MCMC and likelihood-free methods

MCMC and Likelihood-free Methods
Christian P. Robert
Universit´e Paris-Dauphine & CREST
http://www.ceremade.dauphine.fr/~xian
November 2, 2010

Outline
Computational issues in Bayesian statistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Population Monte Carlo
Approximate Bayesian computation
ABC for model choice

Motivation and leading example
The Gibbs Sampler

Latent variables
Latent structures make life harder!
Even simple models may lead to computational complications, as
in latent variable models
f(x|θ) = f (x, x |θ) dx

Latent variables
f(x|θ) = f (x, x |θ) dx
If (x, x ) observed, ﬁne!

Latent variables
f(x|θ) = f (x, x |θ) dx
If (x, x ) observed, ﬁne!
If only x observed, trouble!

Latent variables
Example (Mixture models)
Models of mixtures of distributions:
X ∼ fj with probability pj,
for j = 1, 2, . . . , k, with overall density
X ∼ p1f1(x) + · · · + pkfk(x) .

Latent variables
X ∼ p1f1(x) + · · · + pkfk(x) .
For a sample of independent random variables (X1, · · · , Xn),
sample density
n
i=1
{p1f1(xi) + · · · + pkfk(xi)} .

Latent variables
X ∼ p1f1(x) + · · · + pkfk(x) .
For a sample of independent random variables (X1, · · · , Xn),
sample density
n
i=1
{p1f1(xi) + · · · + pkfk(xi)} .
Expanding this product of sums into a sum of products involves kn
elementary terms: too prohibitive to compute in large samples.

Latent variables
Simple mixture (1)
−1 0 1 2 3
−10123
µ1
µ2
Case of the 0.3N (µ1, 1) + 0.7N (µ2, 1) likelihood

Latent variables
Simple mixture (2)
For the mixture of two normal distributions,
0.3N(µ1, 1) + 0.7N(µ2, 1) ,
likelihood proportional to
n
i=1
[0.3ϕ (xi − µ1) + 0.7 ϕ (xi − µ2)]
containing 2n terms.

Latent variables
Complex maximisation
Standard maximization techniques often fail to ﬁnd the global
maximum because of multimodality or undesirable behavior
(usually at the frontier of the domain) of the likelihood function.
Example
In the special case
f(x|µ, σ) = (1 − ) exp{(−1/2)x2
} +
σ
exp{(−1/2σ2
)(x − µ)2
}
with > 0 known,

Latent variables
Complex maximisation
Standard maximization techniques often fail to ﬁnd the global
maximum because of multimodality or undesirable behavior
(usually at the frontier of the domain) of the likelihood function.
Example
In the special case
f(x|µ, σ) = (1 − ) exp{(−1/2)x2
} +
σ
exp{(−1/2σ2
)(x − µ)2
}
with > 0 known, whatever n, the likelihood is unbounded:
lim
σ→0
L(x1, . . . , xn|µ = x1, σ) = ∞

Latent variables
Unbounded likelihood
−2 0 2 4 6
1234
µ
n= 3
−2 0 2 4 6
1234
µ
σ
n= 6
−2 0 2 4 6
1234
µ
n= 12
−2 0 2 4 6
1234 µ
σ
n= 24
−2 0 2 4 6
1234
µ
n= 48
−2 0 2 4 6
1234
µ
σ
n= 96
Case of the 0.3N (0, 1) + 0.7N (µ, σ) likelihood

Latent variables
Example (Mixture once again)
press for MA Observations from
x1, . . . , xn ∼ f(x|θ) = pϕ(x; µ1, σ1) + (1 − p)ϕ(x; µ2, σ2)

Latent variables
Prior
µi|σi ∼ N (ξi, σ2
i /ni), σ2
i ∼ I G (νi/2, s2
i /2), p ∼ Be(α, β)

Latent variables
Prior
µi|σi ∼ N (ξi, σ2
i /ni), σ2
i ∼ I G (νi/2, s2
i /2), p ∼ Be(α, β)
Posterior
π(θ|x1, . . . , xn) ∝
n
j=1
{pϕ(xj; µ1, σ1) + (1 − p)ϕ(xj; µ2, σ2)} π(θ)
=
n
=0 (kt)
ω(kt)π(θ|(kt))
n

Latent variables
Example (Mixture once again (cont’d))
For a given permutation (kt), conditional posterior distribution
π(θ|(kt)) = N ξ1(kt),
σ2
1
n1 +
× I G ((ν1 + )/2, s1(kt)/2)
×N ξ2(kt),
σ2
2
n2 + n −
× I G ((ν2 + n − )/2, s2(kt)/2)
×Be(α + , β + n − )

Latent variables
Example (Mixture once again (cont’d))
where
¯x1(kt) = 1
t=1 xkt , ˆs1(kt) = t=1(xkt − ¯x1(kt))2,
¯x2(kt) = 1
n−
n
t= +1 xkt , ˆs2(kt) = n
t= +1(xkt − ¯x2(kt))2
and
ξ1(kt) =
n1ξ1 + ¯x1(kt)
n1 +
, ξ2(kt) =
n2ξ2 + (n − )¯x2(kt)
n2 + n −
,
s1(kt) = s2
1 + ˆs2
1(kt) +
n1
n1 +
(ξ1 − ¯x1(kt))2
,
s2(kt) = s2
2 + ˆs2
2(kt) +
n2(n − )
n2 + n −
(ξ2 − ¯x2(kt))2
,
posterior updates of the hyperparameters

Latent variables
Bayes estimator of θ:
δπ
(x1, . . . , xn) =
n
=0 (kt)
ω(kt)Eπ
[θ|x, (kt)]
Too costly: 2n terms

The AR(p) model
AR(p) model
Auto-regressive representation of a time series,
xt|xt−1, . . . ∼ N µ +
p
i=1
i(xt−i − µ), σ2

The AR(p) model
AR(p) model
Auto-regressive representation of a time series,
xt|xt−1, . . . ∼ N µ +
p
i=1
i(xt−i − µ), σ2
Generalisation of AR(1)
Among the most commonly used models in dynamic settings
More challenging than the static models (stationarity
constraints)
Diﬀerent models depending on the processing of the starting
value x0

The AR(p) model
Unknown stationarity constraints
Practical diﬃculty: for complex models, stationarity constraints get
quite involved to the point of being unknown in some cases
Example (AR(1))
Case of linear Markovian dependence on the last value
xt = µ + (xt−1 − µ) + t , t
i.i.d.
∼ N (0, σ2
)
If | | < 1, (xt)t∈Z can be written as
xt = µ +
∞
j=0
j
t−j
and this is a stationary representation.

