Discussion at Adap'skiii, Park City, Utah, Jan. 3-4 2011

1. Adaptive MCMH for Bayesian inference of spatial
autologistic models
discussion by Pierre E. Jacob & Christian P. Robert
Universit´ Paris-Dauphine and CREST
e
http://statisfaction.wordpress.com
Adap’skiii, Park City, Utah, Jan. 03, 2011
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 1/ 7

2. Context: 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
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 2/ 7

3. Context: 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!
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 2/ 7

4. Context: 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!
=⇒ adaptive MCMH
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 2/ 7

5. Illustrations
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
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 3/ 7

6. Compare ABC with adaptive MCMH?
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
ABC can also be applied when the normalizing constant is
unknown, hence it could be compared to adaptive MCMH, e.g. for
a given number of y’s drawn from f , or in terms of execution time.
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 4/ 7

7. ˆ
Some questions on the estimator R(θt , ϑ)
ˆ 1
R(θt , ϑ) =
m0 + m0 θi ∈St {θt } I(||θi − ϑ|| ≤ η)
   
m0 (i) m0 (t)
 g(zj , θt ) g(zj , θt ) 
× I(||θi − ϑ|| ≤ η)
(i)
+
(t)

θi ∈St {θt } j=1 g(zj , ϑ) j=1 g(zj , ϑ)

Why does it bypass the number of repetitions of the θi ’s?
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 5/ 7

8. ˆ
Some questions on the estimator R(θt , ϑ)
ˆ 1
R(θt , ϑ) =
m0 + m0 θi ∈St {θt } I(||θi − ϑ|| ≤ η)
   
m0 (i) m0 (t)
 g(zj , θt ) g(zj , θt ) 
× I(||θi − ϑ|| ≤ η)
(i)
+
(t)

θi ∈St {θt } j=1 g(zj , ϑ) j=1 g(zj , ϑ)

Why does it bypass the number of repetitions of the θi ’s?
(i)
Why resample the yj ’s only to inverse-weight them later?
(i)
Why not stick to the original sample and weight the yj ’s
directly? This would avoid the necessity to choose m0 .
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 5/ 7

9. Combine adaptations?
Adaptive MCMH looks great, but it adds new tuning parameters
compared to standard MCMC!
It would be nice to be able to adapt the proposal Q(θt , ϑ),
e.g. to control the acceptance rate, as in other adaptive
MCMC algorithms.
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 6/ 7

10. Combine adaptations?
Adaptive MCMH looks great, but it adds new tuning parameters
compared to standard MCMC!
It would be nice to be able to adapt the proposal Q(θt , ϑ),
e.g. to control the acceptance rate, as in other adaptive
MCMC algorithms.
Then could η be adaptive, since it does as well depend on the
(unknown) scale of the posterior distribution of θ ?
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 6/ 7

11. Combine adaptations?
Adaptive MCMH looks great, but it adds new tuning parameters
compared to standard MCMC!
It would be nice to be able to adapt the proposal Q(θt , ϑ),
e.g. to control the acceptance rate, as in other adaptive
MCMC algorithms.
Then could η be adaptive, since it does as well depend on the
(unknown) scale of the posterior distribution of θ ?
How would the current theoretical framework cope with these
additional adaptations?
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 6/ 7

12. Combine adaptations?
Adaptive MCMH looks great, but it adds new tuning parameters
compared to standard MCMC!
It would be nice to be able to adapt the proposal Q(θt , ϑ),
e.g. to control the acceptance rate, as in other adaptive
MCMC algorithms.
Then could η be adaptive, since it does as well depend on the
(unknown) scale of the posterior distribution of θ ?
How would the current theoretical framework cope with these
additional adaptations?
In the same spirit, could m (and m0 ) be considered as
algorithmic parameters and be adapted as well? What
criterion would be used to adapt them?
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 6/ 7

13. Sample size and acceptance rate
Another MH algorithm where the “true” acceptance ratio is
replaced by a ratio with an unbiased estimator is the Particle
MCMC algorithm.
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 7/ 7

14. Sample size and acceptance rate
Another MH algorithm where the “true” acceptance ratio is
replaced by a ratio with an unbiased estimator is the Particle
MCMC algorithm.
In PMCMC, the acceptance rate is growing with the number
of particles used to estimate the likelihood, ie when the
estimation is more precise, the acceptance rate increases and
converges towards the acceptance rate of an idealized
standard MH algorithm.
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 7/ 7

15. Sample size and acceptance rate
Another MH algorithm where the “true” acceptance ratio is
replaced by a ratio with an unbiased estimator is the Particle
MCMC algorithm.
In PMCMC, the acceptance rate is growing with the number
of particles used to estimate the likelihood, ie when the
estimation is more precise, the acceptance rate increases and
converges towards the acceptance rate of an idealized
standard MH algorithm.
In adaptive MCMH, is there such a link betweeen m, m0 and
the number of iterations on one side and the acceptance rate
of the algorithm on the other side? Should it be growing when
the estimation becomes more and more precise?
Adaptive MCMH for Bayesian inference of spatial autologistic mo
discussion by Pierre E. Jacob & Christian P. Robert 7/ 7