introduction to monte carlo methods (lecture 2)christophe.chorro.fr/docs/mc2.pdf · when x is...

Introduction to Monte Carlo Methods (Lecture 2)

Christophe Chorro ([email protected])

University Paris 1

June 19 2008

Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 2) June 19 2008 1 / 47

Study Plan

Chapitre 1: Simulation of random variablesInversion of the distribution functionTransformation methodRejection MethodSimulation of Gaussian vectorsSimulation of conditional distribution


Simulation of random variables

Starting from i.i.d uniform random variables on [0, 1] (Un)n∈N obtained

using one sequence of pseudo-random numbers

or several sequences of quasi-random numbers,

we want to generate random samples from a general distribution (exponential,Cauchy, Gaussian, Poisson....).

Inversion of distribution function

Transformation method

Rejection method


Plan

1 Simulation of random variablesInversion of the distribution functionTransformation methodRejection methodSimulation of Gaussian vectorsSimulation of conditional distribution


Inversion Method

Let X be a real random variable, the distribution function of X is defined as

FX (x) = P(X ≤ x); x ∈ R.

Properties: non-decreasing, right C0, limx→+∞

F (x) = 1 and limx→−∞

F (x) = 0.

DefinitionWe define the generalized inverse of FX denoted by F−X where ∀u ∈]0, 1[,

F−X (u) = inf{x | FX (x) ≥ u}.

Remark: When FX is strictly increasing and continuous, F−X = F−1X .


Inversion Method

PropositionIf U ↪→ U([0, 1]) then F−X (U) has the same distribution then X.

Proof: We just have to remark that ∀u ∈]0, 1[, ∀x ∈ R,

F−X (u) ≤ x ⇔ u ≤ FX (x).�

Figure: Illustration of the definition of F−X


Inversion MethodExample 1: Let X be a discrete random variable (Bernoulli, Binomial, Poisson)

such that P(X = xk ) = pk

( ∞∑k=0

pk = 1)

.

One has FX (x) =∞∑

k=0pk 1xk≤x and

F−X (u) = x01u≤p0 +∞∑

k=1

xk 1(k−1Pj=0

pj<u≤kP

j=0pj

).

When X is finite we have the following algorithm:

p = p0, j = 0

while p < U

do j = j + 1, p = p + pj

end

Y = xj


Inversion Method

Example 2: Particular continuous distributions

If X ↪→ E(λ), λ > 0 then FX (x) = (1− e−λx)1x≥0 and

−1λ

Log(1− U) ↪→ E(λ).

If X ↪→ C(a), a > 0, then FX (x) = 1π [Arctan( x

a ) + π2 ] and

a tan(

π(U − 12

)

)↪→ C(a).


Inversion Method

−2000 0 1500

0.00

000.

0010

0.00

20

0 2 4 6 80.

00.

20.

40.

60.

8

Figure: Empirical density of 10000 C(1) (left) and E(1) obtained by Inversion method


Inversion Method

Example 3: When X ↪→ N (0, 1), FX is unknown but we have the followingapproximation:

If u > 0; 5, let t =√−2log(1− u)

F−X (u) ≈ t − c0 + t(c1 + tc2)

1 + t(d1 + t(d2 + td3)).

If u ≤ 0; 5, let t =√−2log(u)

F−X (u) ≈ c0 + t(c1 + tc2)

1 + t(d1 + t(d2 + td3))− t .

Where

c0 = 2.515517, c1 = 0.802853, c3 = 0.010328,

d1 = 1.432788, d2 = 0.189269, d3 = 0.001308.


Inversion Method

−4 −2 0 2 4

0.0

0.1

0.2

0.3

0.4

Figure: Empirical density of 10000 standard normal distribution obtained by Inversionmethod


Plan




Here we try to express X as a function of another random variable Y easy togenerate.

One of the main tool is the following result

PropositionLet D and ∆ be two open sets of Rd and Φ = (Φ1, ...Φd ) : D → ∆ aC1-diffeomorphism. If g : ∆ → R is measurable and bounded then∫

∆

g(v)dv =

∫D

g(Φ(u)) | JΦ(u)) | du

where JΦ(u) = det[(

∂Φi

∂uj(u))

1≤i,j≤d

].