The AR(p) model
Stationary but...
If | | > 1, alternative stationary representation
xt = µ −
∞
j=1
−j
t+j .

The AR(p) model
Stationary but...
If | | > 1, alternative stationary representation
xt = µ −
∞
j=1
−j
t+j .
This stationary solution is criticized as artiﬁcial because xt is
correlated with future white noises ( t)s>t, unlike the case when
| | < 1.
Non-causal representation...

The AR(p) model
Stationarity+causality
Stationarity constraints in the prior as a restriction on the values of
θ.
Theorem
AR(p) model second-order stationary and causal iﬀ the roots of the
polynomial
P(x) = 1 −
p
i=1
ixi
are all outside the unit circle

The AR(p) model
Stationarity constraints
Under stationarity constraints, complex parameter space: each
value of needs to be checked for roots of corresponding
polynomial with modulus less than 1

The AR(p) model
Stationarity constraints
Under stationarity constraints, complex parameter space: each
value of needs to be checked for roots of corresponding
polynomial with modulus less than 1
E.g., for an AR(2) process with
autoregressive polynomial
P(u) = 1 − 1u − 2u2, constraint is
1 + 2 < 1, 1 − 2 < 1
and | 2| < 1
q
−2 −1 0 1 2
−1.0−0.50.00.51.0 θ1
θ2

The MA(q) model
The MA(q) model
Alternative type of time series
xt = µ + t −
q
j=1
ϑj t−j , t ∼ N (0, σ2
)
Stationary but, for identiﬁability considerations, the polynomial
Q(x) = 1 −
q
j=1
ϑjxj
must have all its roots outside the unit circle.

The MA(q) model
Identiﬁability
Example
For the MA(1) model, xt = µ + t − ϑ1 t−1,
var(xt) = (1 + ϑ2
1)σ2
can also be written
xt = µ + ˜t−1 −
1
ϑ1
˜t, ˜ ∼ N (0, ϑ2
1σ2
) ,
Both pairs (ϑ1, σ) & (1/ϑ1, ϑ1σ) lead to alternative
representations of the same model.

The MA(q) model
Properties of MA models
Non-Markovian model (but special case of hidden Markov)
Autocovariance γx(s) is null for |s| > q

The MA(q) model
Representations
x1:T is a normal random variable with constant mean µ and
covariance matrix
Σ =





σ2
γ1 γ2 . . . γq 0 . . . 0 0
γ1 σ2
γ1 . . . γq−1 γq . . . 0 0
...
0 0 0 . . . 0 0 . . . γ1 σ2





,
with (|s| ≤ q)
γs = σ2
q−|s|
i=0
ϑiϑi+|s|
Not manageable in practice [large T’s]

The MA(q) model
Representations (contd.)
Conditional on past ( 0, . . . , −q+1),
L(µ, ϑ1, . . . , ϑq, σ|x1:T , 0, . . . , −q+1) ∝
σ−T
T
t=1
exp



−

xt − µ +
q
j=1
ϑjˆt−j


2
2σ2



,
where (t > 0)
ˆt = xt − µ +
q
j=1
ϑjˆt−j, ˆ0 = 0, . . . , ˆ1−q = 1−q
Recursive deﬁnition of the likelihood, still costly O(T × q)

The MA(q) model
Encompassing approach for general time series models
State-space representation
xt = Gyt + εt , (1)
yt+1 = Fyt + ξt , (2)
(1) is the observation equation and (2) is the state equation

The MA(q) model
Encompassing approach for general time series models
State-space representation
xt = Gyt + εt , (1)
yt+1 = Fyt + ξt , (2)
(1) is the observation equation and (2) is the state equation
Note
This is a special case of hidden Markov model

The MA(q) model
MA(q) state-space representation
For the MA(q) model, take
yt = ( t−q, . . . , t−1, t)
and then
yt+1 =






0 1 0 . . . 0
0 0 1 . . . 0
. . .
0 0 0 . . . 1
0 0 0 . . . 0






yt + t+1







0
0
...
0
1







xt = µ − ϑq ϑq−1 . . . ϑ1 −1 yt .

The MA(q) model
MA(q) state-space representation (cont’d)
Example
For the MA(1) model, observation equation
xt = (1 0)yt
with
yt = (y1t y2t)
directed by the state equation
yt+1 =
0 1
0 0
yt + t+1
1
ϑ1
.

The MA(q) model
c A typology of Bayes computational problems
(i). latent variable models in general

The MA(q) model
(ii). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;

The MA(q) model
(iii). use of a complex sampling model with an intractable
likelihood, as for instance in some graphical models;

The MA(q) model
(iv). use of a huge dataset;

The MA(q) model
(v). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);

The MA(q) model
(v). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);
(vi). use of a particular inferential procedure as for instance, Bayes
factors
Bπ
01(x) =
P(θ ∈ Θ0 | x)
P(θ ∈ Θ1 | x)
π(θ ∈ Θ0)
π(θ ∈ Θ1)
.

Monte Carlo basics
Importance Sampling
Monte Carlo Methods based on Markov Chains
The Metropolis–Hastings algorithm
Random-walk Metropolis-Hastings algorithms
Extensions
The Gibbs Sampler

Monte Carlo basics
General purpose
Given a density π known up to a normalizing constant, and an
integrable function h, compute
Π(h) = h(x)π(x)µ(dx) =
h(x)˜π(x)µ(dx)
˜π(x)µ(dx)
when h(x)˜π(x)µ(dx) is intractable.

Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1, . . . , xN from π and estimate Π(h) by
ˆΠMC
N (h) = N−1
N
i=1
h(xi).
LLN: ˆΠMC
N (h)
as
−→ Π(h)
If Π(h2) = h2(x)π(x)µ(dx) < ∞,
CLT:
√
N ˆΠMC
N (h) − Π(h)
L
N 0, Π [h − Π(h)]2
.

Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1, . . . , xN from π and estimate Π(h) by
ˆΠMC
N (h) = N−1
N
i=1
h(xi).
LLN: ˆΠMC
N (h)
as
−→ Π(h)
If Π(h2) = h2(x)π(x)µ(dx) < ∞,
CLT:
√
N ˆΠMC
N (h) − Π(h)
L
N 0, Π [h − Π(h)]2
.
Caveat announcing MCMC
Often impossible or ineﬃcient to simulate directly from Π

Importance Sampling
Importance Sampling
For Q proposal distribution such that Q(dx) = q(x)µ(dx),
alternative representation
Π(h) = h(x){π/q}(x)q(x)µ(dx).

