Source-linked AI summary
ABC likelihood-freee methods for model choice in Gibbs random fields
Aude Grelaud, Christian Robert, Jean-Michel Marin, Francois Rodolphe, Jean-Francois Taly
TL;DR
Bayesian comparison of Gibbs random-field dependence structures is impeded by unavailable normalising constants. The paper develops an ABC procedure for model choice, exploiting a sufficient statistic shared across models and improving approximation through importance sampling. Applications cover Bernoulli-versus-Markov selection and protein-folding structures, with results supporting accurate posterior and Bayes-factor estimation and discrimination between similar and dissimilar structures.
Problem
Unavailable normalising constants make Gibbs-model posterior probabilities and Bayes factors difficult to compute for model choice.
Method
The paper uses ABC for model choice, exploiting a sufficient statistic across Gibbs models and improving approximation through importance sampling on model distributions.
Results
The method gives good posterior-probability and Bayes-factor fits in the toy example and distinguishes similar from dissimilar protein structures.
Takeaways & Limitations
ABC model choice can address unavailable Gibbs normalising constants without reversible-jump moves, while shared sufficiency can avoid the usual ABC approximation.
Takeaways & Limitations
When more than two models are considered, approximations used for Bayesian model averaging can be dangerous.
Abstract
from arXiv · showhide
Gibbs random fields (GRF) are polymorphous statistical models that can be used to analyse different types of dependence, in particular for spatially correlated data. However, when those models are faced with the challenge of selecting a dependence structure from many, the use of standard model choice methods is hampered by the unavailability of the normalising constant in the Gibbs likelihood. In particular, from a Bayesian perspective, the computation of the posterior probabilities of the models under competition requires special likelihood-free simulation techniques like the Approximate Bayesian Computation (ABC) algorithm that is intensively used in population genetics. We show in this paper how to implement an ABC algorithm geared towards model choice in the general setting of Gibbs random fields, demonstrating in particular that there exists a sufficient statistic across models. The accuracy of the approximation to the posterior probabilities can be further improved by importance sampling on the distribution of the models. The practical aspects of the method are detailed through two applications, the test of an iid Bernoulli model versus a first-order Markov chain, and the choice of a folding structure for two proteins.
1 Introduction
Gibbs random fields model dependence structures, including spatial dependence, but Bayesian model comparison is hindered by unavailable normalising constants. The paper motivates likelihood-free ABC methods for estimating model posterior probabilities and Bayes factors in this setting.
- 1.1 Gibbs random fields: Gibbs random fields represent dependence through potentials defined on cliques of an underlying neighbourhood graph.The graph specifies neighbouring sites and cliques, while the potential uses a scale parameter and sufficient-statistic function.
- 1.2 Bayesian model choice: Bayesian model selection is difficult because the Gibbs likelihood contains a normalising constant that is unavailable or computationally unmanageable in realistic settings.Existing numerical approximations may require heavy computation or lack sufficient accuracy.
- 1.2 Bayesian model choice: Unknown normalising constants make both model posterior probabilities and Bayes factors difficult to compute.The paper therefore targets likelihood-free estimation of these quantities.
- 1.2 Bayesian model choice: The paper proposes ABC-based model selection and discusses improved accuracy through an importance-sampling procedure.The motivating applications include testing an iid sequence against a Markov chain model.
- 1.1 Gibbs random fields: GRFs are used to model dependence in spatially correlated data, including applications in epidemiology and image analysis.
2 Methods
The paper develops ABC for Bayesian model choice among Gibbs random fields, where unknown normalising constants obstruct standard posterior and Bayes-factor calculations. Concatenated model-specific sufficient statistics remain sufficient across GRF models, enabling likelihood-free model selection and further importance-sampling refinement.
- Shared sufficient statistic: The concatenated vector S(x)=(S_0(x),…,S_{M−1}(x)) is sufficient for the joint model index and parameters in Gibbs random fields.This follows from the GRF factorisation and the model-independent cardinality of configurations sharing the same statistic; the property is specific to GRF models.
- Scope of the construction: The shared-statistic construction is not generally available for competing generic exponential-family models.For such models, the concatenated statistic may fail to be sufficient for the model index, unlike in the GRF setting.
- ABC for model choice: ABC replaces unavailable likelihood calculations with tolerance-based simulation, accepting parameter–model draws when simulated and observed summary statistics are sufficiently close.The tolerance ε controls approximation quality: ε=0 is exact when the shared statistic is sufficient, while larger tolerances increase approximation error.
- ABC for model choice: ABC-MC samples a model index, its parameter, and simulated data, then accepts the draw when ρ(S(x0),S(x*))<ε.Posterior model probabilities are estimated from empirical frequencies of accepted model indices.
- Posterior and Bayes-factor estimation: The accepted model frequencies provide estimates of posterior probabilities and can be combined to approximate Bayes factors.A substitute estimator addresses the case where the direct ratio is undefined because no accepted draw visits one competing model.
- Accuracy refinement: The bias of the Bayes-factor estimator is {1−(N+2)(1−p)^(N+1)}/((N+1)p), and it converges to zero as N increases.The two-step correction can stabilize estimates when the first run produces a very large Bayes factor, but it may help little when a competing model is never sampled.
3 Results
The toy comparison shows that ABC can closely approximate posterior model probabilities and Bayes factors, while the protein application uses Gibbs random fields to distinguish candidate structures by similarity to native folds.
- Toy example: The Bernoulli-versus-Markov experiment evaluates ABC against exact posterior probabilities and Bayes factors using 2,000 simulated datasets and 4 × 10^6 proposals.The exact posterior quantities are available because both models have tractable normalising constants.
- Toy example: ABC posterior probabilities fit well across values of P(M = 0|x0), with only slight differences away from probabilities near 0, 1, or 0.5 under the 1% tolerance.The exact-match case used accepted simulations satisfying S(x*) = S(x0).
- Toy example: Bayes-factor estimates fit well in the exact case, with larger discrepancies mainly at extreme Bayes factors and occasional divergences from very small acceptance rates.Near log BF m0/m1(x0) = 0, the reported differences are small.
- Toy example: The median ratio of estimated to exact Bayes factors is very close to 1 for both exact and 1% tolerance cases.The exact case shows more extreme ratio values, whereas the tolerance version permits more simulations to contribute.
- Protein 3D structure prediction: In the protein application, ABC compares four candidate folds—ST1, ST2, ST3, and DT—with native structures represented through Gibbs random-field models.The candidates span predictions from good to very poor, including structures in the FROST uncertainty zone.
- Protein 3D structure prediction: Estimated Bayes factors favor the native structure for every candidate, with weak evidence against ST1 and ST2 but substantial or strong evidence against ST3 and DT.The approach therefore distinguishes similar from dissimilar structures even when threading scores are uncertain.
4 Discussion
The paper concludes that ABC with an auxiliary variable can support model choice in Gibbs random fields despite unavailable normalising constants. A sufficient statistic across models reduces the usual ABC approximation, while additional simulations improve accuracy and Bayes factors rank folding structures in realistic protein applications.
- ABC with an auxiliary variable overcomes unavailable closed-form normalising constants in Gibbs random field and Ising models.
- A sufficient statistic across models can avoid the usual approximation inherent to ABC methods.
- The original ABC approach improves posterior-probability and Bayes-factor accuracy by allowing many more simulations.
- In two protein-folding applications, Bayes factors ranked candidate structures more effectively than standard methods.