Descriptive Complexity Approaches to Inductive Inference

We present a critical review of descriptive complexity approaches to inductive inference. Inductive inference is defined as any process by which a model of the world is formed from observations. The descriptive complexity approach is a formalization of Occam's razor: choose the simplest model consistent with the data. Descriptive complexity as defined by Kolmogorov, Chaitin and Solomonoff is presented as a generalization of Shannon's entropy. We discuss its relationship with randomness and present examples. However, a major result of the theory is negative: descriptive complexity is uncomputable. Rissanen's minimum description length (MDL) principle is presented as a restricted form of the descriptive complexity which avoids the uncomputability problem. We demonstrate the effectiveness of MDL through its application to AR processes. Lastly, we present and discuss LeClerc's application of MDL to the problem of image segmentation.