The hard truth

Kenneth M. Hanson, Gregory S. Cunningham and David R. Wolf [*] Los Alamos National Laboratory, MS P940 Los Alamos, New Mexico 87545 USA kmh@lanl.govcunning@lanl.govwolf@planck.lanl.gov

Abstract

Bayesian analysis provides the foundation for a rich environment in which to explore inferences about models from data and prior knowledge via the posterior probability. If one draws an analogy between \phi = -log(posterior probability) and a physical potential, then the gradient of \phi is analogous to a force, just as in physics. The maximum a posteriori (MAP) solution can be interpreted as the situation in which the forces on all the variables in the problem balance so that the net force on each variable is zero. Further, when the variable x is perturbed from the MAP solution, the derivative \frac{\partial \phi}{\partial x} is the force that drives x back towards the MAP solution. The phrase force of the data'' takes on real meaning in this context.

A quadratic approximation to \phi about the MAP solution implies a linear force law, i.e.\/ the force is proportional to the displacement from equilibrium, as in a simple spring. In this quadratic approximation the curvature of \phi is proportional to the covariance of the MAP estimate. A high curvature is analogous to a stiff spring and therefore represents a rigid'', well-determined solution. Truth is hard''.

We propose to exploit this physical analogy to facilitate the exploration of the reliability of a MAP solution. The rigidity of the solution is indicated by the rate at which the restoring forces increase as a user perturbs a single parameter, or group of parameters. Parameter correlations may be explored by altering some parameters, fixing them, and allowing the remaining parameters to readjust to minimize \phi. The correlations between the fixed set and the others is demonstrated by how much and in what direction the variable parameters change. Ideally, these correlations could be seen through direct interaction with a rapidly-responding dynamical Bayesian system. Alternatively, they may be demonstrated by means of a video loop. As an aside, we use an adjoint method to efficiently calculate the required derivative with respect to the variables of interest.

Another interesting aspect of this technique is the possibility of decomposing the forces into their various components. For example, the force derived from all data (through the likelihood), or even a selected set of data, may be compared to the force derived from the prior. In this way it is possible to examine the influence of the priors on the solution.

We demonstrate these innovative Bayesian tools in a tangible way within the context of geometrically geometrically-defined object models used for tomographic reconstruction from limited projection data.

In the future it may be possible to use the tools of virtual reality, coupled to turbocomputation, to explore the reliability of a Bayesian solution through direct manipulation of the computer model. Force feedback will permit one to actually feel'' the stiffness of a model. Higher dimensional correlations might be felt'' through one's fingers.

[*] -- Supported by the United States Department of Energy under contract number W-7405-ENG-36.
MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk