Bayesian Calibration of Simulators with Structured Discretization Uncertainty

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Bayesian Calibration of Simulators with Structured Discretization Uncertainty Oksana A. Chkrebtii Department of Statistics, The Ohio State University Joint work with Matthew T. Pratola (Statistics, The Ohio State University) Probabilistic Scientific Computing: Statistical inference approaches to numerical analysis and algorithm design, ICERM June 7, 2017 Slide 1/29

Probabilistic Numerics This is an active new field that challenges historical perspectives on numerical analysis. It is important for this community to develop new methods with an eye to overcoming challenges that lay ahead. This talk focuses on calibration for forward problems defined by the solution of ordinary and partial differential equations. If you re not already convinced that probabilistic numerics is useful in this setting... Slide 2/29

Example - galaxy simulation (Kim et al., 2016) Slide 3/29

Example - galaxy simulation (Kim et al., 2016) These are not realizations of a field (the model is deterministic) The initial conditions and inputs are held fixed How do we evaluate these numerical solver outputs? Slide 4/29

Perspectives on Probabilistic Numerics Probability measures on numerical solutions via randomization Conrad et al (2015/16), Lie et al (2017) defined outside of the Bayesian framework, but resulting algorithms overlap Bayesian uncertainty quantification for differential equations Skilling (1991), Chkrebtii et al (2013/16), Arnold et al (2013) defined outside of the numerical analysis framework, but resulting methods can be analogous in some sense Bayesian numerical methods Hennig & Hauberg (2013/14), Schober et al (2014). computationally efficient probabilistic GP based methods; can recover numerical solvers in the mean Slide 5/29

Calibrating stochastic computer models Regardless of the perspective, the deterministic but unknown forward model is replaced by a stochastic process: for fixed inputs, the output is a random variable with (often) unknown distribution. Pratola & Chkrebtii (2017+) describe a hierarchical framework to calibrate stochastic simulators with highly structured output uncertainty/variability. Slide 6/29

Calibration problem We wish to estimate the unknowns, θ Θ, given observations, y(x t ) = A {u(x t, θ)} + ε(x t ), x t X, t = 1,..., T, of the deterministic state u t = u(x t, θ) transformed via an observation process A, and contaminated with stochastic noise ε. The likelihood defines a discrepancy between the model and the data: f (y 1:T θ) ρ {y 1:T A (u 1:T )} Slide 7/29

The Bayesian paradigm Bayesian inference is concerned with modeling degree of belief about an unknown quantity via probability models. For example, we may not know θ Θ but we may have some prior belief about, e.g., its range, most probable values, θ π(θ). We seek to update our prior belief by conditioning on new information, y 1:T Y, e.g., data, model evaluations, via Bayes Rule: p(θ y 1:T ) = p(y 1:T θ) π(θ) p(y1:t θ) π(θ) dθ p(y 1:T θ) π(θ). Slide 8/29

A Hierarchical model representation Hierarchical modelling enables inference over the parameters of a stochastic state, [y 1:T u 1:T, θ] ρ {y 1:T A (u 1:T )} [u 1:T θ] p(u 1:T θ) [θ] π(θ). When p(u 1:T θ) is not known in closed form, exact inference may still be possible via Monte Carlo, using forward-simulation from the model. However, this is often computationally prohibitive. Slide 9/29

For probabilistic numerics If the state is deterministic but defined implicitly by a system of differential equations, our uncertainty about the solution can be modelled probabilistically, [y 1:T u 1:T, θ] ρ [y 1:T A (u 1:T )] [u 1:T θ] a probability measure representing uncertainty in the solution given discretization of size N [θ] π(θ). We use the Bayesian uncertainty quantification approach to model this middle layer. Slide 10/29

Bayesian UQ for differential equations Given θ and for linear operator D consider the initial value problem, { Du = f (x, u), x X, u = u 0 x X. We may have some prior knowledge about smoothness, boundary conditions, etc., described by a prior measure, u π, x X We seek to update our prior knowledge by conditioning on model interrogations, f 1:N via Bayes Rule, p(u(x) f 1:N ) = p(f 1:N u(x)) π(u(x)) p(f1:n u(x)) π (u(x)) du(x) p (f 1:N u(x)) π (u(x)) Slide 11/29

Prior uncertainty in the unknown solution The exact solution function u is deterministic, but unknown. We may describe our prior uncertainty via a probability model defined on the space of suitably smooth derivatives, e.g., u GP(m 0, C 0 ), m 0 : X R, C 0 : X X R with the constraint m 0 = u 0, x X. This yields a joint prior on the fixed but unknown state and its derivative(s) [ ] ([ ] [ u m 0 C 0 C GP Du Dm 0, 0 D ]) DC 0 DC 0 D Slide 12/29

