MUMS: Bayesian, Fiducial, and Frequentist Conference - Model Selection in the Context of Computer Models, Rui Paulo, April 30, 2019
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The document discusses variable selection in computer models, focusing on the importance of identifying active inputs that significantly impact model outputs. It explores the use of Gaussian processes and proposes a jointly robust prior to address model selection and to improve the detection of inert inputs compared to traditional reference priors. Computational challenges and the need for robust emulation techniques in the context of Bayesian models are emphasized, along with results from various test cases demonstrating the efficacy of the proposed approaches.
![Laplace approximation
The jointly robust prior does not allow for a closed form
expression for m(y | δ)
There is a large literature on computing (ratios of) normalizing
constants, often relying on MCMC samples from the posterior
Our goal was to explore the possibility of using the Laplace
approximation to compute m(y | δ), given that there is code
available to obtain ˆβk, k = 1, . . . , p — R package
RobustGaSP [Gu, Palomo and Berger, 2018] — and these
possess nice properties](https://image.slidesharecdn.com/rpaulo-contextofcomputer-190502143729/85/MUMS-Bayesian-Fiducial-and-Frequentist-Conference-Model-Selection-in-the-Context-of-Computer-Models-Rui-Paulo-April-30-2019-12-320.jpg)
![Laplace approximation (cont.d)
For each model indexed by δ = 0,
m(y) ∝ exp[h(β)] dβ
where, with π(β) denoting the JR prior,
h(β) = ln L(β | y) + ln π(β)
and
L(β | y) ∝ (yT
R−1
y)−n/2
|R|−1/2](https://image.slidesharecdn.com/rpaulo-contextofcomputer-190502143729/85/MUMS-Bayesian-Fiducial-and-Frequentist-Conference-Model-Selection-in-the-Context-of-Computer-Models-Rui-Paulo-April-30-2019-13-320.jpg)
![Laplace approximation (and BIC)
Having obtained ˆβ = arg max h(β), expand h(ˆβ) around that
mode to obtain (up to a constant)
m(y) ≈ (2π)k/2
exp[h(ˆβ)] |H|−1/2
H = −
∂2h
∂β∂βT
(ˆβ)
where k = 1T
δ is the number of inputs in model indexed by
δ = 0.
One can obtain explicit formulae for all these quantities
(except for ˆβ!) that are functions of
R−1
,
∂R
∂βj
,
∂2R
∂βi ∂βj
One also has all the quantities to compute BIC-based
posterior marginals:
m(y) ≈ n−k/2
L(ˆβ)](https://image.slidesharecdn.com/rpaulo-contextofcomputer-190502143729/85/MUMS-Bayesian-Fiducial-and-Frequentist-Conference-Model-Selection-in-the-Context-of-Computer-Models-Rui-Paulo-April-30-2019-14-320.jpg)
![Testbeds
Linkletter et al. 2006, Technometrics deals with variable selection
in the context of computer models.
They consider several examples:
Sinusoidal: y(x1, . . . , x10) = sin(x1) + sin(5x2)
Simple linear model:
y(x1, . . . , x10) = 0.2x1 + 0.2x2 + 0.2x3 + 0.2x4
Decreasing impact: y(x1, . . . , x10) = 8
i=1 0.2/2i−1xi
No signal: y(x1, . . . , x10) = 0
The model data is obtained with a 54-run maximin Latin
hypercube design in [0, 1]10 and iid N(0, 0.052) noise is added to
the model runs.](https://image.slidesharecdn.com/rpaulo-contextofcomputer-190502143729/85/MUMS-Bayesian-Fiducial-and-Frequentist-Conference-Model-Selection-in-the-Context-of-Computer-Models-Rui-Paulo-April-30-2019-15-320.jpg)
MUMS: Bayesian, Fiducial, and Frequentist Conference - Model Selection in the Context of Computer Models, Rui Paulo, April 30, 2019
- 1. Variable selection in the context of computer models MUMS Foundations of Model Uncertainty Gonzalo Garcia-Donato1 and Rui Paulo2 1 Universidad de Castilla-La Mancha (Spain), 2 Universidade de Lisboa (Portugal) Duke University, BFF conference, April 30 2019
- 2. Screening Computer model Let y(·) denote the output of the computer model with generic input x ∈ S ⊂ IRp . Screening The question we want to answer is: Which are the inputs that significantly impact the output, the so-called active inputs. The other inputs are called inert.
- 3. Gaussian process prior We place a Gaussian process prior on y(·): y(·) | θ, σ2 ∼ GP(0, σ2 c(·, ·)) with correlation function c(x, x ) = p k=1 ck(xk, xk ) . We will assume that ck is a one-dimensional correlation function with fixed (known) roughness parameter and unknown range parameter γk > 0. We adopt the Matern 5/2 correlation function: with dk = |xk − xk | ck(xk, xk ) = ck(dk) = 1 + √ 5 dk γk + 5 d2 k 3 γ2 k exp − √ 5 dk γk so that θ = (γ1, . . . , γp).
