# 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.gov`

`cunning@lanl.gov`

`wolf@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