random sequence!
Debate on whether or not true
randomness does exist (Laplace’s
demon versus Schroedinger’s
cat), in which case pseudo
random generators are not
random (von Neuman’s state of
sin)

23. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Philosophical foray
¡Paradox!
While avoiding randomness, the deterministic sequence
(u0, u1 = f(u0), . . . , un = f(un−1))
must resemble a random sequence!
Debate on whether or not true
randomness does exist (Laplace’s
demon versus Schroedinger’s
cat), in which case pseudo
random generators are not
random (von Neuman’s state of
sin)

24. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
True random generators
Intel circuit producing “truly random” numbers:
There is no reason physical generators should be
“more” random than congruential (deterministic)
pseudo-random generators, as those are valid
generators, i.e. their distribution is exactly known
(e.g., uniform) and, in the case of parallel
generations, completely independent

25. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
True random generators
Intel generator satisfies all benchmarks of
“randomness” maintained by NIST:
Skepticism about physical devices, when compared
with mathematical functions, because of (a)
non-reproducibility and (b) instability of the device,
which means that proven uniformity at time t does
not induce uniformity at time t + 1

26. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1

27. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1
Example
Take
f(x) = (69069069x + 12345) mod (232
)
and produce
... 518974515 2498053016 1113825472 1109377984 ...
i.e.
... 0.1208332 0.5816233 0.2593327 0.2582972 ...

28. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1

29. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
A standard uniform generator
The congruencial generator on {1, 2, . . . , M}
f(x) = (ax + b) mod (M)
has a period equal to M for proper choices of (a, b) and becomes a
generator on ]0, 1[ when dividing by M + 1

30. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Approximating π
My daughter’s pseudo-code:
N=1000
ˆπ = 0
for I=1,N do
X=RDN(1), Y=RDN(1)
if X2
+ Y2
< 1 then
ˆπ = ˆπ + 1
end if
end for
return 4*ˆπ/N

31. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Approximating π
My daughter’s pseudo-code:
N=1000
ˆπ = 0
for I=1,N do
X=RDN(1), Y=RDN(1)
if X2
+ Y2
< 1 then
ˆπ = ˆπ + 1
end if
end for
return 4*ˆπ/N
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pi = 3.2
100 simulations

32. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Approximating π
My daughter’s pseudo-code:
N=1000
ˆπ = 0
for I=1,N do
X=RDN(1), Y=RDN(1)
if X2
+ Y2
< 1 then
ˆπ = ˆπ + 1
end if
end for
return 4*ˆπ/N
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pi = 3.108
1000 simulations

33. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Approximating π
My daughter’s pseudo-code:
N=1000
ˆπ = 0
for I=1,N do
X=RDN(1), Y=RDN(1)
if X2
+ Y2
< 1 then
ˆπ = ˆπ + 1
end if
end for
return 4*ˆπ/N
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pi = 3.136
10,000 simulations

34. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Approximating π
My daughter’s pseudo-code:
N=1000
ˆπ = 0
for I=1,N do
X=RDN(1), Y=RDN(1)
if X2
+ Y2
< 1 then
ˆπ = ˆπ + 1
end if
end for
return 4*ˆπ/N
106
simulations

35. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Distributions that differ from uniform distributions
Proble
Given a probability distribution with
density f , how can we produce
randomness according to f ?!
implemented algorithms in a
resident software only available for
common distributions
new distributions may require fast
resolution
no approximation allowed
x
f(x)
an arbitrary density

36. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Distributions that differ from uniform distributions
Proble
Given a probability distribution with
density f , how can we produce
randomness according to f ?!
implemented algorithms in a
resident software only available for
common distributions
new distributions may require fast
resolution
no approximation allowed
x
f(x)
an arbitrary density

37. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm (remember π?!)
Given a probability distribution with density f , how can we produce
randomness according to f ?!
The uniform distribution on the
sub-graph region
Sf = {(x, u); 0 ≤ u ≤ f (x)}
produces a marginal distribution
in x with density f .
(”Fundamental theorem of
simulation”)
x
f(x)

38. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm (remember π?!)
Given a probability distribution with density f , how can we produce
randomness according to f ?!
In practice, this means we are
back to throwing dots on a box
containing
Sf = {(x, u); 0 ≤ u ≤ f (x)}
and counting those inside!
x
f(x)
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39. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup
x
f (x) g(x) = M < ∞
2. Generate dots on the subgraph of g, i.e. Y1, Y2, . . . ∼ g, and
U1, U2, . . . ∼ U ([0, 1])
3. Accept only the Yk’s such that
Uk ≤ f (Yk)/Mg(Yk)

40. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup
x
f (x) g(x) = M < ∞
2. Generate dots on the subgraph of g, i.e. Y1, Y2, . . . ∼ g, and
U1, U2, . . . ∼ U ([0, 1])
3. Accept only the Yk’s such that
Uk ≤ f (Yk)/Mg(Yk)

41. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
1. Find a hat on f , i.e. a density g that can be simulated and
such that
sup
x
f (x) g(x) = M < ∞
2. Generate dots on the subgraph of g, i.e. Y1, Y2, . . . ∼ g, and
U1, U2, . . . ∼ U ([0, 1])
3. Accept only the Yk’s such that
Uk ≤ f (Yk)/Mg(Yk)

42. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Accept-Reject Algorithm
Refinement of the above through construction of a proper box:
x
f(x)
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43. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Does not require normalisation constant of f
Does not require an exact upper bound M
Requires on average M Yk’s for one simulated X (efficiency
measure)

44. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Does not require normalisation constant of f
Does not require an exact upper bound M
Requires on average M Yk’s for one simulated X (efficiency
measure)

45. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Does not require normalisation constant of f
Does not require an exact upper bound M
Requires on average M Yk’s for one simulated X (efficiency
measure)

46. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
There exists other ways of exploitating the fundamental idea of
simulation over the subgraph:

47. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
There exists other ways of exploitating the fundamental idea of
simulation over the subgraph:
If direct uniform simulation on
Sf = {(u, x); 0 ≤ u ≤ f (x)}
is too complex [because of unavailable hat] use instead a random
walk on Sf :

48. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
There exists other ways of exploitating the fundamental idea of
simulation over the subgraph:
this means doing random jumps in vertical then horizontal
direction, accounting for the boundaries
0 ≤ u ≤ f (x), i.e. U(0, 1)
f (x) ≥ u, i.e. x ∼ US(u)

49. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.

50. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

51. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

52. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

53. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

54. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

55. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

56. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Slice sampler
Slice sampler algorithm
For t = 1, . . . , T
when at (x(t), ω(t)) simulate
1. ω(t+1) ∼ U[0,f (x(t))]
2. x(t+1) ∼ US(t+1) , where
S(t+1)
= {y; f (y) ≥ ω(t+1)
}.
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y

57. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm
Further generalisation of the fundamental idea to situations when
the slice sampler cannot be easily implemented

58. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm
Further generalisation of the fundamental idea to situations when
the slice sampler cannot be easily implemented
Idea
Create a sequence (Xn) such that, for n ‘large enough’, the density
of Xn is close to f , only using a ‘local’ knowledge of f ...
This is the domain of Markovian
simulation methods: there is a Markov
dependence between the Xn’s
Andre¨ı Markov

59. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm
Further generalisation of the fundamental idea to situations when
the slice sampler cannot be easily implemented
Idea
Create a sequence (Xn) such that, for n ‘large enough’, the density
of Xn is close to f , only using a ‘local’ knowledge of f ...
This is the domain of Markovian
simulation methods: there is a Markov
dependence between the Xn’s
Markov bar, Melbourne

60. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm
Further generalisation of the fundamental idea to situations when
the slice sampler cannot be easily implemented
‘local’ exploration can produce ‘global’ vision:
[ c Civilization 5]

61. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm (2)
If f is the density of interest, we pick a proposal conditional density
q(y|xn)
such that
it connects with the current value xn
it is easy to simulate
it is positive everywhere f is positive

62. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
The Metropolis–Hastings algorithm (2)
If f is the density of interest, we pick a proposal conditional density
q(y|xn)
such that
it connects with the current value xn
it is easy to simulate
it is positive everywhere f is positive

63. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Random walk Metropolis–Hastings
Proposal
Yt = Xn + εn,
where εn ∼ g, independent from Xn, and g symmetrical

64. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Random walk Metropolis–Hastings
Proposal
Yt = Xn + εn,
where εn ∼ g, independent from Xn, and g symmetrical
Motivation
local perturbation of Xn / myopic exploration of its neighbourhood
Yn accepted or rejected depending on the relative values of f (Xn)
and f (Yn)

65. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Random walk Metropolis–Hastings
Proposal
Yt = Xn + εn,
where εn ∼ g, independent from Xn, and g symmetrical

66. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Random walk Metropolis–Hastings
Resulting algorithm
Random walk Metropolis–Hastings
Starting from X(t) = x(t)
1. Generate Yt ∼ g(y − x(t))
2. Take
X(t+1)
=



Yt with proba. min 1,
f (Yt)
f (x(t))
,
x(t) otherwise

67. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion of g
Average probability of acceptance
= min{f (x), f (y)}g(y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]

68. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion of g
Average probability of acceptance
= min{f (x), f (y)}g(y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]

69. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
Always accepts higher point and sometimes lower points
(similar to gradient algorithms)
Depends on the dispersion of g
Average probability of acceptance
= min{f (x), f (y)}g(y − x) dxdy
close to 1 if g has a small variance [Danger!]
far from 1 if g has a large variance [Re-Danger!]

70. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
scale=1

71. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
scale=2

72. Simulation: a ubiquitous tool for statistical computation
Producing randomness by deterministic means
Properties
scale=3

73. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
General purpose of Monte Carlo methods
Given a probability density f known up to a normalizing constant,
f (x) ∝ ˜f (x), and an integrable function h, compute
I(h) = h(x)f (x)dx =
h(x)˜f (x)dx
˜f (x)dx
when h(x)˜f (x)dx is intractable.
[Remember π!!]

74. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
General purpose of Monte Carlo methods
Given a probability density f known up to a normalizing constant,
f (x) ∝ ˜f (x), and an integrable function h, compute
I(h) = h(x)f (x)dx =
h(x)˜f (x)dx
˜f (x)dx
when h(x)˜f (x)dx is intractable.
[Remember π!!]

75. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Monte Carlo basics
Generate a pseudo-random
sample x1, . . . , xN from f and
estimate I(h) by
ˆIMC
N (h) = N−1
N
i=1
h(xi ).
and let N grow to infinity...

76. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Monte Carlo basics
Generate a pseudo-random
sample x1, . . . , xN from f and
estimate I(h) by
ˆIMC
N (h) = N−1
N
i=1
h(xi ).
and let N grow to infinity...
A. Doucet, C. Andrieu, X, A. Philippe, J. Rosenthal, E. Moulines

77. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Monte Carlo basics
Generate a pseudo-random
sample x1, . . . , xN from f and
estimate I(h) by
ˆIMC
N (h) = N−1
N
i=1
h(xi ).
and let N grow to infinity...

78. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Monte Carlo basics
Generate a pseudo-random
sample x1, . . . , xN from f and
estimate I(h) by
ˆIMC
N (h) = N−1
N
i=1
h(xi ).
and let N grow to infinity...
Caveat
Often impossible or inefficient to simulate directly from f

79. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Importance Sampling
For proposal distribution with density q(x), alternative
representations
I(h) = h(x){f /q}(x)q(x)dx
Principle
Generate an iid sample x1, . . . , xN ∼ q and estimate I(h) by
ˆIIS
q,N(h) = N−1
N
i=1
h(xi ){f /q}(xi ).

80. Simulation: a ubiquitous tool for statistical computation
Monte Carlo principles
Importance Sampling
For proposal distribution with density q(x), alternative
representations
I(h) = h(x){f /q}(x)q(x)dx
Principle
Generate an iid sample x1, . . . , xN ∼ q and estimate I(h) by
ˆIIS
q,N(h) = N−1
N
i=1
h(xi ){f /q}(xi ).

81. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Optimisation problems
A (genuine) puzzle
During a dinner with 20 couples sitting at four tables with ten
seats, everyone wants to share a table with everyone. The
assembly decides to switch seats after each serving towards this
goal. What is the minimal number of servings needed to ensure
that every couple shared a table with every other couple at some
point? And what is the optimal switching strategy?
[http://xianblog.wordpress.com/2012/04/12/not-le-monde-puzzle/]
solution to the puzzle

82. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Stochastic minimisation
Example (Egg-box function)
Consider the function
h(x, y) = (x sin(20y) + y sin(20x))2
cosh(sin(10x)x)
+ (x cos(10y) − y sin(10x))2
cosh(cos(20y)y) ,
to be minimised.

83. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Stochastic minimisation
Example (Egg-box function)
Consider the function
h(x, y) = (x sin(20y) + y sin(20x))2
cosh(sin(10x)x)
+ (x cos(10y) − y sin(10x))2
cosh(cos(20y)y) ,
to be minimised. (I know
that the global minimum is 0
for (x, y) = (0, 0).)

84. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Stochastic minimisation
Example (Egg-box function)
Consider the function
h(x, y) = (x sin(20y) + y sin(20x))2
cosh(sin(10x)x)
+ (x cos(10y) − y sin(10x))2
cosh(cos(20y)y) ,
to be minimised. (I know
that the global minimum is 0
for (x, y) = (0, 0).)

85. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Stochastic minimisation
Example (egg-box function (2))
Instead of solving the first order equations
∂h(x, y)
∂x
= 0 ,
∂h(x, y)
∂y
= 0
and of checking that the second order conditions are met, we can
generate a random sequence in R2
θj+1 = θj +
αj
2βj
∆h(θj , βj ζj ) ζj

86. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Stochastic minimisation
Example (egg-box function (2))
we can generate a random sequence in R2
θj+1 = θj +
αj
2βj
∆h(θj , βj ζj ) ζj
where
the ζj ’s are uniform on the unit circle x2 + y2 = 1;
∆h(θ, ζ) = h(θ + ζ) − h(θ − ζ);
(αj ) and (βj ) converge to 0

87. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Calibration parameters
choice 1 2 3 4
αj 1/ log(j + 1) 1/100 log(j + 1) 1/(j + 1) 1/(j + 1)
βj 1/ log(j + 1).1 1/ log(j + 1).1 1/(j + 1).5 1/(j + 1).1
α ↓ 0 slowly, j αj = ∞
β ↓ 0 more slowly,
j (αj /βj )2 < ∞
Scenarios 1-2: not enough
energy
Scenarios 3-4: good
[ c George Casella 1951–2012

88. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
The traveling salesman problem
A classical allocation problem:
Salesman who needs to
visit n cities in a row
Traveling costs between
pairs of cities are known
Search of the optimal
circuit
Aboriginal art, NGA, Melbourne

89. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
The traveling salesman problem
A classical allocation problem:
Salesman who needs to
visit n cities in a row
Traveling costs between
pairs of cities are known
Search of the optimal
circuit
Aboriginal art, NGA, Melbourne

90. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
The traveling salesman problem
A classical allocation problem:
Salesman who needs to
visit n cities in a row
Traveling costs between
pairs of cities are known
Search of the optimal
circuit
Procter & Gamble competition, 1962

91. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
An NP-complete problem
The traveling salesman
problem is an example of
mathematical problems that
require explosive resolution
times
Aboriginal art, NGA, Melbourne

92. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
An NP-complete problem
The traveling salesman
problem is an example of
mathematical problems that
require explosive resolution
times
Number of possible circuits n!
and exact solutions available
in O(2n) time
Exact solution for 15, 112 German cities
found in 2001 in 22.6 CPU years.

93. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
An NP-complete problem
The traveling salesman
problem is an example of
mathematical problems that
require explosive resolution
times
Numerous practical
consequences (networks,
integrated circuit design,
genomic sequencing, &tc.)
Exact solution for the 24, 978 Sweedish cities
found in 2004 in 84.8 CPU years.

94. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
The simulated annealing algorithm:
name is borrowed from metallurgy
metal manufactured by a slow
decrease of temperature
(annealing) stronger than when
manufactured by fast decrease
[ c Joachim Robert, ca. 2006]

95. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
Repeat
Random modifications of parts of the original circuit with cost
C0
Evaluation of the cost C of the new circuit
Acceptation of the new circuit with probability
min exp
C0 − C
T
, 1
[Metropolis et al., 1953]

96. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Resolution via simulation
Repeat
Random modifications of parts of the original circuit with cost
C0
Evaluation of the cost C of the new circuit
Acceptation of the new circuit with probability
min exp
C0 − C
T
, 1
T, temperature, is progressively reduced
[Metropolis et al., 1953]

97. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (1)
Recall the table puzzle
Definition of a target function
Set the story of the 20 couple tables during the 6 courses as a
(20, 6) matrix, e.g.,
S =





1 1 3 2 4 2
2 1 3 3 2 4
...
...
...
...
...
...
3 2 2 2 4 4





and define a penalty function as the number of missed encounters

98. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (1)
Recall the table puzzle
Definition of a target function
I=sample(rep(1:4,5))
for (i in 2:6)
I=cbind(I,sample(rep(1:4,5)))
meet=function(I){
M=outer(I[,1],I[,1],"==")
for (i in 2:6)
M=M+outer(I[,i],I[,i],"==")
M
}
penalty=function(M){ sum(M==0) }
penat=penalty(meet(I))

99. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (2)
Random switch of couples
Pick two couples [among the 20 couples] at random with
probabilities proportional to the number of other couples they
have not seen
prob=apply(meet(I)==0,1,sum)
switch their respective position during one of the 6 courses
accept the switch with Metropolis–Hastings probability
log(runif(1))<(penalty(old)-penalty(new))/gamma

100. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (2)
Random switch of couples
For instance, propose to replace
S =





1 1 3 2 4 2
2 1 3 3 2 4
...
...
...
...
...
...
3 2 2 2 4 4





with S =





1 1 3 2 2 2
2 1 3 3 4 4
...
...
...
...
...
...
3 2 2 2 4 4






101. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (2)
Random switch of couples
for (t in 1:N){
prop=sample(1:20,2,prob=apply(meet(I)==0,1,sum))
cour=sample(1:6,1)
Ip=I
Ip[prop[1],cour]=I[prop[2],cour]
Ip[prop[2],cour]=I[prop[1],cour]
penatp=penalty(meet(Ip))
if (log(runif(1))<(penat-penatp)/gamma){
I=Ip
penat=penatp}
}

102. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
A family meeting (3)
Solution
I
[1,] 1 4 3 2 2 3
[2,] 1 2 4 3 4 4
[3,] 3 2 1 4 1 3
[4,] 1 2 3 1 1 1
[5,] 4 2 4 2 3 3
[6,] 2 4 1 2 4 1
[7,] 4 3 1 1 2 4
[8,] 1 3 2 4 3 1
[9,] 3 3 3 3 4 3
[10,] 4 4 2 3 1 1
[11,] 1 1 1 3 3 2
[12,] 3 4 4 1 3 2
[13,] 4 1 3 4 4 2
[14,] 2 4 3 4 3 4
[15,] 2 3 4 2 1 2
[16,] 2 2 2 3 2 2
[17,] 2 1 2 1 4 3
[18,] 4 3 1 1 2 4
[19,] 3 1 4 4 2 1
[20,] 3 1 2 2 1 4 [ c http://www.metrolyrics.com]

103. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Solving a Sudoku grid as a minimisation problem:
[ c Dan Rice Sudoku blog]

104. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
s=matrix(0,ncol=9,nrow=9)
s[1,c(1,6,7)]=c(8,1,2)
s[2,c(2:3)]=c(7,5)
s[3,c(5,8,9)]=c(5,6,4)
s[4,c(3,9)]=c(7,6)
s[5,c(1,4)]=c(9,7)
s[6,c(1,2,6,8,9)]=c(5,2,9,4,7)
s[7,c(1:3)]=c(2,3,1)
s[8,c(3,5,7,9)]=c(6,2,1,9)

105. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
define a penalty function corresponding to the number of
missed constraints
#local score
scor=function(i,s){
a=((i-1)%%9)+1
b=trunc((i-1)/9)
boxa=3*trunc((a-1)/3)+1
boxb=3*trunc(b/3)+1
return(sum(s[i]==s[9*b+(1:9)])+
sum(s[i]==s[a+9*(0:8)])+
sum(s[i]==s[boxa:(boxa+2),boxb:(boxb+2)])-3)
}

106. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
Given a partly filled Sudoku grid (with a single solution),
define a random Sudoku grid by filling the empty slots at
random
define a penalty function corresponding to the number of
missed constraints
fill “deterministic” slots
make simulated annealing moves
# random moves on the sites
i=sample((1:81)[as.vector(s)==0],sample(1:sum(s==0),1,pro=1/(1:sum(s==0))))
for (r in 1:length(i))
prop[i[r]]=sample((1:9)[pool[i[r]+81*(0:8)]],1)
if (log(runif(1))/lcur<tarcur-target(prop)){
nchange=nchange+(tarcur>target(prop))
cur=prop
points(t,tarcur,col="forestgreen",cex=.3,pch=19)
tarcur=target(cur)
}

107. Simulation: a ubiquitous tool for statistical computation
Simulated annealing
Solving sudokus
many possible variants in the
proposals
rather slow and sometimes
really slow
may get stuck on a penalty of
2 (and never reach zero)
does not compete at all with
nonrandom solvers