- 4. Our approach to screening The information to address the screening question comes in the form of model data: a design D = {x1, . . . , xn} is selected, the computer model is run at each of these configurations and we define y = (y(x1), . . . , y(xn)) . Consider all 2p models for y that result from allowing only a subset of the p inputs to be active: Let δ = (δ1, . . . , δp) ∈ {0, 1}p identify each of the subsets If δ = 0, Mδ : y | σ2, θ ∼ N(0, σ2Rδ) where Rδ = k:δk =1 ck(xki , xkj ) i,j=1,...,n If δ = 0, Mδ : y | σ2, θ ∼ N(0, σ2I) Screening is now tantamount to assessing the support that y lends to each of the Mδ, i.e. a model selection exercise
- 5. Parametrizations In multiple linear regression, the 2p models are formally obtained by setting to zero subsets of the vector of regression coefficients of the full model Here, it depends on the parametrization used for the correlation function: γk — range parameter: xk is inert ⇔ γk → +∞ βk = 1/γk — inverse range parameter: xk is inert ⇔ βk = 0 ξk = ln γk — log-inverse range parameter: xk is inert ⇔ ξk → −∞ These parameterizations are free of the roughness parameter (Gu et al. 2018)
- 6. Bayes factors and posterior inclusion probabilitites Our answer to the screening question is then obtained by considering the marginal likelihoods m(y | δ) ≡ N(y | 0, σ2 Rδ) π(θ, σ2 | δ)dσ2 dθ which allow us to compute Bayes factors and, paired with prior model probabilities, posterior model probabilities, p(δ | y). Marginal posterior inclusion probabilities are particularly interesting in this context: p(xk | y) = δ: δk =1 p(δ | y)
- 7. Difficulties Conceptual: selecting π(θ, σ2 | δ) Computational: computing the integral in m(y | δ) If p is large, enumeration of all models is not practical, so obtaining p(xk | y) is problematic
- 8. Priors for Gaussian processes Berger, De Oliveira and Sans´o, 2001, JASA is a seminal paper c(x, x ) = c(||x − x ||, θ) with θ a unidimensional range parameter Focus on spatial statistics Some of the commonly used priors give rise to improper posteriors Reference prior is derived and posterior propriety is proved π(θ, σ2 ) = π(θ)/σ2 with π(θ) proper has long as the mean of the GP has a constant term P., 2005, AoS Focus on emulation of computer models c(x, x ) = k c(|xk − xk |, θk ) Reference prior is obtained and posterior propriety established when D is a cartesian product Many extensions are considered, e.g. to include a nugget, to focus on spatial models etc. Gu, Wang and Berger, 2018, AoS focuses on robust emulation Gu 2018, BA, the jointly robust prior
- 9. Priors for Gaussian processes The reference prior is proper for many correlation functions as long as there is an unknown mean, but it changes very little if the GP has zero mean It is hence very appealing for variable selection; however, the normalizing constant is unknown (and we need it) it is computationally very costly Gu (2018) jointly robust prior π(β) = C0 p k=1 Ckβk a exp −b p k=1 Ckβk where C0 is known, mimics the tail behavior of the reference prior This talk is about using the jointly robust prior to answer the screening problem
- 10. Robust emulation Emulation: using y(·) | y in lieu of the actual computer model Obtaining the posterior π(θ | y) via MCMC and using it to get to y(·) | y is not always practical The alternative is to compute an estimate of θ and plug it in the conditional posterior predictive of y(·) Problems arise because often a) ˆR ≈ I, prediction reverts to the mean, becoming an impulse function close to y b) ˆR ≈ 11T and numerical instability increases Gu et al. (2018) terms this phenomenon lack of robustness and because of the product nature of the correlation function it happens when a) ˆγk → 0 for at least one k b) ˆγk → +∞ for all k
- 11. Robust emulation (cont.) Gu et al. (2018) shows that if the ˆγk are computed by maximizing f (y | γ) π(γ) where π(γ, σ2, µ) ∝ π(γ)/σ2 is the reference prior, then robustness is guaranteed Remarks: Since γk → +∞ means that xk is inert, this means that the reference prior will not be appropriate to detect through the marginal posterior modes the situation where all the inputs are inert The reference prior will not detect either the situation where some of the inputs are inert (by looking at marginal posterior modes) The mode is not invariant with respect to reparametrization, hence the parametrization matters Gu et al. (2018) discourage the use of βk