Interrogating the model recursively 1 Draw a sample from the marginal predictive distribution on the state at the next discretization grid point s n+1 X, 1 n < N u(s n+1 ) p (u(s n+1 ) f 1:n ) 2 Evaluate the RHS at u(s n+1 ) to obtain a model interrogation, f n+1 = f (s n+1, u(s n+1 )) 3 Model interrogations as noisy measurements of Du: f n+1 Du, f 1:n N ( Du(s n+1 ), Λ(s n ) ) Slide 13/29

Sequential Bayesian updating Updating our knowledge about the true but unknown solution given the new interrogation trajectory f n+1 [ ] ([ ] [ u m Du f n+1 C n+1 C n+1 GP Dm n+1, n+1 D ]) DC n+1 DC n+1 D where, m n+1 = m n + K n (f n+1 m n (s n+1 )) C n+1 = C n K n DC n K n = C n D (DC n + Λ(s n )) 1 This becomes the prior for the next update. Slide 14/29

Bayesian UQ for differential equations Due to the Markov property, we cannot condition the solution on multiple trajectories f j 1:N, j = 1,..., J simultaneously. In fact, the posterior over the unknown solution turns out to be a continuous mixture of Gaussian processes, [u θ, N] = [u, Du f 1:N, θ, N] d(du) df 1:N. Samples from this posterior can be obtained via Monte Carlo. Slide 15/29

Example - Lorenz63 forward model A probability statement over probable trajectories given fixed model parameters and initial conditions for the Lorenz63 model: 1000 draws for the probabilistic forward model for the Lorenz63 system given fixed initial states and model parameters in the chaotic regime. Slide 16/29

Example - Lorenz63 forward model 1000 draws from forward model for Lorenz63 system at four fixed time points. Slide 17/29

For probabilistic numerics If the state is deterministic but defined implicitly by a system of differential equations, our uncertainty about the solution can be modelled probabilistically, [y 1:T u 1:T, θ] ρ [y 1:T A (u 1:T )] [u 1:T θ] a probability measure representing uncertainty in the solution given discretization of size N [θ] π(θ). We use the Bayesian uncertainty quantification approach to model this middle layer. Slide 18/29

Computer model emulation We interrogate the model at M regimes, θ 1:M = (θ 1,..., θ M ). Each interrogation is comprised of an ensemble of K output samples. Let ũ 1:K 1:T (θ 1:M) denote the ensemble of K output simulations for each of the regimes θ 1:M. [y 1:T ũ 1:K 1:T (θ 1:M), u 1:T, δ 1:T, θ] ρ [y 1:T A (u 1:T (θ)) δ 1:T ] [ũ k 1:T (θ m) u k 1:T, θ] N ( u k 1:T (θ m), Λ ) k = 1,..., K [ u k 1:T θ ] generative stochastic model [θ, δ] π(θ, δ). Slide 19/29

A Hierarchical model representation Challenges of MCMC sampling from the posterior include: Emulation based on MK model evaluations is computationally expensive Models are evaluated at multiple spatio-temporal locations and over multiple states Our approach: Dimension reduction over the second (output) layer of the hierarchical model We include the dimension reduction specifications within the hierarchical model, resulting in a fully probabilistic approach Slide 20/29

Example: Exact vs Emulation Based Inference in a Model of Protein Dynamics Slide 21/29

Inference for a model of protein dynamics JAK-STAT chemical signaling pathway model describes concentration of 4 STAT factors by a delay differential equation system on t [0, 60], d dt u(1) (t, θ) = k 1 u (1) (t, θ) EpoR A (t) + 2 k 4 u (4) (t τ, θ) d ( ) 2 dt u(2) (t, θ) = k 1 u (1) (t, θ) EpoR A (t) k 2 u (2) (t, θ) d ( ) 2 dt u(3) (t, θ) = k 3 u (3) (t, θ) + 0.5 k 2 u (2) (t, θ) d dt u(4) (t, θ) = k 3 u (3) (t, θ) k 4 u (4) (t τ, θ) u (i) (t, θ) = φ (i) (t), t [ τ, 0], i = 1,..., 4 Illustration of the JAK-STAT mechanism Slide 22/29

