On National Teacher Day, meet the 2024-25 Kenan Fellows
Jsm09 talk
1. On some computational methods for Bayesian model choice
On some computational methods for Bayesian
model choice
Christian P. Robert
CREST-INSEE and Universit´ Paris Dauphine
e
http://xianblog.wordpress.com
JSM 2009, Washington D.C., August 2, 2009
Joint works with M. Beaumont, N. Chopin,
J.-M. Cornuet, J.-M. Marin and D. Wraith
2. On some computational methods for Bayesian model choice
Outline
1 Evidence
2 Importance sampling solutions
3 Nested sampling
4 ABC model choice
3. On some computational methods for Bayesian model choice
Evidence
Model choice and model comparison
Choice between models
Several models available for the same observation vector x
Mi : x ∼ fi (x|θi ), i∈I
where the family I can be finite or infinite
4. On some computational methods for Bayesian model choice
Evidence
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
define priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
j Θj
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
j Θj
5. On some computational methods for Bayesian model choice
Evidence
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
define priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
j Θj
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
j Θj
6. On some computational methods for Bayesian model choice
Evidence
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
define priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
j Θj
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
j Θj
7. On some computational methods for Bayesian model choice
Evidence
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
define priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
j Θj
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
j Θj
8. On some computational methods for Bayesian model choice
Evidence
Bayes factor
Definition (Bayes factors)
Central quantity
Θi fi (x|θi )πi (θi )dθi mi (x)
Bij (x) == =
Θj fj (x|θj )πj (θj )dθj mj (x)
[Jeffreys, 1939]
9. On some computational methods for Bayesian model choice
Evidence
Evidence
All these problems end up with a similar quantity, the evidence
k = 1, . . .
Zk = πk (θk )Lk (θk ) dθk ,
Θk
aka the marginal likelihood.
10. On some computational methods for Bayesian model choice
Importance sampling solutions
Importance sampling revisited
1 Evidence
2 Importance sampling solutions
Bridge sampling
Harmonic means
One bridge back
3 Nested sampling
4 ABC model choice
11. On some computational methods for Bayesian model choice
Importance sampling solutions
Bridge sampling
Bayes factor approximation
When approximating the Bayes factor
f0 (x|θ0 )π0 (θ0 )dθ0
Θ0
B01 =
f1 (x|θ1 )π1 (θ1 )dθ1
Θ1
use of importance functions 0 and 1 and
n−1
0
n0 i i
i=1 f0 (x|θ0 )π0 (θ0 )/
i
0 (θ0 )
B01 =
n−1
1
n1 i i
i=1 f1 (x|θ1 )π1 (θ1 )/
i
1 (θ1 )
13. On some computational methods for Bayesian model choice
Importance sampling solutions
Bridge sampling
Optimal bridge sampling
The optimal choice of auxiliary function is
n1 + n2
α =
n1 π1 (θ|x) + n2 π2 (θ|x)
leading to
n1
1 π2 (θ1i |x)
˜
n1 n1 π1 (θ1i |x) + n2 π2 (θ1i |x)
i=1
B12 ≈ n2
1 π1 (θ2i |x)
˜
n2 n1 π1 (θ2i |x) + n2 π2 (θ2i |x)
i=1
Back later!
14. On some computational methods for Bayesian model choice
Importance sampling solutions
Bridge sampling
Optimal bridge sampling (2)
Reason:
Var(B12 ) 1 π1 (θ)π2 (θ)[n1 π1 (θ) + n2 π2 (θ)]α(θ)2 dθ
2 ≈ 2 −1
B12 n1 n2 π1 (θ)π2 (θ)α(θ) dθ
δ method
Drag: Dependence on the unknown normalising constants solved
iteratively
15. On some computational methods for Bayesian model choice
Importance sampling solutions
Bridge sampling
Optimal bridge sampling (2)
Reason:
Var(B12 ) 1 π1 (θ)π2 (θ)[n1 π1 (θ) + n2 π2 (θ)]α(θ)2 dθ
2 ≈ 2 −1
B12 n1 n2 π1 (θ)π2 (θ)α(θ) dθ
δ method
Drag: Dependence on the unknown normalising constants solved
iteratively
16. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk from a posterior sample
Use of the [harmonic mean] identity
ϕ(θk ) ϕ(θk ) πk (θk )Lk (θk ) 1
Eπk x = dθk =
πk (θk )Lk (θk ) πk (θk )Lk (θk ) Zk Zk
no matter what the proposal ϕ(·) is.