- 12. Laplace approximation The jointly robust prior does not allow for a closed form expression for m(y | δ) There is a large literature on computing (ratios of) normalizing constants, often relying on MCMC samples from the posterior Our goal was to explore the possibility of using the Laplace approximation to compute m(y | δ), given that there is code available to obtain ˆβk, k = 1, . . . , p — R package RobustGaSP [Gu, Palomo and Berger, 2018] — and these possess nice properties
- 13. Laplace approximation (cont.d) For each model indexed by δ = 0, m(y) ∝ exp[h(β)] dβ where, with π(β) denoting the JR prior, h(β) = ln L(β | y) + ln π(β) and L(β | y) ∝ (yT R−1 y)−n/2 |R|−1/2
- 14. Laplace approximation (and BIC) Having obtained ˆβ = arg max h(β), expand h(ˆβ) around that mode to obtain (up to a constant) m(y) ≈ (2π)k/2 exp[h(ˆβ)] |H|−1/2 H = − ∂2h ∂β∂βT (ˆβ) where k = 1T δ is the number of inputs in model indexed by δ = 0. One can obtain explicit formulae for all these quantities (except for ˆβ!) that are functions of R−1 , ∂R ∂βj , ∂2R ∂βi ∂βj One also has all the quantities to compute BIC-based posterior marginals: m(y) ≈ n−k/2 L(ˆβ)
- 15. Testbeds Linkletter et al. 2006, Technometrics deals with variable selection in the context of computer models. They consider several examples: Sinusoidal: y(x1, . . . , x10) = sin(x1) + sin(5x2) Simple linear model: y(x1, . . . , x10) = 0.2x1 + 0.2x2 + 0.2x3 + 0.2x4 Decreasing impact: y(x1, . . . , x10) = 8 i=1 0.2/2i−1xi No signal: y(x1, . . . , x10) = 0 The model data is obtained with a 54-run maximin Latin hypercube design in [0, 1]10 and iid N(0, 0.052) noise is added to the model runs.
- 16. Sinusoidal function — initial results Upon trying to obtain the Laplace approximation for all 2p − 1 = 1023 models, We encountered models where the mode had entries that were at the boundary: ˆβk = 0 Those modes were of two types: hk < 0 or hk > 0 but non-zero gradient This precludes the use of the usual Laplace approximation, as the mode must be an interior point of the parameter space
- 17. Variable selection: two approaches There are mainly two approaches to variable selection: Estimation-based The full model is assumed to be true and the prior (or penalty) encourages sparcity; a criterion is established for determining whether a variable is included or not Model selection-based, in which formally all possible models are considered, which is what we are trying to accomplish here The issue of multiple comparisons is present, but the priors for the parameters are one might say of a different nature The jointly robust prior differs from the reference prior in that it can detect the case where some of the inputs are inert, whereas the reference prior cannot This is was what we are observing: the jointly robust prior encourages sparcity
- 18. The reference prior revisited... The motivation for the JR prior is that is that it matches the “expontential and polynomial tail decaying rates of the reference prior” Exponential decay when γ → 0 prevents R to be near diagonal Polynomial decay when γ → +∞ allows the likelihood to come into play — large values usually produce better fit Which polynomial? If γi → +∞ for i ∈ E {x1, . . . , xp} and the other γi are bounded, πR (γ) i∈E γ−3 i or πR (β) i∈E βi This motivates considering πJR (β) ∝ p i=1 βi πJR (β)
- 19. Sinusoidal example revisited... This prior pushes the βk away from zero, so no modes at the boundary were found Some of the modes were pushed away from zero an order of magnitude so the inputs deemed more relevant The marginal posteriors of the vectors with some components of the mode at zero did not seem to change much The marginal posteriors of models with some support did not seem to change much either
- 20. Sinusoidal example results Recall that only x1 and x2 are active; the prior on the model space was constant. With our modification of the jointly robust prior, these were the results obtained: x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 p(xk | y) 1 1 0.0808 0.0358 0.5720 0.1917 0.2891 0.7042 0.8249 0.0019 The true model was ranked in 44th place with pmp 0.00011. δ p(δ | y) x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 404 0.3235 1 1 0 0 1 0 0 1 1 0 452 0.2567 1 1 0 0 0 0 1 1 1 0 52 0.1442 1 1 0 0 1 1 0 0 0 0 408 0.0510 1 1 1 0 1 0 0 1 1 0 292 0.0422 1 1 0 0 0 1 0 0 1 0 260 0.03760 1 1 0 0 0 0 0 0 1 0
- 21. The modified prior The culprit is not the inaccuracy of the Laplace approximation correlations in the design the prior or its hyperparameters it’s the model! We need a nugget, for δ = 0, Mδ : y | σ2 , σ2 0, θ ∼ N(0, σ2 Rδ + σ2 0I) The GP prior implies that the emulator is an interpolator at the observed data and that no longer happens when we remove inputs from the correlation matrix, hence defining an inadequate model...