Importance Sampling
Importance Sampling
For Q proposal distribution such that Q(dx) = q(x)µ(dx),
alternative representation
Π(h) = h(x){π/q}(x)q(x)µ(dx).
Principle of importance
Generate an iid sample x1, . . . , xN ∼ Q and estimate Π(h) by
ˆΠIS
Q,N (h) = N−1
N
i=1
h(xi){π/q}(xi).
return to pMC

Importance Sampling
Properties of importance
Then
LLN: ˆΠIS
Q,N (h)
as
−→ Π(h) and if Q((hπ/q)2) < ∞,
CLT:
√
N(ˆΠIS
Q,N (h) − Π(h))
L
N 0, Q{(hπ/q − Π(h))2
} .

Importance Sampling
Properties of importance
Then
LLN: ˆΠIS
Q,N (h)
as
−→ Π(h) and if Q((hπ/q)2) < ∞,
CLT:
√
N(ˆΠIS
Q,N (h) − Π(h))
L
N 0, Q{(hπ/q − Π(h))2
} .
Caveat
If normalizing constant of π unknown, impossible to use ˆΠIS
Q,N
Generic problem in Bayesian Statistics: π(θ|x) ∝ f(x|θ)π(θ).

Importance Sampling
Self-Normalised Importance Sampling
Self normalized version
ˆΠSNIS
Q,N (h) =
N
i=1
{π/q}(xi)
−1 N
i=1
h(xi){π/q}(xi).

Importance Sampling
ˆΠSNIS
Q,N (h) =
N
i=1
{π/q}(xi)
−1 N
i=1
h(xi){π/q}(xi).
LLN : ˆΠSNIS
Q,N (h)
as
−→ Π(h)
and if Π((1 + h2)(π/q)) < ∞,
CLT :
√
N(ˆΠSNIS
Q,N (h) − Π(h))
L
N 0, π {(π/q)(h − Π(h)}2
) .

Importance Sampling
ˆΠSNIS
Q,N (h) =
N
i=1
{π/q}(xi)
−1 N
i=1
h(xi){π/q}(xi).
LLN : ˆΠSNIS
Q,N (h)
as
−→ Π(h)
and if Π((1 + h2)(π/q)) < ∞,
CLT :
√
N(ˆΠSNIS
Q,N (h) − Π(h))
L
N 0, π {(π/q)(h − Π(h)}2
) .
c The quality of the SNIS approximation depends on the
choice of Q

Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f to
approximate the integral
I = h(x)f(x)dx ,

Running Monte Carlo via Markov Chains (MCMC)
It is not necessary to use a sample from the distribution f to
approximate the integral
I = h(x)f(x)dx ,
We can obtain X1, . . . , Xn ∼ f (approx) without directly
simulating from f, using an ergodic Markov chain with
stationary distribution f

Running Monte Carlo via Markov Chains (2)
Idea
For an arbitrary starting value x(0), an ergodic chain (X(t)) is
generated using a transition kernel with stationary distribution f

Idea
Insures the convergence in distribution of (X(t)) to a random
variable from f.
For a “large enough” T0, X(T0) can be considered as
distributed from f
Produce a dependent sample X(T0), X(T0+1), . . ., which is
generated from f, suﬃcient for most approximation purposes.

Idea
Insures the convergence in distribution of (X(t)) to a random
variable from f.
For a “large enough” T0, X(T0) can be considered as
distributed from f
Produce a dependent sample X(T0), X(T0+1), . . ., which is
generated from f, suﬃcient for most approximation purposes.
Problem: How can one build a Markov chain with a given
stationary distribution?

Basics
The algorithm uses the objective (target) density
f
and a conditional density
q(y|x)
called the instrumental (or proposal) distribution

The MH algorithm
Algorithm (Metropolis–Hastings)
Given x(t),
1. Generate Yt ∼ q(y|x(t)).
2. Take
X(t+1)
=
Yt with prob. ρ(x(t), Yt),
x(t) with prob. 1 − ρ(x(t), Yt),
where
ρ(x, y) = min
f(y)
f(x)
q(x|y)
q(y|x)
, 1 .

Features
Independent of normalizing constants for both f and q(·|x)
(ie, those constants independent of x)
Never move to values with f(y) = 0
The chain (x(t))t may take the same value several times in a
row, even though f is a density wrt Lebesgue measure
The sequence (yt)t is usually not a Markov chain

Convergence properties
1. The M-H Markov chain is reversible, with
invariant/stationary density f since it satisﬁes the detailed
balance condition
f(y) K(y, x) = f(x) K(x, y)

balance condition
f(y) K(y, x) = f(x) K(x, y)
2. As f is a probability measure, the chain is positive recurrent

balance condition
f(y) K(y, x) = f(x) K(x, y)
2. As f is a probability measure, the chain is positive recurrent
3. If
Pr
f(Yt) q(X(t)|Yt)
f(X(t)) q(Yt|X(t))
≥ 1 < 1. (1)
that is, the event {X(t+1) = X(t)} is possible, then the chain
is aperiodic

Convergence properties (2)
4. If
q(y|x) > 0 for every (x, y), (2)
the chain is irreducible

4. If
5. For M-H, f-irreducibility implies Harris recurrence

4. If
5. For M-H, f-irreducibility implies Harris recurrence
6. Thus, for M-H satisfying (1) and (2)
(i) For h, with Ef |h(X)| < ∞,
lim
T →∞
1
T
T
t=1
h(X(t)
) = h(x)df(x) a.e. f.
(ii) and
lim
n→∞
Kn
(x, ·)µ(dx) − f
T V
= 0
for every initial distribution µ, where Kn
(x, ·) denotes the
kernel for n transitions.

Random walk Metropolis–Hastings
Use of a local perturbation as proposal
Yt = X(t)
+ εt,
where εt ∼ g, independent of X(t).
The instrumental density is of the form g(y − x) and the Markov
chain is a random walk if we take g to be symmetric g(x) = g(−x)

Algorithm (Random walk Metropolis)
Given x(t)
1. Generate Yt ∼ g(y − x(t))
2. Take
X(t+1)
=



Yt with prob. min 1,
f(Yt)
f(x(t))
,
x(t) otherwise.

Example (Random walk and normal target)
forget History! Generate N(0, 1) based on the uniform proposal [−δ, δ]
[Hastings (1970)]
The probability of acceptance is then
ρ(x(t)
, yt) = exp{(x(t)2
− y2
t )/2} ∧ 1.

Example (Random walk & normal (2))
Sample statistics
δ 0.1 0.5 1.0
mean 0.399 -0.111 0.10
variance 0.698 1.11 1.06
c As δ ↑, we get better histograms and a faster exploration of the
support of f.