Example 1: Box-Muller method

PropositionIf U1 and U2 are two independent uniform random variables on [0, 1] then

G1 =√−2log(U1)cos(2πU2) and G2 =

√−2log(U1)sin(2πU2)

are two independent N (0, 1).

Proof: Let us define the following C1-diffeomorphism

Φ : (x , y) ∈]0, 1[2→ (u =√−2log(x)cos(2πy), v =

√−2log(x)sin(2πy))

fulfilling |JΦ(x , y)| = 2πx . Since u2 + v2 = −2log(x), according to the change

of variables theorem, one has for F ∈ Cb(R2, R),∫]0,1[2

F (Φ(x , y))dxdy =

∫R2−(R+×{0})

F (u, v)1

2πe−

u2+v22 dudv .�


Uniform random variables on particular domain

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−4 −2 0 2 4

−4

−2

02

4

Figure: Simulation of 5000 pairs of independent N (0, 1) by Box-Muller method



Example 2: Gauss Marsiglia method

PropositionIf (U1, U2) is uniformly distributed on {(u1, u2) ∈ R2 | u2

1 + u22 < 1} then

G1 = U1

√−2log(U2

1 + U22 )

U21 + U2

2and G2 = U2

√−2log(U2

1 + U22 )

U21 + U2

2

are two independent N (0, 1).

Proof: Exercise.



Example 3: From exponential to Poisson distribution

First, some reminder on Gamma distribution:

DefinitionWe say that Z follows a gamma distribution of parameters (a, b) ∈ R2

+

(Z ↪→ G(a, b)) if its density is given by

xa−1e−xb

Γ(a)ba 1R+(x) where Γ(a) =

∫ ∞

0xa−1e−xdx .

PropositionIf Z1 and Z2 are two independent random variables such that Z1 ↪→ G(a1, b)and Z2 ↪→ G(a2, b) then

Z1 + Z2 ↪→ G(a1 + a2, b).



PropositionIf (Ei)i≥1 is a sequence of i.i.d E(λ) then

X =∞∑i=1

i 1(iP

k=1Ek≤1<

i+1Pk=1

Ek

) ↪→ P(λ).

Proof: We have ∀n ∈ N,

P(X = i) = P

(i∑

k=1

Ek ≤ 1 <

i+1∑k=1

Ek

)

wherei∑

k=1Ek is a G(i , 1

λ ) independent of Ei+i . The result follows easily. �



Exercise: Explain why the following algorithm generate P(λ) distribution:

a = e−λ, X = 0

U < −− Random

while a ≤ U

do U = U ∗ Random, X = X + 1

end

X


Inversion Method

0 1 2 3 4 5 6 7 8 9 11 13

Values

Num

ber

od r

ealiz

atio

ns

050

100

150

Figure: 1000 realizations of a P(λ) using the preceding algorithm


Plan



Uniform random variables on general domain

PropositionLet (Zn)n≥1 be i.i.d random variables with values in Rd and D ∈ B(Rd ) suchthat P(Z1 ∈ D) > 0. We define

ν1 = inf (k ≥ 1; Zk ∈ D),

νn+1 = inf (k > νn; Zk ∈ D),

Yn = Zνn .

Then (Yn)n≥1 are i.i.d random variables such that

P(Y1 ∈ A) = P(Z1 ∈ A | Z̄1 ∈ D).

Moreover E [ν1] = E [νn+1 − νn] = 1P(Z1∈D) = 1

α .


Uniform random variables on general (bounded)domain

CorollaryLet (Zn)n≥1 be i.i.d uniform random variables on [0, M]d and D ∈ B(Rd ) suchthat D ∈ [0, M]d and P(Z1 ∈ D) > 0. We define

ν1 = inf (k ≥ 1; Zk ∈ D),

νn+1 = inf (k > νn; Zk ∈ D),

Yn = Zνn .