[Gelfand & Dey, 1994; Bartolucci et al., 2006]
Direct exploitation of the MCMC output
17. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk from a posterior sample
Use of the [harmonic mean] identity
ϕ(θk ) ϕ(θk ) πk (θk )Lk (θk ) 1
Eπk x = dθk =
πk (θk )Lk (θk ) πk (θk )Lk (θk ) Zk Zk
no matter what the proposal ϕ(·) is.
[Gelfand & Dey, 1994; Bartolucci et al., 2006]
Direct exploitation of the MCMC output
18. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
RJ-MCMC interpretation
If ϕ is the density of an alternative to model pik (θk )Lk (θk |x), the
acceptance probability for a move from model pik (θk )Lk (θk |x) to
model ϕ(θk ) is based upon
ϕ(θk )
πk (θk )Lk (θk |x)
(under a proper choice of prior weight)
Unbiased estimator of the odds ratio
[Bartolucci, Scaccia, and Mira, 2006]
19. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
An infamous example
When
ϕ(θk ) = πk (θk )
˜
harmonic mean approximation to B12 is
n1
1
1/L1 (θ1i |x)
n1
i=1
B21 = n2 θji ∼ πj (θ|x)
1
1/L2 (θ2i |x)
n2
i=1
[Newton & Raftery, 1994; Raftery et al., 2008]
Infamous: Most often leads to an infinite variance!!!
[Radford Neal’s blog, 2008]
20. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
An infamous example
When
ϕ(θk ) = πk (θk )
˜
harmonic mean approximation to B12 is
n1
1
1/L1 (θ1i |x)
n1
i=1
B21 = n2 θji ∼ πj (θ|x)
1
1/L2 (θ2i |x)
n2
i=1
[Newton & Raftery, 1994; Raftery et al., 2008]
Infamous: Most often leads to an infinite variance!!!
[Radford Neal’s blog, 2008]
21. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
“The Worst Monte Carlo Method Ever”
“The good news is that the Law of Large Numbers guarantees that
this estimator is consistent ie, it will very likely be very close to the
correct answer if you use a sufficiently large number of points from
the posterior distribution.
The bad news is that the number of points required for this
estimator to get close to the right answer will often be greater
than the number of atoms in the observable universe. The even
worse news is that it’s easy for people to not realize this, and to
naively accept estimates that are nowhere close to the correct
value of the marginal likelihood.”
[Radford Neal’s blog, Aug. 23, 2008]
22. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
“The Worst Monte Carlo Method Ever”
“The good news is that the Law of Large Numbers guarantees that
this estimator is consistent ie, it will very likely be very close to the
correct answer if you use a sufficiently large number of points from
the posterior distribution.
The bad news is that the number of points required for this
estimator to get close to the right answer will often be greater
than the number of atoms in the observable universe. The even
worse news is that it’s easy for people to not realize this, and to
naively accept estimates that are nowhere close to the correct
value of the marginal likelihood.”
[Radford Neal’s blog, Aug. 23, 2008]
23. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling
Harmonic mean: Constraint opposed to usual importance sampling
constraints: ϕ(θ) must have lighter (rather than fatter) tails than
πk (θk )Lk (θk ) for the approximation
T (t)
1 ϕ(θk )
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a finite variance.