-1 0 1 2
050100150200250
(a)
-1.5-1.0-0.50.00.5
-2 0 2
0100200300400
(b) -1.5-1.0-0.50.00.5
-3 -2 -1 0 1 2 3
0100200300400
(c)
-1.5-1.0-0.50.00.5
Three samples based on U[−δ, δ] with (a) δ = 0.1, (b) δ = 0.5
and (c) δ = 1.0, superimposed with the convergence of the
means (15, 000 simulations).

π(θ|x) ∝
n
j=1
k
=1
p f(xj|µ , σ ) π(θ)

π(θ|x) ∝
n
j=1
k
=1
p f(xj|µ , σ ) π(θ)
Metropolis-Hastings proposal:
θ(t+1)
=
θ(t) + ωε(t) if u(t) < ρ(t)
θ(t) otherwise
where
ρ(t)
=
π(θ(t) + ωε(t)|x)
π(θ(t)|x)
∧ 1
and ω scaled for good acceptance rate

p
theta
0.0 0.2 0.4 0.6 0.8 1.0
-1012
tau
theta
0.2 0.4 0.6 0.8 1.0 1.2
-1012
p
tau
0.0 0.2 0.4 0.6 0.8 1.0
0.20.40.60.81.01.2
-1 0 1 2
0.01.02.0
theta
0.2 0.4 0.6 0.8
024
tau
0.0 0.2 0.4 0.6 0.8 1.0
0123456
p
Random walk sampling (50000 iterations)
General case of a 3 component normal mixture
[Celeux & al., 2000]

−1 0 1 2 3
−10123
µ1
µ2
X
Random walk MCMC output for .7N(µ1, 1) + .3N(µ2, 1)

Uniform ergodicity prohibited by random walk structure

Uniform ergodicity prohibited by random walk structure
At best, geometric ergodicity:
Theorem (Suﬃcient ergodicity)
For a symmetric density f, log-concave in the tails, and a positive
and symmetric density g, the chain (X(t)) is geometrically ergodic.
[Mengersen & Tweedie, 1996]
no tail eﬀect

Example (Comparison of tail
eﬀects)
Random-walk
Metropolis–Hastings algorithms
based on a N (0, 1) instrumental
for the generation of (a) a
N(0, 1) distribution and (b) a
distribution with density
ψ(x) ∝ (1 + |x|)−3
(a)
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
(a)
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
(b)
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
0 50 100 150 200
-1.5-1.0-0.50.00.51.01.5
90% conﬁdence envelopes of
the means, derived from 500
parallel independent chains
1 + ξ2
1 + (ξ )2
∧ 1 ,

Further convergence properties
Under assumptions skip detailed convergence
(A1) f is super-exponential, i.e. it is positive with positive
continuous ﬁrst derivative such that
lim|x|→∞ n(x) log f(x) = −∞ where n(x) := x/|x|.
In words : exponential decay of f in every direction with rate
tending to ∞
(A2) lim sup|x|→∞ n(x) m(x) < 0, where
m(x) = f(x)/| f(x)|.
In words: non degeneracy of the countour manifold
Cf(y) = {y : f(y) = f(x)}
Q is geometrically ergodic, and
V (x) ∝ f(x)−1/2 veriﬁes the drift condition
[Jarner & Hansen, 2000]

Further [further] convergence properties
skip hyperdetailed convergence
If P ψ-irreducible and aperiodic, for r = (r(n))n∈N real-valued non
decreasing sequence, such that, for all n, m ∈ N,
r(n + m) ≤ r(n)r(m),
and r(0) = 1, for C a small set, τC = inf{n ≥ 1, Xn ∈ C}, and
h ≥ 1, assume
sup
x∈C
Ex
τC −1
k=0
r(k)h(Xk) < ∞,

then,
S(f, C, r) := x ∈ X, Ex
τC −1
k=0
r(k)h(Xk) < ∞
is full and absorbing and for x ∈ S(f, C, r),
lim
n→∞
r(n) Pn
(x, .) − f h = 0.
[Tuominen & Tweedie, 1994]

Comments
[CLT, Rosenthal’s inequality...] h-ergodicity implies CLT
for additive (possibly unbounded) functionals of the chain,
Rosenthal’s inequality and so on...
[Control of the moments of the return-time] The
condition implies (because h ≥ 1) that
sup
x∈C
Ex[r0(τC)] ≤ sup
x∈C
Ex
τC −1
k=0
r(k)h(Xk) < ∞,
where r0(n) = n
l=0 r(l) Can be used to derive bounds for
the coupling time, an essential step to determine computable
bounds, using coupling inequalities
[Roberts & Tweedie, 1998; Fort & Moulines, 2000]

Alternative conditions
The condition is not really easy to work with...
[Possible alternative conditions]
(a) [Tuominen, Tweedie, 1994] There exists a sequence
(Vn)n∈N, Vn ≥ r(n)h, such that
(i) supC V0 < ∞,
(ii) {V0 = ∞} ⊂ {V1 = ∞} and
(iii) PVn+1 ≤ Vn − r(n)h + br(n)IC.

(b) [Fort 2000] ∃V ≥ f ≥ 1 and b < ∞, such that supC V < ∞
and
PV (x) + Ex
σC
k=0
∆r(k)f(Xk) ≤ V (x) + bIC(x)
where σC is the hitting time on C and
∆r(k) = r(k) − r(k − 1), k ≥ 1 and ∆r(0) = r(0).
Result (a) ⇔ (b) ⇔ supx∈C Ex
τC −1
k=0 r(k)f(Xk) < ∞.

Extensions
Langevin Algorithms
Proposal based on the Langevin diffusion Lt is defined by the
stochastic differential equation
dLt = dBt +
1
2
log f(Lt)dt,
where Bt is the standard Brownian motion
Theorem
The Langevin diffusion is the only non-explosive diffusion which is
reversible with respect to f.

