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.

Rao-Blackwellization is an approximation technique for probabilistic inference that flexibly combines exact inference with sampling. It is useful in models where conditioning on some of the variables leaves a simpler inference problem that can be solved tractably. This paper presents Sample Propagation, an efficient implementation of Rao-Blackwellized approximate inference for a large class of models. Sample Propagation tightly integrates sampling with message passing in a junction tree, and is named for its simple, appealing structure: it walks the clusters of a junction tree, sampling some of the current cluster's variables and then passing a message to one of its neighbors. We discuss the application of Sample Propagation to conditional Gaussian inference problems such as switching linear dynamical systems.

Discrete Fourier transforms and other related Fourier methods have been practically implementable due to the fast Fourier transform (FFT). However there are many situations where doing fast Fourier transforms without complete data would be desirable. In this paper it is recognised that formulating the FFT algorithm as a belief network allows suitable priors to be set for the Fourier coefficients. Furthermore efficient generalised belief propagation methods between clusters of four nodes enable the Fourier coefficients to be inferred and the missing data to be estimated in near to O(n log n) time, where n is the total of the given and missing data points. This method is compared with a number of common approaches such as setting missing data to zero or to interpolation. It is tested on generated data and for a Fourier analysis of a damaged audio signal.

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.

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.

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.

In this paper we introduce a new underlying probabilistic model for principal component analysis (PCA). Our formulation interprets PCA as a particular Gaussian process prior on a mapping from a latent space to the observed data-space. We show that if the prior's covariance function constrains the mappings to be linear the model is equivalent to PCA, we then extend the model by considering less restrictive covariance functions which allow nonlinear mappings. This more general Gaussian process latent variable model (GPLVM) is then evaluated as an approach to the visualisation of high dimensional data for three different data-sets. Additionally our nonlinear algorithm can be further kernelised leading to'twin kernel PCA' in which a mapping between feature spaces occurs.

What happens to the optimal interpretation of noisy data when there exists more than one equally plausible interpretation of the data? In a Bayesian model-learning framework the answer depends on the prior expectations of the dynamics of the model parameter that is to be inferred from the data. Local time constraints on the priors are insufficient to pick one interpretation over another. On the other hand, nonlocal time constraints, induced by a 1/f noise spectrum of the priors, is shown to permit learning of a specific model parameter even when there are infinitely many equally plausible interpretations of the data. This transition is inferred by a remarkable mapping of the model estimation problem to a dissipative physical system, allowing the use of powerful statistical mechanical methods to uncover the transition from indeterminate to determinate model learning.

In models that define probabilities via energies, maximum likelihood learning typically involves using Markov Chain Monte Carlo to sample from the model's distribution. If the Markov chain is started at the data distribution, learning often works well even if the chain is only run for a few time steps [3]. But if the data distribution contains modes separated by regions of very low density, brief MCMC will not ensure that different modes have the correct relative energies because it cannot move particles from one mode to another. We show how to improve brief MCMC by allowing long-range moves that are suggested by the data distribution. If the model is approximately correct, these long-range moves have a reasonable acceptance rate.

Recent developments in grid-based and point-based approximation algorithms for POMDPs have greatly improved the tractability of POMDP planning. These approaches operate on sets of belief points by individually learning a value function for each point. In reality, belief points exist in a highly-structured metric simplex, but current POMDP algorithms do not exploit this property. This paper presents a new metric-tree algorithm which can be used in the context of POMDP planning to sort belief points spatially, and then perform fast value function updates over groups of points. We present results showing that this approach can reduce computation in point-based POMDP algorithms for a wide range of problems.