E.g., use finite support kernels for ϕ
24. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling
Harmonic mean: Constraint opposed to usual importance sampling
constraints: ϕ(θ) must have lighter (rather than fatter) tails than
πk (θk )Lk (θk ) for the approximation
T (t)
1 ϕ(θk )
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a finite variance.
E.g., use finite support kernels for ϕ
25. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
HPD supports
In practice, possibility to use the HPD region defined by an MCMC
sample as
Hα = {θ; π (θ|x) ≥ qα [˜ (·|x)]}
˜ ˆ π
where qα [˜ (·|x)] is the empirical α quantile of the observed
ˆ π
π (θi |x))’s
˜
c Construction of ϕ as uniformly supported by Hα or an ellipsoid
approximation whose volume can be computed
26. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
HPD supports
In practice, possibility to use the HPD region defined by an MCMC
sample as
Hα = {θ; π (θ|x) ≥ qα [˜ (·|x)]}
˜ ˆ π
where qα [˜ (·|x)] is the empirical α quantile of the observed
ˆ π
π (θi |x))’s
˜
c Construction of ϕ as uniformly supported by Hα or an ellipsoid
approximation whose volume can be computed
27. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
HPD supports: illustration
Gibbs sample of 103 parameters (θ, σ 2 ) produced for the normal
model, x1 , . . . , xn ∼ N (θ, σ 2 ) with x = 0, s2 = 1 and n = 10,
under Jeffreys’ prior, leads to a straightforward approximation to
the 10% HPD region, replaced by an ellipsoid.
28. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
HPD supports: illustration
Gibbs sample of 103 parameters (θ, σ 2 ) produced for the normal
model, x1 , . . . , xn ∼ N (θ, σ 2 ) with x = 0, s2 = 1 and n = 10,
under Jeffreys’ prior, leads to a straightforward approximation to
the 50% HPD region, replaced by an ellipsoid.
29. On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling (cont’d)
Compare Z1k with a standard importance sampling approximation
T (t) (t)
1 πk (θk )Lk (θk )
Z2k = (t)
T ϕ(θk )
t=1
(t)
where the θk ’s are generated from the density ϕ(·) (with fatter
tails like t’s)
30. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Approximating Zk using a mixture representation
Bridge sampling recall
Design a specific mixture for simulation [importance sampling]
purposes, with density
ϕk (θk ) ∝ ωπk (θk )Lk (θk ) + ϕ(θk ) ,
where ϕ(·) is arbitrary (but normalised)
Note: ω is not a probability weight
31. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Approximating Zk using a mixture representation
Bridge sampling recall
Design a specific mixture for simulation [importance sampling]
purposes, with density
ϕk (θk ) ∝ ωπk (θk )Lk (θk ) + ϕ(θk ) ,
where ϕ(·) is arbitrary (but normalised)
Note: ω is not a probability weight
32. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(t−1) (t−1) (t−1) (t−1) (t−1)
ωπk (θk )Lk (θk ) ωπk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
33. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(t−1) (t−1) (t−1) (t−1) (t−1)
ωπk (θk )Lk (θk ) ωπk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
34. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(t−1) (t−1) (t−1) (t−1) (t−1)
ωπk (θk )Lk (θk ) ωπk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
35. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ 1
ξ=
(t)
ωπk (θk )Lk (θk )
(t) (t) (t)
ωπk (θk )Lk (θk ) + ϕ(θk ) ,
(t)
T
t=1
converges to ωZk /{ωZk + 1}
3k
ˆ ˆ ˆ
Deduce Zˆ from ω Z3k /{ω Z3k + 1} = ξ ie