Inference for a model of protein dynamics States are observed indirectly through a nonlinear transformation: A (1) = k 5 ( u (1) (t; θ) + 2u (3) (t; θ) ) A (2) = k 6 ( u (1) (t; θ) + u (2) (t; θ) + 2u (3) (t; θ) ) A (3) = u (1) (t; θ) Experimental measurements A (4) = u (3) (t; θ) u (2) (t; θ) + u (3) (t; θ) Observations are noisy measurements on the transformed states and forcing function at points t = {t ij } i=1,...,4;j=1,...,ni y(t) = A k4,k 5 u(t; k 1,..., k 6, τ, φ, EpoR A ) + ε(t) Slide 23/29

Results: exact inference Slide 24/29

calibration with discretization uncertainty oksana chkrebtii y1(θ) κ1(χ2(θ) + 2χ3(θ)) + δ1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Results: emulation based inference 50 60 y2(θ) κ2(χ1(θ) + χ2(θ) + 2χ3(θ)) + δ2 0.5 0.5 0 4 8 14 20 30 40 0 4 8 14 20 y1(θ) δ1 30 40 50 60 time (minutes) 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 time (minutes) 50 60 y2(θ) δ2 0.5 0.5 0 4 8 14 20 30 40 time (minutes) Slide 25/29 0 4 8 14 20 30 40 time (minutes) 50 60

Results: emulation based inference 1 θ 1 0.5 02 0 2 4 6 θ 2 1 02 0 2 4 60 0.5 1 1.5 2 θ 3 1 02 0 2 4 60 0.5 1 1.5 20 0.5 1 1.5 2 θ 4 1 0 5 0 2 4 60 0.5 1 1.5 20 0.5 1 1.5 20 0.5 1 1.5 2 θ 5 θ 6 4000 0 2 4 60 0.5 1 1.5 20 0.5 1 1.5 20 0.5 1 1.5 20 2 4 6 300 200 10 5 100 2 0 2 4 60 0.5 1 1.5 20 0.5 1 1.5 20 0.5 1 1.5 20 2 4 100 6 200 300 400 θ 7 1 θ 8 0 5 0 2 4 60 0.5 1 1.5 20 0.5 1 1.5 20 0.5 1 1.5 20 2 4 100 6 200 300 4000 0.5 1 1.5 2 10 5 θ 2 θ 3 θ 4 θ 5 θ 6 0 θ 1 θ 7 10 5 θ 8 1 20 0.5 1 1.5 20 0.5 1 1.5 20 2 4 100 6 200 300 4000 0.5 1 1.5 20 0 2 4 60 0.5 1.5 2 4 6 Kernel density estimates of the marginal stochastically calibrated posterior (gray) with M = 100 model runs, and exact posterior (black) for the JAK-STAT system. Marginal prior densities are shown as dotted lines. Slide 26/29

Thank you! Slide 27/29

References 1 Pratola, M.T., Chkrebtii, O.A. Bayesian Calibration of Multistate Stochastic Simulators. To appear in Statistica Sinica. 2 Chkrebtii, O.A., Campbell, D.A., Calderhead, B., Girolami, M., Bayesian Solution Uncertainty Quantification for Differential Equations, Bayesian Analysis with discussion, 11(4), 1239-1267, 2016. 3 Arnold, A., Calvetti, D, Somersalo, E. Linear multistep methods, particle filtering and sequential Monte Carlo, Inverse Problems, 29, 2013. 4 Conrad, R., Girolami, M., Sarkka, S., Stuart, A., Zygalakis, K. Statistical analysis of differential equations: introducing probability measures on numerical solutions, Statistics and Computing, 2016. 5 Lie, H.C., Stuart, A. M., Sullivan, T. J. Strong convergence rates of probabilisti integrators for ordinary differential equations, arxiv preprint, 2017. Slide 28/29

References 1 Hennig, P., Hauberg, S. Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics, In Proc. of the 17th int. Conf. on Artificial Intelligence and Statistics (AISTATS) (Vol. 33). JMLR, W&CP, 2014. 2 Schober, M., Duvenaud, D. K., Hennig, P. Probabilistic ODE Solvers with Runge-Kutta Means. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, & K. Q. Weinberger (Eds.), Advances in Neural Information Processing Systems 27 (pp. 739747). Curran Associates, Inc, 2014. 3 Skilling, J. Bayesian solution of ordinary differential equations Maximum Entropy and Bayesian Methods, Seattle, 1991. 4 Kim et al., The AGORA High-resolution Galaxy Simulations Comparison Project, The Astrophysical Journal Supplement Series, 2014. 5 Swameye et al., Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling. PNAS, 2003. Slide 29/29

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