A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonably-sized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Explicit bounds are also derived demonstrating that learning multiple tasks within an environment of related tasks can potentially give much better generalization than learning a single task.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

One of the simplest methods is to use linear transformations of the observed data.

There has been much recent work on measuring image statistics and on learning probability distributions on images. We observe that the mapping from images to statistics is many-to-one and show it can be quantified by a phase space factor. This phase space approach throws light on the Minimax Entropy technique for learning Gibbs distributions on images with potentials derived from image statistics and elucidates the ambiguities that are inherent to determining the potentials. In addition, it shows that if the phase factor can be approximated by an analytic distribution then this approximation yields a swift "Minutemax" algorithm that vastly reduces the computation time for Minimax entropy learning. An illustration of this concept, using a Gaussian to approximate the phase factor, gives a good approximation to the results of Zhu and Mumford (1997) in just seconds of CPU time. The phase space approach also gives insight into the multi-scale potentials found by Zhu and Mumford (1997) and suggests that the forms of the potentials are influenced greatly by phase space considerations. Finally, we prove that probability distributions learned in feature space alone are equivalent to Minimax Entropy learning with a multinomial approximation of the phase factor. 1 Introduction Bayesian probability theory gives a powerful framework for visual perception (Knill and Richards 1996). This approach, however, requires specifying prior probabilities and likelihood functions. Learning these probabilities is difficult because it requires estimating distributions on random variables of very high dimensions (for example, images with 200 x 200 pixels, or shape curves of length 400 pixels).

We present a probabilistic method for fusion of images produced by multiple sensors. The approach is based on an image formation model in which the sensor images are noisy, locally linear functions of an underlying, true scene. A Bayesian framework then provides for maximum likelihood or maximum a posteriori estimates of the true scene from the sensor images. Maximum likelihood estimates of the parameters of the image formation model involve (local) second order image statistics, and thus are related to local principal component analysis. We demonstrate the efficacy of the method on images from visible-band and infrared sensors. 1 Introduction Advances in sensing devices have fueled the deployment of multiple sensors in several computational vision systems [1, for example]. Using multiple sensors can increase reliability with respect to single sensor systems.

In supervised learning, we have a set of explicative variables x from which we wish to predict a response variable y.

In previous work [6, 9, 10], we advanced a new technique for direct visual matching of images for the purposes of face recognition and image retrieval, using a probabilistic measure of similarity based primarily on a Bayesian (MAP) analysis of image differences, leading to a "dual" basis similar to eigenfaces [13]. The performance advantage of this probabilistic matching technique over standard Euclidean nearest-neighbor eigenface matching was recently demonstrated using results from DARPA's 1996 "FERET" face recognition competition, in which this probabilistic matching algorithm was found to be the top performer. We have further developed a simple method of replacing the costly com put ion of nonlinear (online) Bayesian similarity measures by the relatively inexpensive computation of linear (offline) subspace projections and simple (online) Euclidean norms, thus resulting in a significant computational speedup for implementation with very large image databases as typically encountered in real-world applications.

This paper presents probabilistic modeling methods to solve the problem of discriminating between five facial orientations with very little labeled data.