T (t) (t) (t) (t) (t)
t=1 ωπk (θk )Lk (θk ) ωπ(θk )Lk (θk ) + ϕ(θk )
ˆ
Z3k =
T (t) (t) (t) (t)
t=1 ϕ(θk ) ωπk (θk )Lk (θk ) + ϕ(θk )
[Bridge sampler redux]
36. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ 1
ξ=
(t)
ωπk (θk )Lk (θk )
(t) (t) (t)
ωπk (θk )Lk (θk ) + ϕ(θk ) ,
(t)
T
t=1
converges to ωZk /{ωZk + 1}
3k
ˆ ˆ ˆ
Deduce Zˆ from ω Z3k /{ω Z3k + 1} = ξ ie
T (t) (t) (t) (t) (t)
t=1 ωπk (θk )Lk (θk ) ωπ(θk )Lk (θk ) + ϕ(θk )
ˆ
Z3k =
T (t) (t) (t) (t)
t=1 ϕ(θk ) ωπk (θk )Lk (θk ) + ϕ(θk )
[Bridge sampler redux]
37. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Mixture illustration
For the normal example, use of the same uniform ϕ over the ellipse
approximating the same HPD region Hα and ω equal to one-tenth
of the true evidence. Gibbs samples of size 104
38. On some computational methods for Bayesian model choice
Importance sampling solutions
One bridge back
Mixture illustration
For the normal example, use of the same uniform ϕ over the ellipse
approximating the same HPD region Hα and ω equal to one-tenth
of the true evidence. Gibbs samples of size 104 and 100 Monte
Carlo replicas
39. On some computational methods for Bayesian model choice
Nested sampling
Nested sampling
1 Evidence
2 Importance sampling solutions
3 Nested sampling
Purpose
Implementation
Error rates
Constraints
4 ABC model choice
40. On some computational methods for Bayesian model choice
Nested sampling
Purpose
Nested sampling: Goal
Skilling’s (2007) technique using the one-dimensional
representation:
1
Z = Eπ [L(θ)] = ϕ(x) dx
0
with
ϕ−1 (l) = P π (L(θ) > l).
Note; ϕ(·) is intractable in most cases.
41. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: First approximation
Approximate Z by a Riemann sum:
N
Z= (xi−1 − xi )ϕ(xi )
i=1
where the xi ’s are either:
deterministic: xi = e−i/N
or random:
x0 = 0, xi+1 = ti xi , ti ∼ Be(N, 1)
so that E[log xi ] = −i/N .
42. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Alternative representation
Another representation is
N −1
Z= {ϕ(xi+1 ) − ϕ(xi )}xi
i=0
which is a special case of
N −1
Z= {L(θ(i+1) ) − L(θ(i) )}π({θ; L(θ) > L(θ(i) )})
i=0
where · · · L(θ(i+1) ) < L(θ(i) ) · · ·
[Lebesgue version of Riemann’s sum]
43. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Alternative representation
Another representation is
N −1
Z= {ϕ(xi+1 ) − ϕ(xi )}xi
i=0
which is a special case of
N −1
Z= {L(θ(i+1) ) − L(θ(i) )}π({θ; L(θ) > L(θ(i) )})
i=0
where · · · L(θ(i+1) ) < L(θ(i) ) · · ·
[Lebesgue version of Riemann’s sum]
44. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Second approximation
Estimate (intractable) ϕ(xi ) by ϕi :
Nested sampling
Start with N values θ1 , . . . , θN sampled from π
At iteration i,
1 Take ϕi = L(θk ), where θk is the point with smallest
likelihood in the pool of θi ’s
2 Replace θk with a sample from the prior constrained to
L(θ) > ϕi : the current N points are sampled from prior
constrained to L(θ) > ϕi .
Note that
1
π({θ; L(θ) > L(θ(i+1) )}) π({θ; L(θ) > L(θ(i) )}) ≈ 1 −
N
45. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Second approximation
Estimate (intractable) ϕ(xi ) by ϕi :
Nested sampling
Start with N values θ1 , . . . , θN sampled from π
At iteration i,
1 Take ϕi = L(θk ), where θk is the point with smallest
likelihood in the pool of θi ’s
2 Replace θk with a sample from the prior constrained to
L(θ) > ϕi : the current N points are sampled from prior
constrained to L(θ) > ϕi .