Extensions
Discretization
Instead, consider the sequence
x(t+1)
= x(t)
+
σ2
2
log f(x(t)
) + σεt, εt ∼ Np(0, Ip)
where σ2 corresponds to the discretization step

Extensions
Discretization
Instead, consider the sequence
x(t+1)
= x(t)
+
σ2
2
log f(x(t)
) + σεt, εt ∼ Np(0, Ip)
where σ2 corresponds to the discretization step
Unfortunately, the discretized chain may be be transient, for
instance when
lim
x→±∞
σ2
log f(x)|x|−1
> 1

Extensions
MH correction
Accept the new value Yt with probability
f(Yt)
f(x(t))
·
exp − Yt − x(t) − σ2
2 log f(x(t))
2
2σ2
exp − x(t) − Yt − σ2
2 log f(Yt)
2
2σ2
∧ 1 .
Choice of the scaling factor σ
Should lead to an acceptance rate of 0.574 to achieve optimal
convergence rates (when the components of x are uncorrelated)
[Roberts & Rosenthal, 1998; Girolami & Calderhead, 2011]

Extensions
Optimizing the Acceptance Rate
Problem of choice of the transition kernel from a practical point of
view
Most common alternatives:
(a) a fully automated algorithm like ARMS;
(b) an instrumental density g which approximates f, such that
f/g is bounded for uniform ergodicity to apply;
(c) a random walk
In both cases (b) and (c), the choice of g is critical,

Extensions
Case of the random walk
Diﬀerent approach to acceptance rates
A high acceptance rate does not indicate that the algorithm is
moving correctly since it indicates that the random walk is moving
too slowly on the surface of f.

Extensions
Case of the random walk
Diﬀerent approach to acceptance rates
A high acceptance rate does not indicate that the algorithm is
moving correctly since it indicates that the random walk is moving
too slowly on the surface of f.
If x(t) and yt are close, i.e. f(x(t)) f(yt) y is accepted with
probability
min
f(yt)
f(x(t))
, 1 1 .
For multimodal densities with well separated modes, the negative
eﬀect of limited moves on the surface of f clearly shows.

Extensions
Case of the random walk (2)
If the average acceptance rate is low, the successive values of f(yt)
tend to be small compared with f(x(t)), which means that the
random walk moves quickly on the surface of f since it often
reaches the “borders” of the support of f

Extensions
Rule of thumb
In small dimensions, aim at an average acceptance rate of 50%. In
large dimensions, at an average acceptance rate of 25%.
[Gelman,Gilks and Roberts, 1995]

Extensions
Rule of thumb
In small dimensions, aim at an average acceptance rate of 50%. In
large dimensions, at an average acceptance rate of 25%.
[Gelman,Gilks and Roberts, 1995]
This rule is to be taken with a pinch of salt!

Extensions
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + t+1 t ∼ N (0, τ2
)
and observables
yt|xt ∼ N (x2
t , σ2
)

Extensions
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + t+1 t ∼ N (0, τ2
)
and observables
yt|xt ∼ N (x2
t , σ2
)
The distribution of xt given xt−1, xt+1 and yt is
exp
−1
2τ2
(xt − ϕxt−1)2
+ (xt+1 − ϕxt)2
+
τ2
σ2
(yt − x2
t )2
.

Extensions
Example (Noisy AR(1) continued)
For a Gaussian random walk with scale ω small enough, the
random walk never jumps to the other mode. But if the scale ω is
suﬃciently large, the Markov chain explores both modes and give a
satisfactory approximation of the target distribution.

Extensions
Markov chain based on a random walk with scale ω = .1.

Extensions
Markov chain based on a random walk with scale ω = .5.

Extensions
MA(2)
Since the constraints on (ϑ1, ϑ2) are well-deﬁned, use of a ﬂat
prior over the triangle as prior.
Simple representation of the likelihood
library(mnormt)
ma2like=function(theta){
n=length(y)
sigma = toeplitz(c(1 +theta[1]^2+theta[2]^2,
theta[1]+theta[1]*theta[2],theta[2],rep(0,n-3)))
dmnorm(y,rep(0,n),sigma,log=TRUE)
}

Extensions
Basic RWHM for MA(2)
Algorithm 1 RW-HM-MA(2) sampler
set ω and ϑ(1)
for i = 2 to T do
generate ˜ϑj ∼ U(ϑ
(i−1)
j − ω, ϑ
(i−1)
j + ω)
set p = 0 and ϑ(i) = ϑ(i−1)
if ˜ϑ within the triangle then
p = exp(ma2like(˜ϑ) − ma2like(ϑ(i−1)))
end if
if U < p then
ϑ(i) = ˜ϑ
end if
end for

Extensions
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,
ϑ2 = 0.2 and scale ω = 0.2
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qqq
q

Extensions
Outcome
ϑ2 = 0.2 and scale ω = 0.5
q q
−2 −1 0 1 2
−1.0−0.50.00.51.0
θ1
θ2
qqqqqqq
qqqq
qqqqq
qq
qqq
q
q
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qq
qq
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q
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qqqq
qqqqqqqqqq
q
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qqqqqq
qqqq
q
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qqq
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qqqqqq
qqqqqqqqqqqqqq
qqq
q
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qqqq
qqqqq
qq
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qqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qq
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qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqq
q
qqqqqqqqq
qqqqqq
q
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q
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qqq
q qq
qq
q
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q
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qq qqqqqqqqqqqqq
qq
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qqqq
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qqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqq
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q
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q
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q
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q
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q
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q
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qq
q
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qqqqqqqq qqqqqq
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q
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
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qq
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q
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q
qqqq qqqqq
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qqqqqqq
q
qqq
qqqqqqqqqqqqqqqqq
q
q
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qqqq
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qqqq
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qqqq
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qq
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qqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqq
q
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q
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q
qq
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qq
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qq
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q
q
qq
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q
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qqqqqq
qq
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q
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qqqqqqq
q

Extensions
Outcome
ϑ2 = 0.2 and scale ω = 2.0
q q
−2 −1 0 1 2
−1.0−0.50.00.51.0
θ1
θ2
qqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

The Gibbs Sampler
The Gibbs Sampler
skip to population Monte Carlo
The Gibbs Sampler
General Principles
Completion
Convergence
The Hammersley-Cliﬀord theorem

The Gibbs Sampler
General Principles
General Principles
A very speciﬁc simulation algorithm based on the target
distribution f:
1. Uses the conditional densities f1, . . . , fp from f

The Gibbs Sampler
General Principles
General Principles
distribution f:
2. Start with the random variable X = (X1, . . . , Xp)

The Gibbs Sampler
General Principles
General Principles
distribution f:
2. Start with the random variable X = (X1, . . . , Xp)
3. Simulate from the conditional densities,
Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp
∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)
for i = 1, 2, . . . , p.