- 22. Adding the nugget By introducing η = σ2 0/σ2, the reference prior is now π(β, η, σ2 ) ∝ π(β, η) σ2 and σ2 can be integrated out The jointly robust prior is π(β, η) = C0 p k=1 Ckβk + η a exp −b p k=1 Ckβk + η And the tail of the reference prior γi → +∞ for i ∈ E {x1, . . . , xp}, η → 0, and the other γi are bounded, πR (γ, η) i∈E γ−3 i or πR (β, η) i∈E βi i.e., there is no penalization in η
- 23. The modified prior For robustness, Gu et al. (2018) recommend the parametrization in terms of (ln β, ln η) = (ξ, τ), so that we set π(β, η) ∝ p k=1 βk p k=1 Ckβk + η a exp −b p k=1 Ckβk + η but do all the calculations (mode and Laplace approximation) in the (ξ, τ) parametrization We next show the results of the four examples (M = 250 simulations) the Laplace-based inclusion probabilities using a flat prior on the model space the Laplace-based inclusion probabilities using a Scott and Berger (2010) prior on the model space the BIC-based inclusion probabilities using a flat prior on the model space
- 24. q q q q q q q q q q q 0 2 4 6 8 10 0.000.020.040.060.08 p P(M_p) Flat vs Scott and Berger
- 25. qq q qqqq q qq q q q q q qqq q q qqqqq q qqqqqq q q q qq qq q qqqqq q q qq q qqqq qqq q qq q qqq q qqqq q q qqqqqq q qq q q q q q qq q qq q q q qq qqq q q q qq q qqq q qq q q q qqq q q q q q qq q q q q qq qqq q q qqqqq q q qqqq qqqq q q q q qqq q q qqq qq q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − flat prior qq q q qq q q q qq q q q q q qqq q q q q q qq q q q q q qq q qq q q q q q q qqqq q q q qqq q q q qq q q q qq q q q qq q q qqq q q q q q q q q q q qqqq q q q qqq qq q q q q q qq q q q q qq q qq q qq q q q q q q q q q q q qq qq q qq q q q q q qqq q q q q qq qq q q qq q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − SB prior qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq q q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqq qqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqq q qqq qqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 BIC − flat prior simple linear model
- 26. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqq q qqqqqq q q q q q qqq qq q qq q q qq q qq q qq q qqqqqq q q qqqq q qq q q q q qqq q q q qq q q q q q q q qqq q q qqqqqqq q q q q q q qq q qqq q q q q q qq qqqqqq q qqqq qqq qq qq qq q qq qqq qq q qqq qq q qq q q q q q q qqqqq q q qqq q q q q q q q qq q qq q q q qq qq qq q q qqqqqqqq q q q qqq q qqqqqqqqqq qq qqq q qqq qq qq q qqqq q q q q q q q qq qqq q qqq q qqqqqqq q q q q qq q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − flat prior qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqq q qqqqqq q q q q q qqqqq q qq q q qq q qqqqq q qqqqqq q q qqqq q qqqqq q qqq qqq qq q q qqq q q qqqq q qqqqqqq q qqqqq qqqqqq q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqq q qq q q q q q q qqqqq qq qqq q q q q qqqqq qqq q q q qqqqqq q q qqqqqqqq q q qqqq q qqqqqqqqqqqqqqqqqqqqqqq q qqqq q q qq q q q qqqqq q qqqqqqqqqqqqqqq qq qq x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − SB prior qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q qq x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 BIC − flat prior sinusoidal model
- 27. q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q qq qqqq qqqqq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − flat prior q q q q q q q q q q q q q q q qqq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − SB prior q q qqqq q qq q q q q q qqqqq q qqqq q qqqq q q q qqq q qq q q q q q q q q q qq qqq q qqqqq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 BIC − flat prior decreasing impact
- 28. q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − flat prior q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 Laplace − SB prior q q q q q q q q q q q q qq q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q x_1 x_3 x_5 x_7 x_9 0.00.20.40.60.81.0 BIC − flat prior no signal
- 29. Work in progress In the simulations, we find models for which H has negative entries in the diagonal. Upon inspection, the marginals seem well behaved, so this is numerical instability — we need more stable ways of computing H Investigate the possibility of using this method in variable selection in the bias function Produce an R package
- 30. Summary Fully automatic formal Bayesian variable selection approach to screening which is computationally very simple Based on a correlation function that possesses atractive properties and on a prior which mimics the behavior of the reference prior Need for a nugget when considering all 2p models BIC behaves quite well and does not require H Thank you for your attention!