The document discusses computational methods for Bayesian statistics when direct simulation from the target distribution is not possible or efficient. It introduces Markov chain Monte Carlo (MCMC) methods, including the Metropolis-Hastings algorithm and Gibbs sampler, which generate dependent samples that approximate the target distribution. The Metropolis-Hastings algorithm uses a proposal distribution to randomly walk through the parameter space. Approximate Bayesian computation (ABC) is also introduced as a method that approximates the posterior distribution when the likelihood is intractable.
1. From MCMC to ABC Methods
From MCMC to ABC Methods
Christian P. Robert
Universit´ Paris-Dauphine, IuF, & CREST
e
http://www.ceremade.dauphine.fr/~xian
O’Bayes 11, Shanghai, June 10, 2011
2. From MCMC to ABC Methods
Outline
Computational issues in Bayesian statistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Approximate Bayesian computation
ABC for model choice
3. From MCMC to ABC Methods
Computational issues in Bayesian statistics
A typology of Bayes computational problems
(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;
4. From MCMC to ABC Methods
Computational issues in Bayesian statistics
A typology of Bayes computational problems
(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;
(ii). use of a complex sampling model with an intractable
likelihood, as for instance in some latent variable or graphical
models or in inverse problems;
5. From MCMC to ABC Methods
Computational issues in Bayesian statistics
A typology of Bayes computational problems
(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;
(ii). use of a complex sampling model with an intractable
likelihood, as for instance in some latent variable or graphical
models or in inverse problems;
(iii). use of a huge dataset;
6. From MCMC to ABC Methods
Computational issues in Bayesian statistics
A typology of Bayes computational problems
(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;
(ii). use of a complex sampling model with an intractable
likelihood, as for instance in some latent variable or graphical
models or in inverse problems;
(iii). use of a huge dataset;
(iv). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);
7. From MCMC to ABC Methods
Computational issues in Bayesian statistics
A typology of Bayes computational problems
(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;
(ii). use of a complex sampling model with an intractable
likelihood, as for instance in some latent variable or graphical
models or in inverse problems;
(iii). use of a huge dataset;
(iv). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);
(v). use of a particular inferential procedure as for instance, Bayes
factors
π P (θ ∈ Θ0 | x) π(θ ∈ Θ0 )
B01 (x) = .
P (θ ∈ Θ1 | x) π(θ ∈ Θ1 )
8. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis-Hastings Algorithm
Computational issues in Bayesian statistics
The Metropolis-Hastings Algorithm
Monte Carlo basics
Monte Carlo Methods based on Markov Chains
The Metropolis–Hastings algorithm
Random-walk Metropolis-Hastings algorithms
The Gibbs Sampler
Approximate Bayesian computation
ABC for model choice
9. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
General purpose
Given a density π known up to a normalizing constant, and an
integrable function h, compute
h(x)˜ (x)µ(dx)
π
Π(h) = h(x)π(x)µ(dx) =
π (x)µ(dx)
˜
when h(x)˜ (x)µ(dx) is intractable.
π
10. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by
N
ˆN
ΠM C (h) = N −1 h(xi ).
i=1
ˆN as
LLN: ΠM C (h) −→ Π(h)
If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞,
√ L
CLT: ˆN
N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 .
11. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Monte Carlo 101
Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by
N
ˆN
ΠM C (h) = N −1 h(xi ).
i=1
ˆN as
LLN: ΠM C (h) −→ Π(h)
If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞,
√ L
CLT: ˆN
N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 .
Caveat announcing MCMC
Often impossible or inefficient to simulate directly from Π
12. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
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).
13. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
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
N
ˆ IS
ΠQ,N (h) = N −1 h(xi ){π/q}(xi ).
i=1
14. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Properties of importance
Then
ˆ as
LLN: ΠIS (h) −→ Π(h)
Q,N and if Q((hπ/q)2 ) < ∞,
√ L
CLT: ˆ Q,N
N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } .
15. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Properties of importance
Then
ˆ as
LLN: ΠIS (h) −→ Π(h)
Q,N and if Q((hπ/q)2 ) < ∞,
√ L
CLT: ˆ Q,N
N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } .
Caveat
ˆ Q,N
If normalizing constant of π unknown, impossible to use ΠIS
Generic problem in Bayesian Statistics: π(θ|x) ∝ f (x|θ)π(θ).
16. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Self-Normalised Importance Sampling
Self normalized version
N −1 N
ˆ Q,N
ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ).
i=1 i=1
17. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Self-Normalised Importance Sampling
Self normalized version
N −1 N
ˆ Q,N
ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ).
i=1 i=1
ˆ as
LLN : ΠSN IS (h) −→ Π(h)
Q,N
and if Π((1 + h2 )(π/q)) < ∞,
√ L
CLT : ˆ Q,N
N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) .
18. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo basics
Self-Normalised Importance Sampling
Self normalized version
N −1 N
ˆ Q,N
ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ).
i=1 i=1
ˆ as
LLN : ΠSN IS (h) −→ Π(h)
Q,N
and if Π((1 + h2 )(π/q)) < ∞,
√ L
CLT : ˆ Q,N
N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) .
c The quality of the SNIS approximation depends on the
choice of Q
19. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
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 ,
20. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
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
21. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
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
22. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Monte Carlo Methods based on Markov Chains
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
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.
23. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
The Metropolis–Hastings algorithm
Basics
The algorithm uses the target density
f
and a conditional density
q(y|x)
called the instrumental (or proposal) distribution
24. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
The MH algorithm
Algorithm (Metropolis–Hastings)
Given x(t) ,
1. Generate Yt ∼ q(y|x(t) ).
2. Take
Yt with prob. ρ(x(t) , Yt ),
X (t+1) =
x(t) with prob. 1 − ρ(x(t) , Yt ),
where
f (y) q(x|y)
ρ(x, y) = min ,1 .
f (x) q(y|x)
25. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
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
26. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties
Under irreducibility,
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)
27. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties
Under irreducibility,
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)
2. As f is a probability measure, the chain is positive recurrent
28. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. If
q(y|x) > 0 for every (x, y), (2)
the chain is irreducible
29. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. If
q(y|x) > 0 for every (x, y), (2)
the chain is irreducible
5. For M-H, f -irreducibility implies Harris recurrence
30. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
The Metropolis–Hastings algorithm
Convergence properties (2)
4. If
q(y|x) > 0 for every (x, y), (2)
the chain is irreducible
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)| < ∞,
T
1
lim h(X (t) ) = h(x)df (x) a.e. f.
T →∞ T t=1
(ii) and
lim K n (x, ·)µ(dx) − f =0
n→∞
TV
for every initial distribution µ, where K n (x, ·) denotes the
kernel for n transitions.
31. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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)
32. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Algorithm (Random walk Metropolis)
Given x(t)
1. Generate Yt ∼ g(y − x(t) )
2. Take
Y f (Yt )
(t+1) t with prob. min 1, ,
X = f (x(t) )
x(t) otherwise.
33. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Optimizing the Acceptance Rate
Problem of choice of the transition kernel from a practical point of
view
Most common alternatives:
1. an instrumental density g which approximates f , such that
f /g is bounded for uniform ergodicity to apply;
2. a random walk
In both cases, the choice of g is critical,
34. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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 .
35. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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
f (yt )
min ,1 1.
f (x(t) )
For multimodal densities with well separated modes, the negative
effect of limited moves on the surface of f clearly shows.
36. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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
37. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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
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]
38. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 )
and observables
yt |xt ∼ N (x2 , σ 2 )
t
39. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Example (Noisy AR(1))
Hidden Markov chain from a regular AR(1) model,
xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 )
and observables
yt |xt ∼ N (x2 , σ 2 )
t
The distribution of xt given xt−1 , xt+1 and yt is
−1 τ2
exp (xt − ϕxt−1 )2 + (xt+1 − ϕxt )2 + (yt − x2 )2
t .
2τ 2 σ2
40. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
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.
41. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Markov chain based on a random walk with scale ω = .1.
42. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Markov chain based on a random walk with scale ω = .5.
43. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
MA(2)
xt = t − θ1 t−1 − θ2 t−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)
}
44. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Basic RWHM for MA(2)
Algorithm 1 RW-HM-MA(2) sampler
set ω and ϑ(1)
for i = 2 to T do
˜ (i−1) (i−1)
generate ϑj ∼ U (ϑj − ω, ϑj + 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
45. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,
ϑ2 = 0.2 and scale ω = 0.2
46. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,
ϑ2 = 0.2 and scale ω = 0.5
47. From MCMC to ABC Methods
The Metropolis-Hastings Algorithm
Random-walk Metropolis-Hastings algorithms
Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,
ϑ2 = 0.2 and scale ω = 2.0
48. From MCMC to ABC Methods
The Gibbs Sampler
The Gibbs Sampler
The Gibbs Sampler
General Principles
Slice sampling
Convergence
49. From MCMC to ABC Methods
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
50. From MCMC to ABC Methods
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
2. Start with the random variable X = (X1 , . . . , Xp )
51. From MCMC to ABC Methods
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
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.
52. From MCMC to ABC Methods
The Gibbs Sampler
General Principles
Algorithm (Gibbs sampler)
(t) (t)
Given x(t) = (x1 , . . . , xp ), generate
(t+1) (t) (t)
1. X1 ∼ f1 (x1 |x2 , . . . , xp );
(t+1) (t+1) (t) (t)
2. X2 ∼ f2 (x2 |x1 , x3 , . . . , xp ),
...
(t+1) (t+1) (t+1)
p. Xp ∼ fp (xp |x1 , . . . , xp−1 )
X(t+1) → X ∼ f
53. From MCMC to ABC Methods
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
54. From MCMC to ABC Methods
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−1 fp fp−1 · · · f1
55. From MCMC to ABC Methods
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
56. From MCMC to ABC Methods
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
2. requires some knowledge of f
57. From MCMC to ABC Methods
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
2. requires some knowledge of f
3. is, by construction, multidimensional
58. From MCMC to ABC Methods
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
2. requires some knowledge of f
3. is, by construction, multidimensional
4. does not apply to problems where the number of parameters
varies as the resulting chain is not irreducible.
59. From MCMC to ABC Methods
The Gibbs Sampler
General Principles
A wee mixture problem
4
3
2
µ2
1
0
−1
−1 0 1 2 3 4
µ1
Gibbs started at random
60. From MCMC to ABC Methods
The Gibbs Sampler
General Principles
A wee mixture problem
Gibbs stuck at the wrong mode
4
3
3
2
2
µ2
1
µ2
1
0
0
−1
−1
−1 0 1 2 3 4
µ1
Gibbs started at random −1 0 1 2 3
µ1
61. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Slice sampler as generic Gibbs
If f (θ) can be written as a product
k
fi (θ),
i=1
62. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Slice sampler as generic Gibbs
If f (θ) can be written as a product
k
fi (θ),
i=1
it can be completed as
k
I0≤ωi ≤fi (θ) ,
i=1
leading to the following Gibbs algorithm:
63. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Algorithm (Slice sampler)
Simulate
(t+1)
1. ω1 ∼ U[0,f1 (θ(t) )] ;
...
