One frequently estimates density functions for which there is little prior knowledge on the shape of the density and for which one wants a flexible and robust estimator (allowing multimodality if it exists). In this context, the methods of choice tend to be finite mixture models and kernel density estimation methods. For mixture modeling, mixtures of Gaussian components are frequently assumed and model choice reduces to the problem of choosing the number k of Gaussian components in the model (Titterington, Smith and Makov, 1986). For kernel density estimation, kernel shapes are typically chosen from a selection of simple unimodal densities such as Gaussian, triangular, or Cauchy densities, and kernel bandwidths are selected in a data-driven manner (Silverman 1986; Scott 1994). As argued by Draper (1996), model uncertainty can contribute significantly to pre- - Also with the Jet Propulsion Laboratory 525-3660, California Institute of Technology, Pasadena, CA 91109 Stacked Density Estimation 669 dictive error in estimation. While usually considered in the context of supervised learning, model uncertainty is also important in unsupervised learning applications such as density estimation. Even when the model class under consideration contains the true density, if we are only given a finite data set, then there is always a chance of selecting the wrong model. Moreover, even if the correct model is selected, there will typically be estimation error in the parameters of that model.

Our model consists of a classical energy mechanism whose output is divided by nonclassical gain control and texture contrast mechanisms. We apply this model to review movies, a stimulus sequence that replicates the stimulation a cell receives during free viewing of natural images. Data were collected from three cells using five different review movies, and the model was fit separately to the data from each movie. For the energy mechanism alone we find modest but significant correlations (rE 0.41, 0.43, 0.59, 0.35) between model and data. These correlations are improved somewhat when we allow for suppressive surround effects (rE G 0.42, 0.56, 0.60, 0.37). In one case the inclusion of a delayed suppressive surround dramatically improves the fit to the data by modifying the time course of the model's response.

Singular value decomposition (SVD) can be viewed as a method for unsupervised training of a network that associates two classes of events reciprocally by linear connections through a single hidden layer. SVD was used to learn and represent relations among very large numbers of words (20k-60k) and very large numbers of natural text passages (lk-70k) in which they occurred. The result was 100-350 dimensional "semantic spaces" in which any trained or newly aibl word or passage could be represented as a vector, and similarities were measured by the cosine of the contained angle between vectors. Good accmacy in simulating human judgments and behaviors has been demonstrated by performance on multiple-choice vocabulary and domain knowledge tests, emulation of expert essay evaluations, and in several other ways. Examples are also given of how the kind of knowledge extracted by this method can be applied.

Conditioning experiments probe the ways that animals make predictions about rewards and punishments and use those predictions to control their behavior. One standard model of conditioning paradigms which involve many conditioned stimuli suggests that individual predictions should be added together. Various key results show that this model fails in some circumstances, and motivate an alternative model, in which there is attentional selection between different available stimuli. The new model is a form of mixture of experts, has a close relationship with some other existing psychological suggestions, and is statistically well-founded.

We explore methods for incorporating prior knowledge about a problem at hand in Support Vector learning machines. We show that both invariances under group transfonnations and prior knowledge about locality in images can be incorporated by constructing appropriate kernel functions.

We discuss a solution to the problem of separating waveforms produced bymultiple cells in an extracellular neural recording. We take an explicitly probabilistic approach, using latent-variable models ofvarying sophistication to describe the distribution of waveforms producedby a single cell. The models range from a single Gaussian distribution of waveforms for each cell to a mixture of hidden Markov models. We stress the overall statistical structure of the approach, allowing the details of the generative model chosen to depend on the specific neural preparation.

Each chart is primarily divided into the three major noise conditions, cf.

We derive a learning algorithm for inferring an overcomplete basis by viewing it as probabilistic model of the observed data.

We present a new approach to reinforcement learning in which the policies consideredby the learning process are constrained by hierarchies of partially specified machines. This allows for the use of prior knowledge to reduce the search space and provides a framework in which knowledge can be transferred across problems and in which component solutions can be recombined to solve larger and more complicated problems. Our approach can be seen as providing a link between reinforcement learning and"behavior-based" or "teleo-reactive" approaches to control. We present provably convergent algorithms for problem-solving and learning withhierarchical machines and demonstrate their effectiveness on a problem with several thousand states.

Friedman's combination rule is quite intuitive: assign to the class that wins the most pairwise comparisons.