Conditional Expectation as the Basis for Bayesian Updating Hermann - PowerPoint PPT Presentation

Conditional Expectation as the Basis for Bayesian Updating Hermann G. Matthies Bojana V. Rosi´ c, Elmar Zander, Alexander Litvinenko, Oliver Pajonk Institute of Scientific Computing, TU Braunschweig Brunswick, Germany wire@tu-bs.de http://www.wire.tu-bs.de 15 Cond-Exp.tex,v 2.9 2017/07/18 00:35:08 hgm Exp

2 Overview 1. BIG DATA 2. Parameter identification 3. Stochastic identification — Bayes’s theorem 4. Conditional probability and conditional expectation 5. Updating — filtering. TU Braunschweig Institute of Scientific Computing

3 Representation of knowledge Data from measurements, sensors, observations ⇒ one form of knowledge about a system. ‘Big Data’ considers only data — looking for patterns, interpolating, etc. Mathematical / computational models of a system represent another form of knowledge — ‘structural’ knowledge — about a system. These models are often generated based on general physical laws (e.g. conservation laws), a very compressed form of knowledge. These two views on systems are not in competition, they are complementary. The challenge is to combine these forms of knowledge — in form of a synthesis. Knowledge may be uncertain. TU Braunschweig Institute of Scientific Computing

4 Big Data 16th century Johannes Kepler Isaac Newton Tycho Brahe Pierre-Simon Laplace (1571 – 1630) (1643 – 1727) (1546 – 1601) (1749 – 1827) Description Understanding Data Perfection I. Newton: The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics. Kepler’s 2nd law: (adapted from M. Ortiz) TU Braunschweig Institute of Scientific Computing

5 BIG DATA Mathematically speaking, big data algorithms (feature / pattern recognition) are regression (generalised interpolation) methods. Often based on deep artificial neural networks (deep ANNs), combining many inputs (= high-dimensional data). Deep networks are connected to sparse tensor decompositions (buzzword: deep-learning). Although often spectacularly successful, as knowledge representation, it is difficult to extract insight. But there is a connection of such regression to Bayesian updating. TU Braunschweig Institute of Scientific Computing

6 Inference Our uncertain knowledge about some situation is described by probabilities. Now we obtain new information. How does it change our knowledge — the probabilistic description? Answered by T. Bayes and P.-S. Laplace more than 250 years ago. Thomas Bayes Pierre-Simon Laplace (1701 – 1761) (1749 – 1827) TU Braunschweig Institute of Scientific Computing

7 Synopsis of Bayesian inference We have a some knowledge about an event A , but it can not be observed directly. After some new information B (an observation, a measurement), our knowledge has to be made consistent with the new information, i.e. we are looking for conditional probabilities P ( A|B ) . The idea is to change our present model by just so much — as little as possible — so that it becomes consistent. For this we have to predict — with our present knowledge / model — the probability of all possible observations and compare with the actual observation. TU Braunschweig Institute of Scientific Computing

8 Model inverse problem Geometry flow = 0 Sources 2 flow out 1.5 Dirichlet b.c. 0 1 0.5 0.5 1 1.5 0 2 Aquifier 2D Model Governing model equations: ̺ ∂u ∈ G ⊂ R d . ∂t − ∇ · ( κ · ∇ u ) = f Parameter q = log κ . Conductivity field κ , initial condition u 0 , and state u ( t ) may be unknown. They have to be determined from observations Y ( q ; u ) . TU Braunschweig Institute of Scientific Computing

9 A possible realisation of κ ( x, ω ) TU Braunschweig Institute of Scientific Computing

10 Measurement patches 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 0 1 −1 0 1 447 measurement patches 239 measurement patches 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 0 1 −1 0 1 120 measurement patches 10 measurement patches TU Braunschweig Institute of Scientific Computing

11 Convergence plot of updates 0 10 447 pt 239 pt 120 pt Relative error ε a 60 pt 10 pt −1 10 −2 10 0 1 2 3 4 Number of sequential updates TU Braunschweig Institute of Scientific Computing

12 Forecast and assimilated pdfs 6 κ f κ a 4 PDF 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 κ Forecast and assimilated probability density functions (pdfs) for κ at a point where κ t = 2 . TU Braunschweig Institute of Scientific Computing

13 Setting for identification General idea: We observe / measure a system, whose structure we know in principle. The system behaviour depends on some quantities (parameters), which we do not know ⇒ uncertainty. We model (uncertainty in) our knowledge in a Bayesian setting: as a probability distribution on the parameters. We start with what we know a priori, then perform a measurement. This gives new information, to update our knowledge (identification). Update in probabilistic setting works with conditional probabilities ⇒ Bayes’s theorem. Repeated measurements lead to better identification. TU Braunschweig Institute of Scientific Computing