(t+1)
k. ωk ∼ U[0,fk (θ(t) )] ;
k+1. θ(t+1) ∼ UA(t+1) , with
(t+1)
A(t+1) = {y; fi (y) ≥ ωi , i = 1, . . . , k}.
64. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2
65. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3
66. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3, 4
67. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3, 4, 5
68. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3, 4, 5, 10
69. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3, 4, 5, 10, 50
70. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y
0.004
0.002
0.000
0.0 0.2 0.4 0.6 0.8 1.0
x
Number of Iterations 2, 3, 4, 5, 10, 50, 100
71. From MCMC to ABC Methods
The Gibbs Sampler
Slice sampling
Good slices, tough 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.
72. From MCMC to ABC Methods
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) (y > 0 for every i = 1, · · · , p, implies that
i)
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.
73. From MCMC to ABC Methods
The Gibbs Sampler
Convergence
Properties of the Gibbs sampler (2)
Consequences
(i) If h(y)g(y)dy < ∞, then
T
1
lim h1 (Y (t) ) = h(y)g(y)dy a.e. g.
nT →∞ T
t=1
(ii) If, in addition, (Y (t) ) is aperiodic, then
lim K n (y, ·)µ(dx) − f =0
n→∞
TV
for every initial distribution µ.
74. From MCMC to ABC Methods
The Gibbs Sampler
Convergence
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
g2 (y2 |y1 )
g(y1 , y2 ) = .
g2 (v|y1 )/g1 (y1 |v) dv
[Hammersley & Clifford, circa 1970]
75. From MCMC to ABC Methods
The Gibbs Sampler
Convergence
General HC decomposition
Under the positivity condition, the joint distribution g satisfies
p
g j (y j |y 1 , . . . , y j−1 ,y j+1
, . . . , y p)
g(y1 , . . . , yp ) ∝
g j (y j |y 1 , . . . , y j−1 ,y , . . . , y p)
j=1 j+1
for every permutation on {1, 2, . . . , p} and every y ∈ Y .
76. From MCMC to ABC Methods
Approximate Bayesian computation
Approximate Bayesian computation
Computational issues in Bayesian statistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Approximate Bayesian computation
ABC basics
Alphabet soup
Calibration of ABC
ABC for model choice
77. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Untractable likelihoods
Cases when the likelihood function f (y|θ) is unavailable and when
the completion step
f (y|θ) = f (y, z|θ) dz
Z
is impossible or too costly because of the dimension of z
c MCMC cannot be implemented!
78. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Illustrations
Example
Stochastic volatility model: for Highest weight trajectories
t = 1, . . . , T,
0.4
0.2
yt = exp(zt ) t , zt = a+bzt−1 +σηt ,
0.0
−0.2
T very large makes it difficult to
−0.4
include z within the simulated 0 200 400
t
600 800 1000
parameters
79. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Illustrations
Example
Potts model: if y takes values on a grid Y of size k n and
f (y|θ) ∝ exp θ Iyl =yi
l∼i
where l∼i denotes a neighbourhood relation, n moderately large
prohibits the computation of the normalising constant
80. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Illustrations
Example
Inference on CMB: in cosmology, study of the Cosmic Microwave
Background via likelihoods immensely slow to computate (e.g
WMAP, Plank), because of numerically costly spectral transforms
[Data is a Fortran program]
[Kilbinger et al., 2010, MNRAS]
81. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Illustrations
Example
Phylogenetic tree: in population
genetics, reconstitution of a common
ancestor from a sample of genes via
a phylogenetic tree that is close to
impossible to integrate out
[100 processor days with 4
parameters]
[Cornuet et al., 2009, Bioinformatics]
82. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
The ABC method
Bayesian setting: target is π(θ)f (x|θ)
83. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
The ABC method
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
84. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
The ABC method
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
ABC algorithm
For an observation y ∼ f (y|θ), under the prior π(θ), keep jointly
simulating
θ ∼ π(θ) , z ∼ f (z|θ ) ,
until the auxiliary variable z is equal to the observed value, z = y.
[Tavar´ et al., 1997]
e
85. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Why does it work?!
The proof is trivial:
f (θi ) ∝ π(θi )f (z|θi )Iy (z)
z∈D
∝ π(θi )f (y|θi )
= π(θi |y) .
[Accept–Reject 101]
86. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
A as approximative
When y is a continuous random variable, equality z = y is replaced
with a tolerance condition,
(y, z) ≤
where is a distance
87. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
A as approximative
When y is a continuous random variable, equality z = y is replaced
with a tolerance condition,
(y, z) ≤
where is a distance
Output distributed from
π(θ) Pθ { (y, z) < } ∝ π(θ| (y, z) < )
88. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
ABC algorithm
Algorithm 2 Likelihood-free rejection sampler 2
for i = 1 to N do
repeat
generate θ from the prior distribution π(·)
generate z from the likelihood f (·|θ )
until ρ{η(z), η(y)} ≤
set θi = θ
end for
where η(y) defines a (not necessarily sufficient) statistic
89. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Output
The likelihood-free algorithm samples from the marginal in z of:
π(θ)f (z|θ)IA ,y (z)
π (θ, z|y) = ,
A ,y ×Θ π(θ)f (z|θ)dzdθ
where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
90. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Output
The likelihood-free algorithm samples from the marginal in z of:
π(θ)f (z|θ)IA ,y (z)
π (θ, z|y) = ,
A ,y ×Θ π(θ)f (z|θ)dzdθ
where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
The idea behind ABC is that the summary statistics coupled with a
small tolerance should provide a good approximation of the
posterior distribution:
π (θ|y) = π (θ, z|y)dz ≈ π(θ|y) .
91. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
MA example
Back to the MA(q) model
q
xt = t + ϑi t−i
i=1
Simple prior: uniform over the inverse [real and complex] roots in
q
Q(u) = 1 − ϑi ui
i=1
under the identifiability conditions
92. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
MA example
Back to the MA(q) model
q
xt = t+ ϑi t−i
i=1
Simple prior: uniform prior over the identifiability zone, e.g.
triangle for MA(2)
93. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
MA example (2)
ABC algorithm thus made of
1. picking a new value (ϑ1 , ϑ2 ) in the triangle
2. generating an iid sequence ( t )−q<t≤T
3. producing a simulated series (xt )1≤t≤T
94. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
MA example (2)
ABC algorithm thus made of
1. picking a new value (ϑ1 , ϑ2 ) in the triangle
2. generating an iid sequence ( t )−q<t≤T
3. producing a simulated series (xt )1≤t≤T
Distance: basic distance between the series
T
ρ((xt )1≤t≤T , (xt )1≤t≤T ) = (xt − xt )2
t=1
or distance between summary statistics like the q autocorrelations
T
τj = xt xt−j
t=j+1
95. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Comparison of distance impact
Evaluation of the tolerance on the ABC sample against both
distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
96. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Comparison of distance impact
Evaluation of the tolerance on the ABC sample against both
distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
97. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
Comparison of distance impact
Evaluation of the tolerance on the ABC sample against both
distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
98. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
ABC advances
Simulating from the prior is often poor in efficiency
99. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
ABC advances
Simulating from the prior is often poor in efficiency
Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y...
[Marjoram et al, 2003; Bortot et al., 2007; Sisson et al., 2007]
100. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
ABC advances
Simulating from the prior is often poor in efficiency
Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y...
[Marjoram et al, 2003; Bortot et al., 2007; Sisson et al., 2007]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
101. From MCMC to ABC Methods
Approximate Bayesian computation
ABC basics
ABC advances
Simulating from the prior is often poor in efficiency
Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y...
[Marjoram et al, 2003; Bortot et al., 2007; Sisson et al., 2007]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
.....or even by including in the inferential framework [ABCµ ]
[Ratmann et al., 2009]
102. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-NP
Better usage of [prior] simulations by
adjustement: instead of throwing away
θ such that ρ(η(z), η(y)) > , replace
θs with locally regressed
ˆ
θ∗ = θ − {η(z) − η(y)}T β
[Csill´ry et al., TEE, 2010]
e
ˆ
where β is obtained by [NP] weighted least square regression on
(η(z) − η(y)) with weights
Kδ {ρ(η(z), η(y))}
[Beaumont et al., 2002, Genetics]
103. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-MCMC
Markov chain (θ(t) ) created via the transition function
θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y
π(θ )Kω (t) |θ )
θ (t+1)
= and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,
(t) ω (θ
θ otherwise,
104. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-MCMC
Markov chain (θ(t) ) created via the transition function
θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y
π(θ )Kω (t) |θ )
θ (t+1)
= and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,
(t) ω (θ
θ otherwise,
has the posterior π(θ|y) as stationary distribution
[Marjoram et al, 2003]
105. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-MCMC (2)
Algorithm 3 Likelihood-free MCMC sampler
Use Algorithm 2 to get (θ(0) , z(0) )
for t = 1 to N do
Generate θ from Kω ·|θ(t−1) ,
Generate z from the likelihood f (·|θ ),
Generate u from U[0,1] ,
π(θ )Kω (θ (t−1) |θ )
if u ≤ I
π(θ (t−1) Kω (θ |θ (t−1) ) A ,y (z ) then
set (θ(t) , z(t) ) = (θ , z )
else
(θ(t) , z(t) )) = (θ(t−1) , z(t−1) ),
end if
end for
106. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Why does it work?
Acceptance probability that does not involve the calculation of the
likelihood and
π (θ , z |y) Kω (θ(t−1) |θ )f (z(t−1) |θ(t−1) )
×
π (θ(t−1) , z(t−1) |y) Kω (θ |θ(t−1) )f (z |θ )
π(θ ) f (z |θ ) IA ,y (z )
= (t−1) ) f (z(t−1) |θ (t−1) )I (t−1) )
π(θ A ,y (z
Kω (θ(t−1) |θ ) f (z(t−1) |θ(t−1) )
×
Kω (θ |θ(t−1) ) f (z |θ )
π(θ )Kω (θ(t−1) |θ )
= IA (z ) .
π(θ(t−1) Kω (θ |θ(t−1) ) ,y
107. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ
[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]
Use of a joint density
f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( )
where y is the data, and ξ( |y, θ) is the prior predictive density of
ρ(η(z), η(y)) given θ and x when z ∼ f (z|θ)
108. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ
[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]
Use of a joint density
f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( )
where y is the data, and ξ( |y, θ) is the prior predictive density of
ρ(η(z), η(y)) given θ and x when z ∼ f (z|θ)
Warning! Replacement of ξ( |y, θ) with a non-parametric kernel
approximation.
109. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ details
Multidimensional distances ρk (k = 1, . . . , K) and errors
k = ρk (ηk (z), ηk (y)), with
ˆ 1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b
ˆ
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
110. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ details
Multidimensional distances ρk (k = 1, . . . , K) and errors
k = ρk (ηk (z), ηk (y)), with
ˆ 1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b
ˆ
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
ABCµ involves acceptance probability
ˆ
π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ )
ˆ
π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)
111. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ multiple errors
[ c Ratmann et al., PNAS, 2009]
112. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABCµ for model choice
[ c Ratmann et al., PNAS, 2009]
113. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Questions about ABCµ
For each model under comparison, marginal posterior on used to
assess the fit of the model (HPD includes 0 or not).
114. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Questions about ABCµ
For each model under comparison, marginal posterior on used to
assess the fit of the model (HPD includes 0 or not).
Is the data informative about ? [Identifiability]
How is the prior π( ) impacting the comparison?
How is using both ξ( |x0 , θ) and π ( ) compatible with a
standard probability model? [remindful of Wilkinson]
Where is the penalisation for complexity in the model
comparison?
[X, Mengersen & Chen, 2010, PNAS]
115. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
A PMC version
Generate a sample at iteration t by
N
(t) (t−1) (t−1)
πt (θ ) ∝
ˆ ωj Kt (θ(t) |θj )
j=1
modulo acceptance of the associated xt , and use an importance
(t)
weight associated with an accepted simulation θi
(t) (t) (t)
ωi ∝ π(θi ) πt (θi ) .
ˆ
c Still likelihood free
[Beaumont et al., 2009]
116. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
The ABC-PMC algorithm
Given a decreasing sequence of approximation levels 1 ≥ ... ≥ T,
1. At iteration t = 1,
For i = 1, ..., N
(1) (1)
Simulate θi ∼ π(θ) and x ∼ f (x|θi ) until (x, y) < 1
(1)
Set ωi = 1/N
(1)
Take τ 2 as twice the empirical variance of the θi ’s
2. At iteration 2 ≤ t ≤ T ,
For i = 1, ..., N , repeat
(t−1) (t−1)
Pick θi from the θj ’s with probabilities ωj
(t) 2 (t)
generate θi |θi ∼ N (θi , σt ) and x ∼ f (x|θi )
until (x, y) < t
(t) (t) N (t−1) −1 (t) (t−1)
Set ωi ∝ π(θi )/ j=1 ωj ϕ σt θi − θj )
2 (t)
Take τt+1 as twice the weighted empirical variance of the θi ’s
117. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Sequential Monte Carlo
SMC is a simulation technique to approximate a sequence of
related probability distributions πn with π0 “easy” and πT as
target.
Iterated IS: particles moved from time n to time n via kernel Kn
and use of a sequence of extended targets πn˜
n
πn (z0:n ) = πn (zn )
˜ Lj (zj+1 , zj )
j=0
where the Lj ’s are backward Markov kernels [check that πn (zn ) is
a marginal]
[Del Moral, Doucet & Jasra, Series B, 2006]
118. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Sequential Monte Carlo (2)
Algorithm 4 SMC sampler
(0)
sample zi ∼ γ0 (x) (i = 1, . . . , N )
(0) (0) (0)
compute weights wi = π0 (zi ))/γ0 (zi )
for t = 1 to N do
if ESS(w(t−1) ) < NT then
resample N particles z (t−1) and set weights to 1
end if
(t−1) (t−1)
generate zi ∼ Kt (zi , ·) and set weights to
(t) (t) (t−1)
(t) (t−1) πt (zi ))Lt−1 (zi ), zi ))
wi = Wi−1 (t−1) (t−1) (t)
πt−1 (zi ))Kt (zi ), zi ))
end for
119. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-SMC
[Del Moral, Doucet & Jasra, 2009]
True derivation of an SMC-ABC algorithm
Use of a kernel Kn associated with target π n and derivation of the
backward kernel
π n (z )Kn (z , z)
Ln−1 (z, z ) =
πn (z)
Update of the weights
M
m=1 IA (xm )
in
win ∝ wi(n−1) M
n
m=1 IA n−1
(xm
i(n−1) )
when xm ∼ K(xi(n−1) , ·)
in
120. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
ABC-SMCM
Modification: Makes M repeated simulations of the pseudo-data z
given the parameter, rather than using a single [M = 1]
simulation, leading to weight that is proportional to the number of
accepted zi s
M
1
ω(θ) = Iρ(η(y),η(zi ))<
M
i=1
[limit in M means exact simulation from (tempered) target]
121. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Properties of ABC-SMC
The ABC-SMC method properly uses a backward kernel L(z, z ) to
simplify the importance weight and to remove the dependence on
the unknown likelihood from this weight. Update of importance
weights is reduced to the ratio of the proportions of surviving
particles
122. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Properties of ABC-SMC
The ABC-SMC method properly uses a backward kernel L(z, z ) to
simplify the importance weight and to remove the dependence on
the unknown likelihood from this weight. Update of importance
weights is reduced to the ratio of the proportions of surviving
particles
Adaptivity in ABC-SMC algorithm only found in on-line
construction of the thresholds t , slowly enough to keep a large
number of accepted transitions
123. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Semi-automatic ABC
Fearnhead and Prangle (2010) study ABC and the selection of the
summary statistic in close proximity to Wilkinson’s proposal
ABC then considered from a purely inferential viewpoint and
calibrated for estimation purposes.
