Bayesian excursion set estimation with GPs Dario Azzimonti 1 Joint works with: David Ginsbourger, Cl´ ement Chevalier, Julien Bect, Yann Richet 1 Institute of Mathematical Statistics and Actuarial Science University of Bern, Switzerland (Visiting UQOD group, Idiap Research Institute) Uncertainty Quantification Workshop Sheffield, UK September 15, 2016 dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 1 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates Outline Introduction Expectations of random closed sets Vorob’ev expectation Distance average approach Quasi-realizations for excursion sets estimation Approximate field Optimal design Implementation Assessing uncertainties with the distance transform Conservative estimates Definition Computational issues GanMC method Test case dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 2 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates Outline Introduction Expectations of random closed sets Vorob’ev expectation Distance average approach Quasi-realizations for excursion sets estimation Approximate field Optimal design Implementation Assessing uncertainties with the distance transform Conservative estimates Definition Computational issues GanMC method Test case dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 3 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates The framework In this talk we focus on the problem of determining the set Γ ⋆ = { x ∈ D : f ( x ) ∈ T } = f − 1 ( T ) where D ⊂ R d is compact, f : D − → R k is measurable, T ⊂ R k . Here: k = 1, f is continuous, and T = ( −∞ , t ] for a fixed t ∈ R . Γ ⋆ = { x ∈ D : f ( x ) ≤ t } is denoted the excursion set of f below t . Objective Estimate Γ ⋆ and quantify uncertainty on it when f is evaluated only at a few points X n = { x 1 , . . . , x n } ⊂ D . dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 4 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates The framework: IRSN test case Test case: Moret test case (k_eff) ◮ k eff function of PuO 2 density and 5 H 2 O thickness, D = [0 . 2 , 5 . 2] × [0 , 5]; Water thickness (cm3) ◮ continuous function, expensive to 4 evaluate; 3 ◮ n = 20 observations (black triangles); 2 Objective: estimate Γ ⋆ = { x ∈ D : f ( x ) ≤ t } and evaluate the 1 uncertainty of the estimate. 0 1 2 3 4 5 PuO2 density (cm3) Acknowledgements: Yann Richet, Institut de Radioprotection et de Sˆ uret´ e Nucleaire. dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 5 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates The framework: an example Function f : D ⊂ R d → R Gaussian random field realization ◮ expensive to evaluate; ◮ continuous. Evaluated at X n = ( x 1 , . . . , x n ) (black triangles) with values f n = ( f ( x 1 ) , . . . , f ( x n )). Objective: estimate Γ ⋆ = { x ∈ D : f ( x ) ≤ t } and evaluate the uncertainty of the estimate. dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 6 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates Bayesian approach Bayesian framework: f is seen as one realization of a (GRF) ( Z x ) x ∈ D with prior mean m and covariance kernel k . Given the function evaluations f n the posterior field has a Gaussian distribution Z | ( Z ( X n ) = f n ) with mean and covariance kernel m n ( x ) = m ( x ) + k ( x , X n ) k ( X n , X n ) − 1 ( f n − m ( X n )) k n ( x , y ) = k ( x , y ) − k ( x , X n ) k ( X n , X n ) − 1 k ( X n , y ) Γ ⋆ is a realization of Γ = { x ∈ D : Z x ≤ t } = Z − 1 (( −∞ , t ]) dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 7 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates A prior on the space of functions Assume: f realization of ( Z x ) x ∈ D , Gaussian Random Field (GRF) Prior: ( Z x ) x ∈ D with Posterior GRF realization ◮ a.s. continuous paths; ◮ Mat´ ern covariance kernel k ( ν = 3 / 2); ◮ constant mean function m . Given n = 15 evaluations f n at X n Posterior field: Z | Z X n = f n with mean m n and covariance k n . dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 8 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates Distribution of excursion sets The posterior field defines posterior distribution on excursion sets. Γ = { x ∈ D : Z x ≤ t } Excursion set realization dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 9 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates How to summarize the distribution on sets? The posterior excursion set is a random closed set. Here we focus on Expectations of random closed sets 1 ◮ Vorob’ev expectation ◮ distance average expectation Conservative estimates , based on Vorob’ev quantiles. 1. for more definitions of expectation see Molchanov, I. (2005). Theory of Random Sets. Springer. dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 10 / 43
Introduction Expectations of random closed sets Quasi-realizations for excursion sets estimation Conservative estimates Main references: E. Vazquez and M. P. Martinez. (2006). Estimation of the volume of an excursion set of a Gaussian process using intrinsic kriging. Tech Report. arXiv:math/0611273. Ranjan, P., Bingham, D., and Michailidis, G. (2008). Sequential experiment design for contour estimation from complex computer codes. Technometrics, 50(4):527541. Bect, J., Ginsbourger, D., Li, L., Picheny, V., and Vazquez, E. (2012). Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput., 22 (3):773793. Chevalier, C., Bect, J., Ginsbourger, D., Vazquez, E., Picheny, V., and Richet, Y. (2014). Fast kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics. Chevalier, C., Ginsbourger, D., Bect, J., and Molchanov, I. (2013). Estimating and quantifying uncertainties on level sets using the Vorobev expectation and deviation with Gaussian process models. mODa 10. Bolin, D. and Lindgren, F. (2015), French, J. P. and Sain, S. R. (2013) and references therein... dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 11 / 43
Introduction Expectations of random closed sets Vorob’ev expectation Quasi-realizations for excursion sets estimation Distance average approach Conservative estimates Outline Introduction Expectations of random closed sets Vorob’ev expectation Distance average approach Quasi-realizations for excursion sets estimation Approximate field Optimal design Implementation Assessing uncertainties with the distance transform Conservative estimates Definition Computational issues GanMC method Test case dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 12 / 43
Introduction Expectations of random closed sets Vorob’ev expectation Quasi-realizations for excursion sets estimation Distance average approach Conservative estimates Vorob’ev quantiles The function p n : x ∈ D → p n ( x ) = P n ( x ∈ Γ ) ∈ [0 , 1] is the coverage function of Γ , where P n ( · ) = P ( · | Z X n = f n ). Coverage probability function In the Gaussian case ◮ fast to compute � � m n ( x ) − t √ p n ( x ) = Φ k n ( x , x ) ◮ marginal statement ◮ creates a family of set estimates Q ρ = { x ∈ D : p n ( x ) ≥ ρ } dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 13 / 43
Introduction Expectations of random closed sets Vorob’ev expectation Quasi-realizations for excursion sets estimation Distance average approach Conservative estimates Vorob’ev expectation Consider a Borel measure µ on D . From the family of “quantiles” Q ρ we can choose Q � ρ such that µ ( Q � ρ ) = E [ µ ( Γ )]. Vorob'ev expectation Properties: ◮ based on the measure µ ; ◮ for some choices of µ , fast to compute; ◮ no confidence statements on the set. Chevalier, C., Ginsbourger, D., Bect, J., and Molchanov, I. (2013). Estimating and quantifying uncertainties on level sets using the Vorobev expectation and deviation with Gaussian process models. mODa 10 dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 14 / 43
Introduction Expectations of random closed sets Vorob’ev expectation Quasi-realizations for excursion sets estimation Distance average approach Conservative estimates Distance average approach Consider the distance function d : ( x , Γ ) → d ( x , Γ ) . Posterior GRF realization Distance transform of set realization Γ is random therefore d ( x , Γ ) is a random variable for each x ∈ D . dario.azzimonti@stat.unibe.ch Bayesian set estimation with GPs 15 / 43
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