Note that
1
π({θ; L(θ) > L(θ(i+1) )}) π({θ; L(θ) > L(θ(i) )}) ≈ 1 −
N
46. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Second approximation
Estimate (intractable) ϕ(xi ) by ϕi :
Nested sampling
Start with N values θ1 , . . . , θN sampled from π
At iteration i,
1 Take ϕi = L(θk ), where θk is the point with smallest
likelihood in the pool of θi ’s
2 Replace θk with a sample from the prior constrained to
L(θ) > ϕi : the current N points are sampled from prior
constrained to L(θ) > ϕi .
Note that
1
π({θ; L(θ) > L(θ(i+1) )}) π({θ; L(θ) > L(θ(i) )}) ≈ 1 −
N
47. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Second approximation
Estimate (intractable) ϕ(xi ) by ϕi :
Nested sampling
Start with N values θ1 , . . . , θN sampled from π
At iteration i,
1 Take ϕi = L(θk ), where θk is the point with smallest
likelihood in the pool of θi ’s
2 Replace θk with a sample from the prior constrained to
L(θ) > ϕi : the current N points are sampled from prior
constrained to L(θ) > ϕi .
Note that
1
π({θ; L(θ) > L(θ(i+1) )}) π({θ; L(θ) > L(θ(i) )}) ≈ 1 −
N
48. On some computational methods for Bayesian model choice
Nested sampling
Implementation
Nested sampling: Third approximation
Iterate the above steps until a given stopping iteration j is
reached: e.g.,
observe very small changes in the approximation Z;
reach the maximal value of L(θ) when the likelihood is
bounded and its maximum is known;
truncate the integral Z at level , i.e. replace
1 1
ϕ(x) dx with ϕ(x) dx
0
49. On some computational methods for Bayesian model choice
Nested sampling
Error rates
Approximation error
Error = Z − Z
j 1
= (xi−1 − xi )ϕi − ϕ(x) dx = − ϕ(x) dx
i=1 0 0
j 1
+ (xi−1 − xi )ϕ(xi ) − ϕ(x) dx (Quadrature Error)
i=1
j
+ (xi−1 − xi ) {ϕi − ϕ(xi )} (Stochastic Error)
i=1
[Dominated by Monte Carlo]
50. On some computational methods for Bayesian model choice
Nested sampling
Error rates
A CLT for the Stochastic Error
The (dominating) stochastic error is OP (N −1/2 ):
D
N 1/2 Z − Z → N (0, V )
with
V =− sϕ (s)tϕ (t) log(s ∨ t) ds dt.
s,t∈[ ,1]
[Proof based on Donsker’s theorem]
51. On some computational methods for Bayesian model choice
Nested sampling
Error rates
What of log Z?
If the interest lies within log Z, Slutsky’s transform of the CLT:
D
N −1/2 log Z − log Z → N 0, V /Z2
Note: The number of simulated points equals the number of
iterations j, and is a multiple of N : if one stops at first iteration j
such that e−j/N < , then:
j = N − log .
52. On some computational methods for Bayesian model choice
Nested sampling
Error rates
What of log Z?
If the interest lies within log Z, Slutsky’s transform of the CLT:
D
N −1/2 log Z − log Z → N 0, V /Z2
Note: The number of simulated points equals the number of
iterations j, and is a multiple of N : if one stops at first iteration j
such that e−j/N < , then:
j = N − log .
53. On some computational methods for Bayesian model choice
Nested sampling
Error rates
What of log Z?
If the interest lies within log Z, Slutsky’s transform of the CLT:
D
N −1/2 log Z − log Z → N 0, V /Z2
Note: The number of simulated points equals the number of
iterations j, and is a multiple of N : if one stops at first iteration j
such that e−j/N < , then:
j = N − log .
54. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Sampling from constr’d priors
Exact simulation from the constrained prior is intractable in most
cases
Skilling (2007) proposes to use MCMC but this introduces a bias
(stopping rule)
If implementable, then slice sampler can be devised at the same
cost [or less]
55. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Sampling from constr’d priors
Exact simulation from the constrained prior is intractable in most
cases
Skilling (2007) proposes to use MCMC but this introduces a bias
(stopping rule)
If implementable, then slice sampler can be devised at the same
cost [or less]
56. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Sampling from constr’d priors
Exact simulation from the constrained prior is intractable in most
cases
Skilling (2007) proposes to use MCMC but this introduces a bias
(stopping rule)
If implementable, then slice sampler can be devised at the same
cost [or less]
57. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Banana illustration
Case of a banana target made of a twisted 2D normal:
x2 = x2 + βx2 − 100β
1
[Haario, Sacksman, Tamminen, 1999]
β = .03, Σ = diag(100, 1)
58. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Banana illustration (2)
Use of nested sampling with N = 1000, 50 MCMC steps with size
0.1, compared with a population Monte Carlo (PMC) based on 10
iterations with 5000 points per iteration and final sample of 50000
points, using nine Student’s t components with 9 df
[Wraith, Kilbinger, Benabed et al., 2009, Physica Rev. D]
Evidence estimation
59. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Banana illustration (2)
Use of nested sampling with N = 1000, 50 MCMC steps with size
0.1, compared with a population Monte Carlo (PMC) based on 10
iterations with 5000 points per iteration and final sample of 50000
points, using nine Student’s t components with 9 df
[Wraith, Kilbinger, Benabed et al., 2009, Physica Rev. D]
E[X1 ] estimation
60. On some computational methods for Bayesian model choice
Nested sampling
Constraints
Banana illustration (2)
Use of nested sampling with N = 1000, 50 MCMC steps with size
0.1, compared with a population Monte Carlo (PMC) based on 10
iterations with 5000 points per iteration and final sample of 50000
points, using nine Student’s t components with 9 df
[Wraith, Kilbinger, Benabed et al., 2009, Physica Rev. D]
E[X2 ] estimation
61. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
Approximate Bayesian Computation
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) , x ∼ f (x|θ ) ,
until the auxiliary variable x is equal to the observed value, x = y.
[Pritchard et al., 1999]
62. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
Approximate Bayesian Computation
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) , x ∼ f (x|θ ) ,
until the auxiliary variable x is equal to the observed value, x = y.
[Pritchard et al., 1999]
63. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
Approximate Bayesian Computation
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) , x ∼ f (x|θ ) ,
until the auxiliary variable x is equal to the observed value, x = y.
[Pritchard et al., 1999]
64. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
A as approximative
When y is a continuous random variable, equality x = y is replaced
with a tolerance condition,
(x, y) ≤
where is a distance between summary statistics
Output distributed from
π(θ) Pθ { (x, y) < } ∝ π(θ| (x, y) < )
65. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
A as approximative
When y is a continuous random variable, equality x = y is replaced
with a tolerance condition,
(x, y) ≤
where is a distance between summary statistics
Output distributed from
π(θ) Pθ { (x, y) < } ∝ π(θ| (x, y) < )
66. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
Dynamic area
67. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC improvements
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
[Ratmann et al., 2009]
68. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC improvements
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
[Ratmann et al., 2009]
69. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC improvements
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
[Ratmann et al., 2009]
70. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC improvements
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
[Ratmann et al., 2009]
71. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-MCMC
Markov chain (θ(t) ) created via the transition function
θ ∼ K(θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y
(t+1) π(θ )K(θ(t) |θ )
θ = and u ∼ U(0, 1) ≤ π(θ(t) )K(θ |θ(t) ) ,
(t)
θ otherwise,
has the posterior π(θ|y) as stationary distribution
[Marjoram et al, 2003]
72. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-MCMC
Markov chain (θ(t) ) created via the transition function
θ ∼ K(θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y
(t+1) π(θ )K(θ(t) |θ )
θ = and u ∼ U(0, 1) ≤ π(θ(t) )K(θ |θ(t) ) ,
(t)
θ otherwise,
has the posterior π(θ|y) as stationary distribution
[Marjoram et al, 2003]
73. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC
Another sequential version producing a sequence of Markov
(t) (t)
transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T )
ABC-PRC Algorithm
(t−1)
1 Pick a θ is selected at random among the previous θi ’s
(t−1)
with probabilities ωi (1 ≤ i ≤ N ).