14 Mathematical formulation of model Consider operator equation, physical system modelled by A : u ∈ U , d u + A ( u ; q ) d t = g d t + B ( u ; q ) d W U — space of states, g a forcing, W noise, q ∈ Q unknown parameters. Well-posed problem: for q, g and initial cond. u ( t 0 ) = u 0 a unique solution u ( t ) , given by the flow or solution operator, S : ( u 0 , t 0 , q, g, W, t ) �→ u ( t ; q ) = S ( u 0 , t 0 , q, g, W, t ) . Set extended state ξ = ( u, q ) ∈ X = U × Q , advance from ξ n − 1 = ( u n − 1 , q n − 1 ) at time t n − 1 to ξ n = ( u n , q n ) at t n , ξ n =( u n , q n ) = ( S ( u n − 1 , t n − 1 , q n , g, W, t n ) , q n ) =: f ( ξ n − 1 , w n − 1 ) . This is the model for the system observed at times t n . Applies also to stationary case A ( u ; q ) = g . TU Braunschweig Institute of Scientific Computing

15 Mathematical formulation of observation Measurement operator Y with values in Y : η n = Y ( u n ; q ) = Y ( S ( u n − 1 , t n − 1 , q, g, W, t n ); q ) . But observed at time t n , it is noisy y n with noise ǫ n y n = H ( η n , ǫ n ) = H ( Y ( u n ; q ) , ǫ n ) =: h ( ξ n , ǫ n ) = h ( f ( ξ n − 1 , w n − 1 ) , ǫ n ) . For given g, w , measurement η = Y ( u ( q ); q ) is just a function of q . This function is usually not invertible ⇒ ill-posed problem, measurement η does not contain enough information. Parameters q and initial state u 0 uncertain, modelled as RVs q ∈ Q = Q ⊗ S ⇒ u ∈ U = U ⊗ S , with e.g. S = L 2 ( Ω, P ) a RV-space. Bayesian setting allows updating of information about ξ = ( u, q ) . The problem of updating becomes well-posed. TU Braunschweig Institute of Scientific Computing

16 Mathematical formulation of filtering We want to track the extended state ξ n by a tracking equation for a RV x n through observations ˆ y n . • Prediction / forecast state is a RV x n,f = f ( x n − 1 , w n − 1 ) ; • Forecast observation is a RV y n = h ( x n,f , ǫ n ) , actual observation ˆ y n , • Updated / assimilated x n = x n,f + Ξ ( x n,f , y n , ˆ y n ) , • Hopefully x n ≈ ξ n , and the update map Ξ has to be determined. x n,i := Ξ ( x n,f , y n , ˆ y n ) is called the innovation. We concentrate on one step from forecast to assimilated variables. • Forecast state x f := x n,f , forecast observation y f := y n , • Actual observation ˆ y and assimilated variable x a := x f + Ξ ( x f , y f , ˆ y ) = x n = x n,f + Ξ ( x n,f , y n,f , ˆ y n ) . This is the filtering or update equation. TU Braunschweig Institute of Scientific Computing

17 Setting for updating Knowledge prior to new observation is also called forecast: the state u f ∈ U = U ⊗ S and parameters q f ∈ Q = Q ⊗ S modelled as random variables (RVs), also the extended state x f = ( u f , q f ) ∈ X = X ⊗ S and the measurement y ( x f , ε ) ∈ Y = Y ⊗ S . Then an observation ˆ y is performed, and is compared to predicted measurement y ( x f , ε ) . Bayes’s theorem gives only probability distribution of posterior or assimilated extended state x a . Here we want more: a filter x a := x f + Ξ ( x f , y f , ˆ y ) . TU Braunschweig Institute of Scientific Computing

18 Using Bayes’s theorem Classically, Bayes’s theorem gives conditional probability P ( I x |M y ) = P ( M y |I x ) P ( I x ) for P ( M y ) > 0 . P ( M y ) Well-known special form with densities of RVs x, y (w.r.t. some background measure µ ): = π ( y | x ) ( y | x ) π ( x | y ) ( x | y ) = π xy ( x, y ) π x ( x ); π y ( y ) Z y � with marginal density Z y := π y ( y ) = X π xy ( x, y ) µ (d x ) (from German Zustandssumme) — only valid when π xy ( x, y ) exists. Problems / paradoxa appear when P ( M y ) = 0 (and P ( M y |I x ) = 0 ) e.g. Borel-Kolmogorov paradox. Problem is limit P ( M y ) → 0 , or when no joint density π xy ( x, y ) exists. TU Braunschweig Institute of Scientific Computing