1. 1 Graphical Models in Data Assimilation Problems Alexander Ihler UC Irvine ihler@ics.uci.edu Collaborators: Sergey Kirshner Andrew Robertson Padhraic Smyth

2. 2 Outline Graphical models –Convenient description of structure among random variables Use this structure to –Organize inference computations Finding optimal (ML, etc.) estimates Calculate data likelihood Simulation / drawing samples –Suggest sub-optimal (approximate) inference computations e.g. when optimal computations too expensive Some examples from data assimilation –Markov chains, Kalman filtering –Rainfall models Mixtures of trees Loopy graphs –Image analysis (de-noising, smoothing, etc.)

3. 3 set of nodes set of edges connecting nodes Nodes are associated with random variables An undirected graph is defined by Graph Separation Conditional Independence Graphical Models

4. 4 Graphical Models: Factorization Sufficient condition –Distribution factors into product of “potential functions” defined on cliques of G –Condition also necessary if distribution strictly positive Examples

5. 5 Graphical Models: Inference Many possible inference goals –Given a few observed RVs, compute: Marginal distributions Joint, Maximum a-posteriori (MAP) values Data likelihood of observed variables Samples from posterior Use graph structure to do computations efficiently –Example: compute posterior marginal p(x2 | x5=X5)

6. 6 Combine the observations from all nodes in the graph through a series of local message-passing operations neighborhood of node s (adjacent nodes) message sent from node t to node s (“sufficient statistic” of t’s knowledge about s) Finding marginals via Belief Propagation (aka sum-product; other goals have similar algorithms)

7. 7 II. Message Propagation: Transform distribution from node t to node s using the pairwise interaction potential Integrate over to form distribution summarizing node t ’s knowledge about BP Message Updates I. Message Product: Multiply incoming messages (from all nodes but s ) with the local observation to form a distribution over

8. 8 Example: sequential estimation Well-known example –Markov Chain –Jointly Gaussian uncertainty Gives integrals a simple, closed form –Optimal inference (in many senses) given by Kalman filter –Convert large (T) problem to collection of smaller problems –“exact” non-Gaussian: particle & ensemble filtering & extensions –Same general results hold for any tree-structured graph Partial elimination ordering of nodes –Complexity limited by dimension of each variable

9. 9 Exact estimation in non-trees Often our variables aren’t so well-behaved –May be able to convert using variable augmentation Often the case in Bayesian parameter estimation –Treat parameters as variables, include them in the graph –(increases nonlinearities!) But, dimensionality problem –Computation increases (maybe a lot!) Jointly Gaussian,  d 3 Otherwise often exponential in d –Can trade off graph complexity with dimensionality… 

10. 10 Example: rainfall data 41 stations in India Rainfall occurrence & amounts for ~30 years Some stations/days missing Tasks –Impute missing entries –Simulate realistic rainfall –Short term predictions –… Can’t deal with joint distribution – too large to even manipulate Conditional independence structure? –Unlikely to be tree-structured

11. 11 Example: rainfall data “True” relationships –not tree-like at all –High tree-width Need some approximations –Approximate model, exact inference –Correct model, approximate inference Even harder: –May get multiple observation modalities (satellite data, etc.) –Have own statistical structure & relationships to stations

12. 12 Example: rainfall data Consider a single time-slice Option 1: mixtures of trees –Add “hidden” variable indicating which of several trees –(Generally) marginalize over this variable Option 2: use loopy graph, ignore loops in inference –Utility depends on task: –Works well for filling in missing data –Perhaps less well for other tasks + ++

13. 13 Multi-scale models Another example of graph structure Efficient computation if tree-structured Again, don’t really believe any particular tree –Perhaps average over (use mixture of) several (see e.g. Willsky 2002) (also w/ loops, similar to multi-grid)

14. 14 Summary Explicit structure among variables –Prior knowledge / learned from data –Structure organizes computation, suggests approximations –Can provide computational efficiency –(often naïve distribution too large to represent / estimate) Offers some choice –Where to put the complexity? –Simple graph structure with high-dimensional variables –Complex graph structure with more manageable variables Approximate structure, exact computations Improved structures, approximate computations