We introduce a class of nonstationary covariance functions for Gaussian process (GP) regression. Nonstationary covariance functions allow the model to adapt to functions whose smoothness varies with the inputs. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the regression function is controlled by a parameter, freeing one from fixing the differentiability in advance. In experiments, the nonstationary GP regression model performs well when the input space is two or three dimensions, outperforming a neural network model and Bayesian free-knot spline models, and competitive with a Bayesian neural network, but is outperformed in one dimension by a state-of-the-art Bayesian free-knot spline model.

We describe a nonparametric Bayesian approach to generalizing from few labeled examples, guided by a larger set of unlabeled objects and the assumption of a latent tree-structure to the domain. The tree (or a distribution over trees) may be inferred using the unlabeled data. A prior over concepts generated by a mutation process on the inferred tree(s) allows efficient computation of the optimal Bayesian classification function from the labeled examples. We test our approach on eight real-world datasets.

Recently, relevance vector machines (RVM) have been fashioned from a sparse Bayesian learning (SBL) framework to perform supervised learning using a weight prior that encourages sparsity of representation. The methodology incorporates an additional set of hyperparameters governing the prior, one for each weight, and then adopts a specific approximation to the full marginalization over all weights and hyperparameters. Despite its empirical success however, no rigorous motivation for this particular approximation is currently available. To address this issue, we demonstrate that SBL can be recast as the application of a rigorous variational approximation to the full model by expressing the prior in a dual form. This formulation obviates the necessity of assuming any hyperpriors and leads to natural, intuitive explanations of why sparsity is achieved in practice.

Approximation structure plays an important role in inference on loopy graphs. As a tractable structure, tree approximations have been utilized in the variational method of Ghahramani & Jordan (1997) and the sequential projection method of Frey et al. (2000). However, belief propagation represents each factor of the graph with a product of single-node messages. In this paper, belief propagation is extended to represent factors with tree approximations, by way of the expectation propagation framework. That is, each factor sends a "message" to all pairs of nodes in a tree structure. The result is more accurate inferences and more frequent convergence than ordinary belief propagation, at a lower cost than variational trees or double-loop algorithms.

We address the problem of learning topic hierarchies from data. The model selection problem in this domain is daunting--which of the large collection of possible trees to use? We take a Bayesian approach, generating an appropriate prior via a distribution on partitions that we refer to as the nested Chinese restaurant process. This nonparametric prior allows arbitrarily large branching factors and readily accommodates growing data collections. We build a hierarchical topic model by combining this prior with a likelihood that is based on a hierarchical variant of latent Dirichlet allocation. We illustrate our approach on simulated data and with an application to the modeling of NIPS abstracts.

In the problem of probability forecasting the learner's goal is to output, given a training set and a new object, a suitable probability measure on the possible values of the new object's label. An online algorithm for probability forecasting is said to be well-calibrated if the probabilities it outputs agree with the observed frequencies. We give a natural nonasymptotic formalization of the notion of well-calibratedness, which we then study under the assumption of randomness (the object/label pairs are independent and identically distributed). It turns out that, although no probability forecasting algorithm is automatically well-calibrated in our sense, there exists a wide class of algorithms for "multiprobability forecasting" (such algorithms are allowed to output a set, ideally very narrow, of probability measures) which satisfy this property; we call the algorithms in this class "Venn probability machines". Our experimental results demonstrate that a 1-Nearest Neighbor Venn probability machine performs reasonably well on a standard benchmark data set, and one of our theoretical results asserts that a simple Venn probability machine asymptotically approaches the true conditional probabilities regardless, and without knowledge, of the true probability measure generating the examples.

Consider a number of moving points, where each point is attached to a joint of the human body and projected onto an image plane. Johannson showed that humans can effortlessly detect and recognize the presence of other humans from such displays. This is true even when some of the body points are missing (e.g. because of occlusion) and unrelated clutter points are added to the display. We are interested in replicating this ability in a machine. To this end, we present a labelling and detection scheme in a probabilistic framework. Our method is based on representing the joint probability density of positions and velocities of body points with a graphical model, and using Loopy Belief Propagation to calculate a likely interpretation of the scene. Furthermore, we introduce a global variable representing the body's centroid. Experiments on one motion-captured sequence suggest that our scheme improves on the accuracy of a previous approach based on triangulated graphical models, especially when very few parts are visible. The improvement is due both to the more general graph structure we use and, more significantly, to the introduction of the centroid variable.

We present a Bayesian approach to color constancy which utilizes a non-Gaussian probabilistic model of the image formation process. The parameters of this model are estimated directly from an uncalibrated image set and a small number of additional algorithmic parameters are chosen using cross validation. The algorithm is empirically shown to exhibit RMS error lower than other color constancy algorithms based on the Lambertian surface reflectance model when estimating the illuminants of a set of test images. This is demonstrated via a direct performance comparison utilizing a publicly available set of real world test images and code base.

In this paper we present Discriminative Random Fields (DRF), a discriminative framework for the classification of natural image regions by incorporating neighborhood spatial dependencies in the labels as well as the observed data. The proposed model exploits local discriminative models and allows to relax the assumption of conditional independence of the observed data given the labels, commonly used in the Markov Random Field (MRF) framework. The parameters of the DRF model are learned using penalized maximum pseudo-likelihood method. Furthermore, the form of the DRF model allows the MAP inference for binary classification problems using the graph min-cut algorithms. The performance of the model was verified on the synthetic as well as the real-world images. The DRF model outperforms the MRF model in the experiments.

When we model a higher order functions, such as learning and memory, we face a difficulty of comparing neural activities with hidden variables that depend on the history of sensory and motor signals and the dynamics of the network. Here, we propose novel method for estimating hidden variables of a learning agent, such as connection weights from sequences of observable variables. Bayesian estimation is a method to estimate the posterior probability of hidden variables from observable data sequence using a dynamic model of hidden and observable variables. In this paper, we apply particle filter for estimating internal parameters and metaparameters of a reinforcement learning model. We verified the effectiveness of the method using both artificial data and real animal behavioral data.