Use of a randomised (or ‘noisy’) version of the summary statistics
η (y) = η(y) + τ
˜
Derivation of a well-calibrated version of ABC, i.e. an algorithm
that gives proper predictions for the distribution associated with
this randomised summary statistic.
124. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Summary statistics
Optimality of the posterior expectations of the parameters of
interest as summary statistics!
125. From MCMC to ABC Methods
Approximate Bayesian computation
Alphabet soup
Summary statistics
Optimality of the posterior expectations of the parameters of
interest as summary statistics!
Use of the standard quadratic loss function
(θ − θ0 )T A(θ − θ0 ) .
126. From MCMC to ABC Methods
Approximate Bayesian computation
Calibration of ABC
Which summary?
Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics [except when done by the
experimenters in the field]
127. From MCMC to ABC Methods
Approximate Bayesian computation
Calibration of ABC
Which summary?
Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics [except when done by the
experimenters in the field]
Starting from a large collection of summary statistics is available,
Joyce and Marjoram (2008) consider the sequential inclusion into
the ABC target, with a stopping rule based on a likelihood ratio
test.
128. From MCMC to ABC Methods
Approximate Bayesian computation
Calibration of ABC
Which summary?
Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics [except when done by the
experimenters in the field]
Starting from a large collection of summary statistics is available,
Joyce and Marjoram (2008) consider the sequential inclusion into
the ABC target, with a stopping rule based on a likelihood ratio
test.
Does not taking into account the sequential nature of the tests
Depends on parameterisation
Order of inclusion matters.
129. From MCMC to ABC Methods
ABC for model choice
ABC for model choice
Computational issues in Bayesian statistics
The Metropolis-Hastings Algorithm
The Gibbs Sampler
Approximate Bayesian computation
ABC for model choice
Model choice
Gibbs random fields
Model choice via ABC
Illustrations
Generic ABC model choice
130. From MCMC to ABC Methods
ABC for model choice
Model choice
Bayesian model choice
Several models M1 , M2 , . . . are considered simultaneously for a
dataset y and the model index M is part of the inference.
Use of a prior distribution. π(M = m), plus a prior distribution on
the parameter conditional on the value m of the model index,
πm (θ m )
Goal is to derive the posterior distribution of M , challenging
computational target when models are complex.
131. From MCMC to ABC Methods
ABC for model choice
Model choice
Generic ABC for model choice
Algorithm 5 Likelihood-free model choice sampler (ABC-MC)
for t = 1 to T do
repeat
Generate m from the prior π(M = m)
Generate θ m from the prior πm (θ m )
Generate z from the model fm (z|θ m )
until ρ{η(z), η(y)} <
Set m(t) = m and θ (t) = θ m
end for
132. From MCMC to ABC Methods
ABC for model choice
Model choice
ABC estimates
Posterior probability π(M = m|y) approximated by the frequency
of acceptances from model m
T
1
Im(t) =m .
T
t=1
Issues with implementation:
should tolerances be the same for all models?
should summary statistics vary across models (incl. their
dimension)?
should the distance measure ρ vary as well?
133. From MCMC to ABC Methods
ABC for model choice
Model choice
ABC estimates
Posterior probability π(M = m|y) approximated by the frequency
of acceptances from model m
T
1
Im(t) =m .
T
t=1
Issues with implementation:
should tolerances be the same for all models?
should summary statistics vary across models (incl. their
dimension)?
should the distance measure ρ vary as well?
Extension to a weighted polychotomous logistic regression estimate
of π(M = m|y), with non-parametric kernel weights
[Cornuet et al., DIYABC, 2009]
134. From MCMC to ABC Methods
ABC for model choice
Model choice
The Great ABC controversy
On-going controvery in phylogeographic genetics about the validity
of using ABC for testing
Against: Templeton, 2008,
2009, 2010a, 2010b, 2010c
argues that nested hypotheses
cannot have higher probabilities
than nesting hypotheses (!)
135. From MCMC to ABC Methods
ABC for model choice
Model choice
The Great ABC controversy
On-going controvery in phylogeographic genetics about the validity
of using ABC for testing
Replies: Fagundes et al., 2008,
Against: Templeton, 2008, Beaumont et al., 2010, Berger et
2009, 2010a, 2010b, 2010c al., 2010, Csill`ry et al., 2010
e
argues that nested hypotheses point out that the criticisms are
cannot have higher probabilities addressed at [Bayesian]
than nesting hypotheses (!) model-based inference and have
nothing to do with ABC...
