# The g Factor: Relating Distributions on Features to Distributions on Images

### James M. Coughlan and Alan L. Yuille

#### Smith-Kettlewell Eye Research Institute

We introduce the g-factor which relates probability distributions on features to distributions on images. It arises when we seek to learn distributions from image data but {\it it depends only on our choice of features and lattice quantization} and is independent of the training image data. We show that simple, and plausible, approximations of the $g$-factor can throw light on aspects of Minimax Entropy Learning (MEL) \cite{Zhu97}, which learns probability distributions on images in terms of Markov Random Fields with clique potentials. Analyzing the $g$-factor allows us to determine when the clique potentials decouple for different features. Moreover, when the approximations of the $g$-factor are valid then the clique potentials in MEL can be computed analytically. Finally, we describe ways to extend these approximations by computing approximations to the $g$-factor offline, thereby enabling rapid methods for computing the clique potentials from new image data. Overall, we seek to give understanding of how MEL relates to alternative methods of learning on images. (In this paper the features we are considering will be extracted from the image by filters -- hence we almost always use the terms features'' and filters'' synonymously.)