Then (Yn)n≥1 are i.i.d uniform random variables on D.

Moreover E [ν1] = E [νn+1 − νn] = vol(D)Md .



Example: The following algorithm generate a uniform random variable X onD(0, 1) (see Gauss Marsiglia method.)

U < −− 2Random− 1; V < −− 2Random− 1

while U2 + V2 ≥ 1

do U < −− 2Random− 1; V < −− 2Random− 1

end

X = (U, V)

Average number of simulation π4 ≈ 0, 7854



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−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

Figure: Realization of 10000 uniform random variable in the unit disk



Sketch of the proof of the proposition:

P(ν1 = k) = P(Z1 6∈ D)k−1P(Z1 ∈ D) = α(1− α)k−1; k ≥ 1.

P(Zν1 ∈ A) =∞∑

k=1P(Zν1 ∈ A ∩ ν1 = k) = P(Z1∈A∩D)

P(Z1∈D) .

By induction we have

P

(n⋂

i=1

Zνi ∈ Ai

)=

n∏i=1

P(Z1 ∈ Ai ∩ D)

P(Z1 ∈ D).�



PropositionLet f be a density on Rd and

Df = {(x , u) ∈ Rd × R+ | 0 ≤ u ≤ f (x)}.

Let U ↪→ U([0, 1]) and X be a random variable independent of U with densityf . Then, (X , Uf (X )) ↪→ U(Df ).

Proof: Exercise.



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−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

Figure: On the domain

(x , u) ∈ R× R+ | 0 ≤ u ≤q

12π

e−x2

2

ff, realization of 1000

uniform random variables


Rejection method (I)

PropositionLet f be a density on Rd and

Df = {(x , u) ∈ Rd × R+ | 0 ≤ u ≤ f (x)}.

Let (X , U) be a random variable on Rd × R+, then,

(X , U) ↪→ U(Df ) ⇔ X has density f and ∀x ∈ R+, U | x ↪→ U([0, f (x)]).

Proof: One has1Df (x , u) = f (x)

1f (x)

1[0,f (x)](u).�

Exercise: When Z is a random variable that have a density fX with compactsupport and bounded, find an algorithm to simulate it.


Rejection method (II)

Aim: We want to simulate a random variable X with a known density fX .

Pb: DfX = {(x , u) ∈ Rd × R+ | 0 ≤ u ≤ fX (x)} may be unbounded...

Idea: Find another random variable Y easy to simulate with a known densityfY such that

fX ≤ cfY ; c ∈ R.



Let f de a density on R and g another density such that f ≤ cg.

PropositionLet (Zn)n≥1 a sequence i.i.d random variables with density g and (Un)n≥1 asequence of i.i.d uniform random variables on [0, 1] independent of (Zn)n≥1.We define

ν1 = inf (k ≥ 1; f (Zk ) > cUk g(Zk ))

Y = Zν1 .

Then Y has the density f .

Moreover E [ν1] = c.



Proof of the proposition:

P(f (Zk ) > cUk g(Zk )) = 1c

P(ν1 = k) = (1− 1c )k−1 1

c ; k ≥ 1.

P(Y ≤ t) =∞∑

k=1(1− 1

c )k−1∫ t−∞ g(z)

∫ f (z)cg(z)

0 dudz =∫ t−∞ f (z)dz.



Application 1: From Laplace to Gaussian distribution

Remind that the density of the Laplace distribution is given by

g(x) =12

e−|x|

and that

1√2π

e−x22 ≤

√2eπ

12

e−|x|.



The following algorithm generate a standard gaussian random variable X :

U < −− Random; V < −− Random; W < −− Random

if U < 12 then X < −− (−log(V)) else X < −− log(V)

while 1√2π

e−X22 ≤ W

√2eπ

12 e−|X| do

U < −− Random; V < −− Random; W < −− Random

if U < 12 then X < −− (−log(V)) else X < −− log(V)

end

X

Average number of simulation√

2eπ ≈ 1.31



Exercise: a) Find an algorithm to generate a gaussian distribution from acauchy one.

b) Is it possible to do the converse?