The Gibbs Sampler
General Principles
Algorithm (Gibbs sampler)
Given x(t) = (x
(t)
1 , . . . , x
(t)
p ), generate
1. X
(t+1)
1 ∼ f1(x1|x
(t)
2 , . . . , x
(t)
p );
2. X
(t+1)
2 ∼ f2(x2|x
(t+1)
1 , x
(t)
3 , . . . , x
(t)
p ),
. . .
p. X
(t+1)
p ∼ fp(xp|x
(t+1)
1 , . . . , x
(t+1)
p−1 )
X(t+1) → X ∼ f

The Gibbs Sampler
General Principles
Properties
The full conditionals densities f1, . . . , fp are the only densities used
for simulation. Thus, even in a high dimensional problem, all of
the simulations may be univariate

The Gibbs Sampler
General Principles
Properties
The full conditionals densities f1, . . . , fp are the only densities used
for simulation. Thus, even in a high dimensional problem, all of
the simulations may be univariate
The Gibbs sampler is not reversible with respect to f. However,
each of its p components is. Besides, it can be turned into a
reversible sampler, either using the Random Scan Gibbs sampler
see section or running instead the (double) sequence
f1 · · · fp−1fpfp−1 · · · f1

The Gibbs Sampler
General Principles
A Very Simple Example: Independent N(µ, σ2
)
Observations
When Y1, . . . , Yn
iid
∼ N(y|µ, σ2) with both µ and σ unknown, the
posterior in (µ, σ2) is conjugate outside a standard familly

The Gibbs Sampler
General Principles
A Very Simple Example: Independent N(µ, σ2
)
Observations
When Y1, . . . , Yn
iid
∼ N(y|µ, σ2) with both µ and σ unknown, the
posterior in (µ, σ2) is conjugate outside a standard familly
But...
µ|Y 0:n, σ2 ∼ N µ 1
n
n
i=1 Yi, σ2
n )
σ2|Y 1:n, µ ∼ IG σ2 n
2 − 1, 1
2
n
i=1(Yi − µ)2
assuming constant (improper) priors on both µ and σ2
Hence we may use the Gibbs sampler for simulating from the
posterior of (µ, σ2)

The Gibbs Sampler
General Principles
R Gibbs Sampler for Gaussian posterior
n = length(Y);
S = sum(Y);
mu = S/n;
for (i in 1:500)
S2 = sum((Y-mu)^2);
sigma2 = 1/rgamma(1,n/2-1,S2/2);
mu = S/n + sqrt(sigma2/n)*rnorm(1);

The Gibbs Sampler
General Principles
Example of results with n = 10 observations from the
N(0, 1) distribution
Number of Iterations 1

The Gibbs Sampler
General Principles
Number of Iterations 1, 2

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5, 10

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5, 10, 25

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100

The Gibbs Sampler
General Principles
Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100, 500

The Gibbs Sampler
General Principles
Limitations of the Gibbs sampler
Formally, a special case of a sequence of 1-D M-H kernels, all with
acceptance rate uniformly equal to 1.
The Gibbs sampler
1. limits the choice of instrumental distributions

The Gibbs Sampler
General Principles
The Gibbs sampler
2. requires some knowledge of f

The Gibbs Sampler
General Principles
The Gibbs sampler
3. is, by construction, multidimensional

The Gibbs Sampler
General Principles
The Gibbs sampler
3. is, by construction, multidimensional
4. does not apply to problems where the number of parameters
varies as the resulting chain is not irreducible.

The Gibbs Sampler
Completion
Latent variables are back
The Gibbs sampler can be generalized in much wider generality
A density g is a completion of f if
Z
g(x, z) dz = f(x)

The Gibbs Sampler
Completion
Latent variables are back
The Gibbs sampler can be generalized in much wider generality
A density g is a completion of f if
Z
g(x, z) dz = f(x)
Note
The variable z may be meaningless for the problem

The Gibbs Sampler
Completion
Purpose
g should have full conditionals that are easy to simulate for a
Gibbs sampler to be implemented with g rather than f
For p > 1, write y = (x, z) and denote the conditional densities of
g(y) = g(y1, . . . , yp) by
Y1|y2, . . . , yp ∼ g1(y1|y2, . . . , yp),
Y2|y1, y3, . . . , yp ∼ g2(y2|y1, y3, . . . , yp),
. . . ,
Yp|y1, . . . , yp−1 ∼ gp(yp|y1, . . . , yp−1).

The Gibbs Sampler
Completion
The move from Y (t) to Y (t+1) is deﬁned as follows:
Algorithm (Completion Gibbs sampler)
Given (y
(t)
1 , . . . , y
(t)
p ), simulate
1. Y
(t+1)
1 ∼ g1(y1|y
(t)
2 , . . . , y
(t)
p ),
2. Y
(t+1)
2 ∼ g2(y2|y
(t+1)
1 , y
(t)
3 , . . . , y
(t)
p ),
. . .
p. Y
(t+1)
p ∼ gp(yp|y
(t+1)
1 , . . . , y
(t+1)
p−1 ).

The Gibbs Sampler
Completion
Example (Mixtures all over again)
Hierarchical missing data structure:
If
X1, . . . , Xn ∼
k
i=1
pif(x|θi),
then
X|Z ∼ f(x|θZ), Z ∼ p1I(z = 1) + . . . + pkI(z = k),
Z is the component indicator associated with observation x

The Gibbs Sampler
Completion
Example (Mixtures (2))
Conditionally on (Z1, . . . , Zn) = (z1, . . . , zn) :
π(p1, . . . , pk, θ1, . . . , θk|x1, . . . , xn, z1, . . . , zn)
∝ pα1+n1−1
1 . . . pαk+nk−1
k
×π(θ1|y1 + n1¯x1, λ1 + n1) . . . π(θk|yk + nk ¯xk, λk + nk),
with
ni =
j
I(zj = i) and ¯xi =
j; zj=i
xj/ni.

The Gibbs Sampler
Completion
Algorithm (Mixture Gibbs sampler)
1. Simulate
θi ∼ π(θi|yi + ni¯xi, λi + ni) (i = 1, . . . , k)
(p1, . . . , pk) ∼ D(α1 + n1, . . . , αk + nk)
2. Simulate (j = 1, . . . , n)
Zj|xj, p1, . . . , pk, θ1, . . . , θk ∼
k
i=1
pijI(zj = i)
with (i = 1, . . . , k)
pij ∝ pif(xj|θi)
and update ni and ¯xi (i = 1, . . . , k).

The Gibbs Sampler
Completion
A wee problem
−1 0 1 2 3 4
−101234
µ1
µ2
Gibbs started at random

The Gibbs Sampler
Completion
A wee problem
−1 0 1 2 3 4
−101234
µ1
µ2
Gibbs started at random
Gibbs stuck at the wrong mode
−1 0 1 2 3
−10123
µ1
µ2

The Gibbs Sampler
Completion
Slice sampler as generic Gibbs
If f(θ) can be written as a product
k
i=1
fi(θ),

The Gibbs Sampler
Completion
Slice sampler as generic Gibbs
If f(θ) can be written as a product
k
i=1
fi(θ),
it can be completed as
k
i=1
I0≤ωi≤fi(θ),
leading to the following Gibbs algorithm:

The Gibbs Sampler
Completion
Algorithm (Slice sampler)
Simulate
1. ω
(t+1)
1 ∼ U[0,f1(θ(t))];
. . .
k. ω
(t+1)
k ∼ U[0,fk(θ(t))];
k+1. θ(t+1) ∼ UA(t+1) , with
A(t+1)
= {y; fi(y) ≥ ω
(t+1)
i , i = 1, . . . , k}.