136. From MCMC to ABC Methods
ABC for model choice
Gibbs random fields
Gibbs random fields
Gibbs distribution
The rv y = (y1 , . . . , yn ) is a Gibbs random field associated with
the graph G if
1
f (y) = exp − Vc (yc ) ,
Z
c∈C
where Z is the normalising constant, C is the set of cliques of G
and Vc is any function also called potential
U (y) = c∈C Vc (yc ) is the energy function
137. From MCMC to ABC Methods
ABC for model choice
Gibbs random fields
Gibbs random fields
Gibbs distribution
The rv y = (y1 , . . . , yn ) is a Gibbs random field associated with
the graph G if
1
f (y) = exp − Vc (yc ) ,
Z
c∈C
where Z is the normalising constant, C is the set of cliques of G
and Vc is any function also called potential
U (y) = c∈C Vc (yc ) is the energy function
c Z is usually unavailable in closed form
138. From MCMC to ABC Methods
ABC for model choice
Gibbs random fields
Potts model
Potts model
Vc (y) is of the form
Vc (y) = θS(y) = θ δyl =yi
l∼i
where l∼i denotes a neighbourhood structure
139. From MCMC to ABC Methods
ABC for model choice
Gibbs random fields
Potts model
Potts model
Vc (y) is of the form
Vc (y) = θS(y) = θ δyl =yi
l∼i
where l∼i denotes a neighbourhood structure
In most realistic settings, summation
Zθ = exp{θ T S(x)}
x∈X
involves too many terms to be manageable and numerical
approximations cannot always be trusted
[Cucala, Marin, CPR & Titterington, 2009]
140. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Bayesian Model Choice
Comparing a model with potential S0 taking values in Rp0 versus a
model with potential S1 taking values in Rp1 can be done through
the Bayes factor corresponding to the priors π0 and π1 on each
parameter space
exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 )
0
Bm0 /m1 (x) =
exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 )
1
141. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Bayesian Model Choice
Comparing a model with potential S0 taking values in Rp0 versus a
model with potential S1 taking values in Rp1 can be done through
the Bayes factor corresponding to the priors π0 and π1 on each
parameter space
exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 )
0
Bm0 /m1 (x) =
exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 )
1
Use of Jeffreys’ scale to select most appropriate model
142. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Neighbourhood relations
Choice to be made between M neighbourhood relations
m
i∼i (0 ≤ m ≤ M − 1)
with
Sm (x) = I{xi =xi }
m
i∼i
driven by the posterior probabilities of the models.
143. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Model index
Formalisation via a model index M that appears as a new
parameter with prior distribution π(M = m) and
π(θ|M = m) = πm (θm )
144. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Model index
Formalisation via a model index M that appears as a new
parameter with prior distribution π(M = m) and
π(θ|M = m) = πm (θm )
Computational target:
P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) ,
Θm
145. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Sufficient statistics
By definition, if S(x) sufficient statistic for the joint parameters
(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .
146. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Sufficient statistics
By definition, if S(x) sufficient statistic for the joint parameters
(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .
For each model m, own sufficient statistic Sm (·) and
S(·) = (S0 (·), . . . , SM −1 (·)) also sufficient.
147. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
Sufficient statistics
By definition, if S(x) sufficient statistic for the joint parameters
(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .
For each model m, own sufficient statistic Sm (·) and
S(·) = (S0 (·), . . . , SM −1 (·)) also sufficient.
For Gibbs random fields,
1 2
x|M = m ∼ fm (x|θm ) = fm (x|S(x))fm (S(x)|θm )
1
= f 2 (S(x)|θm )
n(S(x)) m
where
n(S(x)) = {˜ ∈ X : S(˜ ) = S(x)}
x x
c S(x) is therefore also sufficient for the joint parameters
[Specific to Gibbs random fields!]
148. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
ABC model choice Algorithm
ABC-MC
Generate m∗ from the prior π(M = m).
∗
Generate θm∗ from the prior πm∗ (·).
Generate x∗ from the model fm∗ (·|θm∗ ).
∗
Compute the distance ρ(S(x0 ), S(x∗ )).
Accept (θm∗ , m∗ ) if ρ(S(x0 ), S(x∗ )) < .
∗
Note When = 0 the algorithm is exact
149. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
ABC approximation to the Bayes factor
Frequency ratio:
ˆ
P(M = m0 |x0 ) π(M = m1 )
BF m0 /m1 (x0 ) = ×
ˆ
P(M = m1 |x0 ) π(M = m0 )
{mi∗ = m0 } π(M = m1 )
= × ,
{mi∗ = m1 } π(M = m0 )
150. From MCMC to ABC Methods
ABC for model choice
Model choice via ABC
ABC approximation to the Bayes factor
Frequency ratio:
ˆ
P(M = m0 |x0 ) π(M = m1 )
BF m0 /m1 (x0 ) = ×
ˆ
P(M = m1 |x0 ) π(M = m0 )
{mi∗ = m0 } π(M = m1 )
= × ,
{mi∗ = m1 } π(M = m0 )
replaced with
1 + {mi∗ = m0 } π(M = m1 )
BF m0 /m1 (x0 ) = ×
1 + {mi∗ = m1 } π(M = m0 )
to avoid indeterminacy (also Bayes estimate).
151. From MCMC to ABC Methods
ABC for model choice
Illustrations
Toy example
iid Bernoulli model versus two-state first-order Markov chain, i.e.
n
f0 (x|θ0 ) = exp θ0 I{xi =1} {1 + exp(θ0 )}n ,
i=1
versus
n
1
f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } {1 + exp(θ1 )}n−1 ,
2
i=2
with priors θ0 ∼ U (−5, 5) and θ1 ∼ U (0, 6) (inspired by “phase
transition” boundaries).
152. From MCMC to ABC Methods
ABC for model choice
Illustrations
Toy example (2)
(left) Comparison of the true BF m0 /m1 (x0 ) with BF m0 /m1 (x0 )
(in logs) over 2, 000 simulations and 4.106 proposals from the
prior. (right) Same when using tolerance corresponding to the
1% quantile on the distances.
153. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Back to sufficiency
If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and
η2 (x) sufficient statistic for model m = 2 and parameter θ2 ,
(η1 (x), η2 (x)) is not always sufficient for (m, θm )
154. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Back to sufficiency
If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and
η2 (x) sufficient statistic for model m = 2 and parameter θ2 ,
(η1 (x), η2 (x)) is not always sufficient for (m, θm )
c Potential loss of information at the testing level
[X, Cornuet, Marin, and Pillai, 2011]
155. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 (T → ∞)
ABC approximation
T
t=1 Imt =1 Iρ{η(zt ),η(y)}≤
B12 (y) = T
,
t=1 Imt =2 Iρ{η(zt ),η(y)}≤
where the (mt , z t )’s are simulated from the (joint) prior
156. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 (T → ∞)
ABC approximation
T
t=1 Imt =1 Iρ{η(zt ),η(y)}≤
B12 (y) = T
,
t=1 Imt =2 Iρ{η(zt ),η(y)}≤
where the (mt , z t )’s are simulated from the (joint) prior
As T go to infinity, limit
Iρ{η(z),η(y)}≤ π1 (θ 1 )f1 (z|θ 1 ) dz dθ 1
B12 (y) =
Iρ{η(z),η(y)}≤ π2 (θ 2 )f2 (z|θ 2 ) dz dθ 2
η
Iρ{η,η(y)}≤ π1 (θ 1 )f1 (η|θ 1 ) dη dθ 1
= η ,
Iρ{η,η(y)}≤ π2 (θ 2 )f2 (η|θ 2 ) dη dθ 2
η η
where f1 (η|θ 1 ) and f2 (η|θ 2 ) distributions of η(z)
157. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 ( → 0)
When goes to zero,
η
η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η ,
π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2
158. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 ( → 0)
When goes to zero,
η
η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η ,
π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2
Bayes factor based on the sole observation of η(y)
159. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 (under sufficiency)
If η(y) sufficient statistic for both models,
fi (y|θ i ) = gi (y)fiη (η(y)|θ i )
Thus
η
Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η
Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2
η
g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η
= η = B (y) .
g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12
[Didelot, Everitt, Johansen & Lawson, 2011]
160. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Limiting behaviour of B12 (under sufficiency)
If η(y) sufficient statistic for both models,
fi (y|θ i ) = gi (y)fiη (η(y)|θ i )
Thus
η
Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η
Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2
η
g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η
= η = B (y) .
g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12
[Didelot, Everitt, Johansen & Lawson, 2011]
c No discrepancy only when cross-model sufficiency
161. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Poisson/geometric example
Sample
x = (x1 , . . . , xn )
from either a Poisson P(λ) or from a geometric G(p) Then
n
S= yi = η(x)
i=1
sufficient statistic for either model but not simultaneously
Discrepancy ratio
g1 (x) S!n−S / i yi !
=
g2 (x) 1 n+S−1
S
162. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Poisson/geometric discrepancy
η
Range of B12 (x) versus B12 (x) B12 (x): The values produced have
nothing in common.
163. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Formal recovery
Creating an encompassing exponential family
T T
f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)}
leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x))
[Didelot, Everitt, Johansen & Lawson, 2011]
164. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Formal recovery
Creating an encompassing exponential family
T T
f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)}
leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x))
[Didelot, Everitt, Johansen & Lawson, 2011]
In the Poisson/geometric case, if i xi ! is added to S, no
discrepancy
165. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Formal recovery
Creating an encompassing exponential family
T T
f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)}
leads to a sufficient statistic (η1 (x), η2 (x), t1 (x), t2 (x))
[Didelot, Everitt, Johansen & Lawson, 2011]
Only applies in genuine sufficiency settings...
c Inability to evaluate loss brought by summary statistics
166. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Meaning of the ABC-Bayes factor
In the Poisson/geometric case, if E[yi ] = θ0 > 0,
η (θ0 + 1)2 −θ0
lim B12 (y) = e
n→∞ θ0
167. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
MA(q) divergence
Evolution [against ] of ABC Bayes factor, in terms of frequencies of
visits to models MA(1) (left) and MA(2) (right) when equal to
10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample
of 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factor
equal to 17.71.
168. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
MA(q) divergence
Evolution [against ] of ABC Bayes factor, in terms of frequencies of
visits to models MA(1) (left) and MA(2) (right) when equal to
10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sample
of 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21
equal to .004.
169. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
A population genetics evaluation
Population genetics example with
3 populations
2 scenari
15 individuals
5 loci
single mutation parameter
170. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
A population genetics evaluation
Population genetics example with
3 populations
2 scenari
15 individuals
5 loci
single mutation parameter
24 summary statistics
2 million ABC proposal
importance [tree] sampling alternative
171. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Stability of importance sampling
172. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Comparison with ABC
Use of 24 summary statistics and DIY-ABC logistic correction
173. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Comparison with ABC
Use of 15 summary statistics and DIY-ABC logistic correction
174. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
Comparison with ABC
Use of 24 summary statistics and DIY-ABC logistic correction
175. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
The only safe cases
Besides specific models like Gibbs random fields,
using distances over the data itself escapes the discrepancy...
[Toni & Stumpf, 2010; Sousa et al., 2009]
176. From MCMC to ABC Methods
ABC for model choice
Generic ABC model choice
The only safe cases
Besides specific models like Gibbs random fields,
using distances over the data itself escapes the discrepancy...
[Toni & Stumpf, 2010; Sousa et al., 2009]
...and so does the use of more informal model fitting measures
[Ratmann et al., 2009, 2011]