Exercise: a) Show that f (x) = eλ−x−1eλ−1−λ

is a density on R.

b) Show that f (x) ≤ λ−xλ

eλ−1eλ−1−λ

.

c) Find an algorithm to simulate a random variable with density f .



Application 2: The G(a, 1) distribution (0 < a < 1)

The density of the gamma G(a, 1) distribution is given by

f (x) =xa−1e−x

Γ(a)1R+(x)

and we can show that for 0 < a < 1

f (x) ≤ a + eaeΓ(a)

g(x)

where g is a density given by

g(x) =ae

a + e[xa−11[0,1[(x) + e−x1x≥1.

]



The distribution function G associated to g is explicitly known and G−1 isgiven on ]0, 1[ by

G−1(z) =

[a + e

ez] 1

a

1{z< ea+e} − log

[a + e

ae(1− z)

]1{z≥ e

a+e}.

Thus, we may

generate a random variable of density g by the inversion method,

generate a G(a, 1) by the rejection method.

Exercise: a) How to generate a general G(a, b) (a > 0, b > 0) distribution?

b) How to generate a β(a, b) (a > 0, b > 0) distribution?



0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Figure: Empirical density of 100000 realizations of a G(1, 0.5) by rejection method


Plan



Simulation of Gaussian vectors

DefinitionLet X = (X1, ..., Xn) be a random vector of Rn, X is said to be a Gaussianvector if for all x = (x1, ..., xn) in Rn, the real random variable < x , X > isGaussian.

PropositionIf X is a Gaussian vector of Rn then, ∀x ∈ Rn,

ΦX (x) = ei<x,m>e−xt Σx

2

wherem = (E [X1], ..., E [Xn])

andΣ = [cov(Xi , Xj)]1≤i,j≤n.

In this case, we say that X follows a N (m,Σ).



PropositionLet G = (G1, ..., Gn) be a n-sample of a N (0, 1), m ∈ Rn and A be a n × nmatrix. Then

m + AG ↪→ N (m, AAt).

Algorithm to generate a N (m,Σ)

Generate G = (G1, ..., Gn) (for example by Box-Muller).Find A such that AAt = Σ (Cholesky decomposition).Compute m+A G .



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−4 −2 0 2 4

−3

−2

−1

01

23

Figure: 10000 realisations of a gaussian vector of covariance matrix

1116

7√

316

7√

316

2516

!


Plan



Conditional distribution

We consider a random variable X with density fX . For a < b such thatP(a < X < b) > 0 and g : R → R bounded, we want to evaluate

E [g(X ) | a < X < b] =E [g(X )1{a<X<b}]

P(a < X < b).

First Idea: Compute the numerator and the denominator by Monte Carlomethods.

Pb: In practice P(a < X < b) > 0 is very small ⇒ High number ofsimulations...

Second Idea: Find Y with a distribution equals to the conditional distribution ofX given {a < X < b} and use MC Method.

Pb: How to find Y?


Simulation of conditional distribution by rejectionmethod

PropositionLet (Zn)n≥1 a sequence i.i.d random variables with density fX . We define

ν = inf (k ≥ 1; a < Zk < b)

Y = Zν .

Then Y follows the conditional distribution of X given {a < X < b}.

Moreover E [ν] = 1P(a<X<b) .

Pb: In practice P(a < X < b) > 0 is very small ⇒ High number ofsimulations...


Simulation of conditional distribution by inversion

It is easy to see that the distribution of Y has to be equal to

FY (y) =

0 if y ≤ a

FX (y)−FX (a)FX (b)−FX (a) if a < y < b

1 if y ≥ b.

PropositionIf U ↪→ U([0, 1]), then

F−X (FX (a) + (FX (b)− FX (a))U)

follows the conditional distribution of X given {a < X < b}.


Inversion Method

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure: Empirical density of 10000 realization of X given 1 < X < 2 (by the inversionmethod) when X ↪→ E(3)


introduction to monte carlo methods (lecture 2)christophe.chorro.fr/docs/mc2.pdf · when x is...

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