The Gibbs Sampler
Completion
Example of results with a truncated N(−3, 1) distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3, 4

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3, 4, 5

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3, 4, 5, 10

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3, 4, 5, 10, 50

The Gibbs Sampler
Completion
0.0 0.2 0.4 0.6 0.8 1.0
0.0000.0020.0040.0060.0080.010
x
y
Number of Iterations 2, 3, 4, 5, 10, 50, 100

The Gibbs Sampler
Completion
Good slices
The slice sampler usually enjoys good theoretical properties (like
geometric ergodicity and even uniform ergodicity under bounded f
and bounded X ).
As k increases, the determination of the set A(t+1) may get
increasingly complex.

The Gibbs Sampler
Completion
Example (Stochastic volatility core distribution)
Diﬃcult part of the stochastic volatility model
π(x) ∝ exp − σ2
(x − µ)2
+ β2
exp(−x)y2
+ x /2 ,
simpliﬁed in exp − x2 + α exp(−x)

The Gibbs Sampler
Completion
Example (Stochastic volatility core distribution)
Diﬃcult part of the stochastic volatility model
π(x) ∝ exp − σ2
(x − µ)2
+ β2
exp(−x)y2
+ x /2 ,
simpliﬁed in exp − x2 + α exp(−x)
Slice sampling means simulation from a uniform distribution on
A = x; exp − x2
+ α exp(−x) /2 ≥ u
= x; x2
+ α exp(−x) ≤ ω
if we set ω = −2 log u.
Note Inversion of x2 + α exp(−x) = ω needs to be done by
trial-and-error.

The Gibbs Sampler
Completion
0 10 20 30 40 50 60 70 80 90 100
−0.1
−0.05
0
0.05
0.1
Lag
Correlation
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5
0
0.2
0.4
0.6
0.8
1
Density
Histogram of a Markov chain produced by a slice sampler
and target distribution in overlay.

The Gibbs Sampler
Convergence
Properties of the Gibbs sampler
Theorem (Convergence)
For
(Y1, Y2, · · · , Yp) ∼ g(y1, . . . , yp),
if either
[Positivity condition]
(i) g(i)(yi) > 0 for every i = 1, · · · , p, implies that
g(y1, . . . , yp) > 0, where g(i) denotes the marginal distribution
of Yi, or
(ii) the transition kernel is absolutely continuous with respect to g,
then the chain is irreducible and positive Harris recurrent.

The Gibbs Sampler
Convergence
Properties of the Gibbs sampler (2)
Consequences
(i) If h(y)g(y)dy < ∞, then
lim
nT→∞
1
T
T
t=1
h1(Y (t)
) = h(y)g(y)dy a.e. g.
(ii) If, in addition, (Y (t)) is aperiodic, then
lim
n→∞
Kn
(y, ·)µ(dx) − f
TV
= 0
for every initial distribution µ.

The Gibbs Sampler
Convergence
Slice sampler
fast on that slice
For convergence, the properties of Xt and of f(Xt) are identical
Theorem (Uniform ergodicity)
If f is bounded and suppf is bounded, the simple slice sampler is
uniformly ergodic.
[Mira & Tierney, 1997]

The Gibbs Sampler
Convergence
A small set for a slice sampler
no slice detail
For > ,
C = {x ∈ X; < f(x) < }
is a small set:
Pr(x, ·) ≥ µ(·)
where
µ(A) =
1
0
λ(A ∩ L( ))
λ(L( ))
d
if L( ) = {x ∈ X; f(x) > }‘
[Roberts & Rosenthal, 1998]

The Gibbs Sampler
Convergence
Slice sampler: drift
Under diﬀerentiability and monotonicity conditions, the slice
sampler also veriﬁes a drift condition with V (x) = f(x)−β, is
geometrically ergodic, and there even exist explicit bounds on the
total variation distance

The Gibbs Sampler
Convergence
Slice sampler: drift
Under diﬀerentiability and monotonicity conditions, the slice
sampler also veriﬁes a drift condition with V (x) = f(x)−β, is
geometrically ergodic, and there even exist explicit bounds on the
total variation distance
Example (Exponential Exp(1))
For n > 23,
||Kn
(x, ·) − f(·)||TV ≤ .054865 (0.985015)n
(n − 15.7043)

The Gibbs Sampler
Convergence
Slice sampler: convergence
no more slice detail
Theorem
For any density such that
∂
∂
λ ({x ∈ X; f(x) > }) is non-increasing
then
||K523
(x, ·) − f(·)||TV ≤ .0095

The Gibbs Sampler
Convergence
A poor slice sampler
Example
Consider
f(x) = exp {−||x||} x ∈ Rd
Slice sampler equivalent to
one-dimensional slice sampler on
π(z) = zd−1
e−z
z > 0
or on
π(u) = e−u1/d
u > 0
Poor performances when d large
(heavy tails)
0 200 400 600 800 1000
-2-101
1 dimensional run
correlation
0 10 20 30 40
0.00.20.40.60.81.0
1 dimensional acf
0 200 400 600 800 1000
1015202530
10 dimensional run
correlation
0 10 20 30 40
0.00.20.40.60.81.0
10 dimensional acf
0 200 400 600 800 1000
0204060
20 dimensional run
correlation
0 10 20 30 40
0.00.20.40.60.81.0
20 dimensional acf
0 200 400 600 800 1000
0100200300400
100 dimensional run
correlation
0 10 20 30 40
0.00.20.40.60.81.0
100 dimensional acf
Sample runs of log(u) and
ACFs for log(u) (Roberts

The Gibbs Sampler
Hammersley-Cliﬀord theorem
An illustration that conditionals determine the joint distribution
Theorem
If the joint density g(y1, y2) have conditional distributions
g1(y1|y2) and g2(y2|y1), then
g(y1, y2) =
g2(y2|y1)
g2(v|y1)/g1(y1|v) dv
.
[Hammersley & Cliﬀord, circa 1970]

The Gibbs Sampler
General HC decomposition
Under the positivity condition, the joint distribution g satisﬁes
g(y1, . . . , yp) ∝
p
j=1
g j
(y j
|y 1 , . . . , y j−1
, y j+1
, . . . , y p
)
g j
(y j
|y 1 , . . . , y j−1
, y j+1
, . . . , y p
)
for every permutation on {1, 2, . . . , p} and every y ∈ Y .

