Fisher Information Explorer

Live Script exploring optimization of experiments described by linear models using a fully symbolic Fisher Information Matrix.

Optimization of experiments concerns choosing how best to deploy measurements to achieve some goal. For example, suppose you embark on observing the vertical positions of falling objects at some location with mobile phone video. You might presume systematic effects like air drag are negligible and that any one motion is governed by the model

which is a linear function of three parameters: initial position y, initial velocity , and the acceleration of gravity g. A camera calibration relates mean pixel coordinates to space coordinates and characterizes the essential uncorrelated uncertainties due to finite pixel size and correlated uncertainties due to camera vibration. Your goal might be to estimate the mean initial velocity independent of and g, or to estimate g independent of and . If you select only n video frames to analyze, what relative frame times will best determine one or more of the model parameters? How might the answer depend on the uncertainties?

This Live Script generates symbolic formulae to answer such questions for general models linear in parameters like this one when uncertainties are correlated and described by a multivariate normal distribution. The best estimates of the parameters in such a case are themselves described by a multivariate normal distribution with a calculable covariance matrix. That matrix is the inverse of the Fisher Information Matrix and is a function of the error model covariance parameters, the measurement times, and the model functions. It is a basis for addressing the general and quite complex field of experimental optimization.

The Live Script shows how to construct and exploit a fully symbolic Fisher Information Matrix and provides a simple simulation to explore optimization. The script may interest students and instructors in physics, statistics, and other fields. 'Try this' suggestions, coding 'Challenges', and references are included for further exploration.