RELACS Projects: Causal Modeling

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Collaborators

• Jaric Zola at UB CSE
• Atri Ruda at UB CSE

Projects

• Causal Modeling using Probabilistic Graphical Models

  Problem

  Learning models of both the physical and social world, if based only on correlation, is subject to errors from confounding variables. Causal theories can indicate unambigusously the dependence of some variables on others, but this requires modifications to the dependence structure during learning (experiments).

  Approach

  Judea Pearl's theory of causation relies on modeling controlled experiments by surgery of probabilistic graphical models (PGM). We can assemble large scale PGMs using DMPlex (and perhaps even DMNetwork). Solving these models can perhaps be accomplished by variants of SGD.

  Impact

  Correctly infering physical models would allow us to produce constitutive models from data, as well as suggest experiments to disambiguate dependence on primal variables. Causal social models could be used for threat assessment, as well as determining the implications of tax policy. Our framework would allow models with billions of actors and hundreds of billions of dependencies, putting models of complex real-world scenarios within reach.

  Plan

  We would like to mimic the experiment with images ([1] Visual Causal Feature Learning, arXiv:1412.2309v2) using PDEs. The horizontal and verticel lines in the images are features that can be separately controlled. Thus we need two separate, but perhaps coupled PDEs. Consider a thermomechanical Darcy flow in a box, where we specify the temperature and input flux on one wall, and measure the heat flux and pressure on another. Since temperature is advected, increasing the input flow will increase the output heat flux, but we can decrease the input temperature to offset this effect. Suppose that a measurable material property (stress?) at the output is a function of the heat flux and a common cause of the input flow. This is our analogue of the common cause of the output and horizontal lines in [1]. Our output will be correlated to both output pressure and heat flux, but it will only be caused by heat flux. Now to compute this we need:

  1. A duct-like domain
  2. Darcy flow in this domain, with a specfied input velocity, no flow on sides, and stress-free output.
  3. Add advection-diffusion temperature equations to this system, with specified temperature at input, insulating on the sides, and free flux at the output.
  4. Measure the heat flux and pressure at the output
  5. Create a function of \(\mu\) that sets the input boundary conditions, and the measured state function \(m(\phi, \mu)\) where \(\phi\) is the output heat flux
  6. Run a bunch of simulations to see the correlation of \(m\) with \(\phi\) and \(p_{out}\).
  7. Change \(v_{in}, T_{in}\) and remeasure the output state. The idea would be to vary them independently, which we should be able to do using adjoints.
  8. Realize (perhaps using SGS) that the output state is caused by \(\phi\) and \(\mu\), but not \(p_{out}\)

  The important thing about this model is that it is a piece of a larger physical system, rather than being the whole story itself. We usually image our PDEs to be everything that is important in a system, but this is naive. Here we acknowledge that it can accurately model some of the system, but other factors may be in play between that subsystem and our measurement.

  Another approach is to work with Profs. Zola and Ruda to encode their PGM framework in PETSc.

  Software
  1. Tetrad

Finished Projects