Sequential importance sampling
The Gibbs Sampler

Importance sampling (revisited)
basic importance
Approximation of integrals
I = h(x)π(x)dx
by unbiased estimators
ˆI =
1
n
n
i=1
ih(xi)
when
x1, . . . , xn
iid
∼ q(x) and i
def
=
π(xi)
q(xi)

Iterated importance sampling
As in Markov Chain Monte Carlo (MCMC) algorithms,
introduction of a temporal dimension :
x
(t)
i ∼ qt(x|x
(t−1)
i ) i = 1, . . . , n, t = 1, . . .
and
ˆIt =
1
n
n
i=1
(t)
i h(x
(t)
i )
is still unbiased for
(t)
i =
πt(x
(t)
i )
qt(x
(t)
i |x
(t−1)
i )
, i = 1, . . . , n

Fundamental importance equality
Preservation of unbiasedness
E h(X(t)
)
π(X(t))
qt(X(t)|X(t−1))
= h(x)
π(x)
qt(x|y)
qt(x|y) g(y) dx dy
= h(x) π(x) dx
for any distribution g on X(t−1)

Sequential variance decomposition
Furthermore,
var ˆIt =
1
n2
n
i=1
var
(t)
i h(x
(t)
i ) ,
if var
(t)
i exists, because the x
(t)
i ’s are conditionally uncorrelated
Note
This decomposition is still valid for correlated [in i] x
(t)
i ’s when
incorporating weights
(t)
i

Simulation of a population
The importance distribution of the sample (a.k.a. particles) x(t)
qt(x(t)
|x(t−1)
)
can depend on the previous sample x(t−1) in any possible way as
long as marginal distributions
qit(x) = qt(x(t)
) dx
(t)
−i
can be expressed to build importance weights
it =
π(x
(t)
i )
qit(x
(t)
i )

Special case of the product proposal
If
qt(x(t)
|x(t−1)
) =
n
i=1
qit(x
(t)
i |x(t−1)
)
[Independent proposals]
then
var ˆIt =
1
n2
n
i=1
var
(t)
i h(x
(t)
i ) ,

Validation
skip validation
E
(t)
i h(X
(t)
i )
(t)
j h(X
(t)
j )
= h(xi)
π(xi)
qit(xi|x(t−1))
π(xj)
qjt(xj|x(t−1))
h(xj)
qit(xi|x(t−1)
) qjt(xj|x(t−1)
) dxi dxj g(x(t−1)
)dx(t−1)
= Eπ [h(X)]2
whatever the distribution g on x(t−1)

Self-normalised version
In general, π is unscaled and the weight
(t)
i ∝
π(x
(t)
i )
qit(x
(t)
i )
, i = 1, . . . , n ,
is scaled so that
i
(t)
i = 1

Self-normalised version properties
Loss of the unbiasedness property and the variance
decomposition
Normalising constant can be estimated by
t =
1
tn
t
τ=1
n
i=1
π(x
(τ)
i )
qiτ (x
(τ)
i )
Variance decomposition (approximately) recovered if t−1 is
used instead

Sampling importance resampling
Importance sampling from g can also produce samples from the
target π
[Rubin, 1987]

target π
[Rubin, 1987]
Theorem (Bootstraped importance sampling)
If a sample (xi )1≤i≤m is derived from the weighted sample
(xi, i)1≤i≤n by multinomial sampling with weights i, then
xi ∼ π(x)

target π
[Rubin, 1987]
Theorem (Bootstraped importance sampling)
If a sample (xi )1≤i≤m is derived from the weighted sample
(xi, i)1≤i≤n by multinomial sampling with weights i, then
xi ∼ π(x)
Note
Obviously, the xi ’s are not iid

Iterated sampling importance resampling
This principle can be extended to iterated importance sampling:
After each iteration, resampling produces a sample from π
[Again, not iid!]

Iterated sampling importance resampling
This principle can be extended to iterated importance sampling:
After each iteration, resampling produces a sample from π
[Again, not iid!]
Incentive
Use previous sample(s) to learn about π and q

Generic Population Monte Carlo
Algorithm (Population Monte Carlo Algorithm)
For t = 1, . . . , T
For i = 1, . . . , n,
1. Select the generating distribution qit(·)
2. Generate ˜x
(t)
i ∼ qit(x)
3. Compute
(t)
i = π(˜x
(t)
i )/qit(˜x
(t)
i )
Normalise the
(t)
i ’s into ¯
(t)
i ’s
Generate Ji,t ∼ M((¯
(t)
i )1≤i≤N ) and set xi,t = ˜x
(t)
Ji,t

D-kernels in competition
A general adaptive construction:
Construct qi,t as a mixture of D diﬀerent transition kernels
depending on x
(t−1)
i
qi,t =
D
=1
pt, K (x
(t−1)
i , x),
D
=1
pt, = 1 ,
and adapt the weights pt, .

D-kernels in competition
A general adaptive construction:
Construct qi,t as a mixture of D diﬀerent transition kernels
depending on x
(t−1)
i
qi,t =
D
=1
pt, K (x
(t−1)
i , x),
D
=1
pt, = 1 ,
and adapt the weights pt, .
Darwinian example
Take pt, proportional to the survival rate of the points
(a.k.a. particles) x
(t)
i generated from K

Implementation
Algorithm (D-kernel PMC)
For t = 1, . . . , T
generate (Ki,t)1≤i≤N ∼ M ((pt,k)1≤k≤D)
for 1 ≤ i ≤ N, generate
˜xi,t ∼ KKi,t (x)
compute and renormalize the importance weights ωi,t
generate (Ji,t)1≤i≤N ∼ M ((ωi,t)1≤i≤N )
take xi,t = ˜xJi,t,t and pt+1,d = N
i=1 ¯ωi,tId(Ki,t)

Links with particle ﬁlters
Sequential setting where π = πt changes with t: Population
Monte Carlo also adapts to this case
Can be traced back all the way to Hammersley and Morton
(1954) and the self-avoiding random walk problem
Gilks and Berzuini (2001) produce iterated samples with (SIR)
resampling steps, and add an MCMC step: this step must use
a πt invariant kernel
Chopin (2001) uses iterated importance sampling to handle
large datasets: this is a special case of PMC where the qit’s
are the posterior distributions associated with a portion kt of
the observed dataset

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