2 Generate
(t) (t)
θi ∼ Kt (θ|θ ) , x ∼ f (x|θi ) ,
3 Check that (x, y) < , otherwise start again.
[Sisson et al., 2007]
74. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC
Another sequential version producing a sequence of Markov
(t) (t)
transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T )
ABC-PRC Algorithm
(t−1)
1 Pick a θ is selected at random among the previous θi ’s
(t−1)
with probabilities ωi (1 ≤ i ≤ N ).
2 Generate
(t) (t)
θi ∼ Kt (θ|θ ) , x ∼ f (x|θi ) ,
3 Check that (x, y) < , otherwise start again.
[Sisson et al., 2007]
75. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
Inspired from Del Moral et al. (2006), who use backward kernels
Lt−1 in SMC to achieve unbiasedness with a recent (2008)
ABC-SMC version
76. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
Inspired from Del Moral et al. (2006), who use backward kernels
Lt−1 in SMC to achieve unbiasedness with a recent (2008)
ABC-SMC version
77. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
Inspired from Del Moral et al. (2006), who use backward kernels
Lt−1 in SMC to achieve unbiasedness with a recent (2008)
ABC-SMC version
78. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
Inspired from Del Moral et al. (2006), who use backward kernels
Lt−1 in SMC to achieve unbiasedness with a recent (2008)
ABC-SMC version
79. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC bias
Lack of unbiasedness of the method
Joint density of the accepted pair (θ(t−1) , θ(t) ) proportional to
π(θ(t−1) |y)Kt (θ(t) |θ(t−1) )f (y|θ(t) ) ,
For an arbitrary function h(θ), E[ωt h(θ(t) )] proportional to
π(θ (t) )Lt−1 (θ (t−1) |θ (t) )
ZZ
(t) (t−1) (t) (t−1) (t) (t−1) (t)
h(θ ) π(θ |y)Kt (θ |θ )f (y|θ )dθ dθ
π(θ (t−1) )Kt (θ (t) |θ (t−1) )
π(θ (t) )Lt−1 (θ (t−1) |θ (t) )
ZZ
(t) (t−1) (t−1)
∝ h(θ ) π(θ )f (y|θ )
π(θ (t−1) )Kt (θ (t) |θ (t−1) )
(t) (t−1) (t) (t−1) (t)
× Kt (θ |θ )f (y|θ )dθ dθ
Z Z ff
(t) (t) (t−1) (t) (t−1) (t−1) (t)
∝ h(θ )π(θ |y) Lt−1 (θ |θ )f (y|θ )dθ dθ .
80. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
ABC-PRC bias
Lack of unbiasedness of the method
Joint density of the accepted pair (θ(t−1) , θ(t) ) proportional to
π(θ(t−1) |y)Kt (θ(t) |θ(t−1) )f (y|θ(t) ) ,
For an arbitrary function h(θ), E[ωt h(θ(t) )] proportional to
π(θ (t) )Lt−1 (θ (t−1) |θ (t) )
ZZ
(t) (t−1) (t) (t−1) (t) (t−1) (t)
h(θ ) π(θ |y)Kt (θ |θ )f (y|θ )dθ dθ
π(θ (t−1) )Kt (θ (t) |θ (t−1) )
π(θ (t) )Lt−1 (θ (t−1) |θ (t) )
ZZ
(t) (t−1) (t−1)
∝ h(θ ) π(θ )f (y|θ )
π(θ (t−1) )Kt (θ (t) |θ (t−1) )
(t) (t−1) (t) (t−1) (t)
× Kt (θ |θ )f (y|θ )dθ dθ
Z Z ff
(t) (t) (t−1) (t) (t−1) (t−1) (t)
∝ h(θ )π(θ |y) Lt−1 (θ |θ )f (y|θ )dθ dθ .
81. On some computational methods for Bayesian model choice
ABC model choice
ABC method(s)
A mixture example (0)
Toy model of Sisson et al. (2007): if
θ ∼ U(−10, 10) , x|θ ∼ 0.5 N (θ, 1) + 0.5 N (θ, 1/100) ,
then the posterior distribution associated with y = 0 is the normal
mixture
θ|y = 0 ∼ 0.5 N (0, 1) + 0.5 N (0, 1/100)
restricted to [−10, 10].
Furthermore, true target available as
π(θ||x| < ) ∝ Φ( −θ)−Φ(− −θ)+Φ(10( −θ))−Φ(−10( +θ)) .
83. On some computational methods for Bayesian model choice
ABC model choice
ABC-PMC
A PMC version
Use of the same kernel idea as ABC-PRC but with IS correction
Generate a sample at iteration t by
N
(t−1) (t−1)
πt (θ(t) ) ∝
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]
84. On some computational methods for Bayesian model choice
ABC model choice
ABC-PMC
A mixture example (2)
Recovery of the target, whether using a fixed standard deviation of
τ = 0.15 or τ = 1/0.15,
85. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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
86. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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
87. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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{θT S(x)}
x∈X
involves too many terms to be manageable and numerical
approximations cannot always be trusted
[Cucala, Marin, CPR & Titterington, 2009]
88. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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{θT S(x)}
x∈X
involves too many terms to be manageable and numerical
approximations cannot always be trusted
[Cucala, Marin, CPR & Titterington, 2009]
89. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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
T
exp{θ0 S0 (x)}/Zθ0 ,0 π0 (dθ0 )
Bm0 /m1 (x) = T
exp{θ1 S1 (x)}/Zθ1 ,1 π1 (dθ1 )
90. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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.
91. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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
92. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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
93. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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!]
94. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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!]
95. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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!]
96. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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∗ )) < .
∗
[Cornuet, Grelaud, Marin & Robert, 2009]
Note When = 0 the algorithm is exact
97. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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).
98. On some computational methods for Bayesian model choice
ABC model choice
ABC for model choice in GRFs
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).
99. On some computational methods for Bayesian model choice
ABC 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).
100. On some computational methods for Bayesian model choice
ABC model choice
Illustrations
Toy example (2)
10
5
5
BF01
BF01
0
^
^
0
−5
−5
−10
−40 −20 0 10 −40 −20 0 10
BF01 BF01
(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.
101. On some computational methods for Bayesian model choice
ABC model choice
Illustrations
Protein folding
Superposition of the native structure (grey) with the ST1
structure (red.), the ST2 structure (orange), the ST3 structure
(green), and the DT structure (blue).
102. On some computational methods for Bayesian model choice
ABC model choice
Illustrations
Protein folding (2)
% seq . Id. TM-score FROST score
1i5nA (ST1) 32 0.86 75.3
1ls1A1 (ST2) 5 0.42 8.9
1jr8A (ST3) 4 0.24 8.9
1s7oA (DT) 10 0.08 7.8
Characteristics of dataset. % seq. Id.: percentage of identity with
the query sequence. TM-score.: similarity between predicted and
native structure (uncertainty between 0.17 and 0.4) FROST score:
quality of alignment of the query onto the candidate structure
(uncertainty between 7 and 9).
103. On some computational methods for Bayesian model choice
ABC model choice
Illustrations
Protein folding (3)
NS/ST1 NS/ST2 NS/ST3 NS/DT
BF 1.34 1.22 2.42 2.76
P(M = NS|x0 ) 0.573 0.551 0.708 0.734
Estimates of the Bayes factors between model NS and models
ST1, ST2, ST3, and DT, and corresponding posterior
probabilities of model NS based on an ABC-MC algorithm using
1.2 106 simulations and a tolerance equal to the 1% quantile of
the distances.