We consider unsupervised estimation of mixtures of discrete graphical models, where the class variable is hidden and each mixture component can have a potentially different Markov graph structure and parameters over the observed variables. We propose a novel method for estimating the mixture components with provable guarantees. Our output is a tree-mixture model which serves as a good approximation to the underlying graphical model mixture. The sample and computational requirements for our method scale as poly(p, r), for an r-component mixture of p-variate graphical models, for a wide class of models which includes tree mixtures and mixtures over bounded degree graphs.

We present a dynamic nonlinear generative model for visual motion based on a latent representation of binary-gated Gaussian variables. Trained on sequences of images, the model learns to represent different movement directions in different variables. We use an online approximate inference scheme that can be mapped to the dynamics of networks of neurons.

We consider the problem of estimating Shannon's entropy H in the under-sampled regime, where the number of possible symbols may be unknown or countably infinite. Dirichlet and Pitman-Yor processes provide tractable prior distributions over the space of countably infinite discrete distributions, and have found major applications in Bayesian non-parametric statistics and machine learning. Here we show that they provide natural priors for Bayesian entropy estimation, due to the analytic tractability of the moments of the induced posterior distribution over entropy H. We derive formulas for the posterior mean and variance of H given data. However, we show that a fixed Dirichlet or Pitman-Yor process prior implies a narrow prior on H, meaning the prior strongly determines the estimate in the under-sampled regime. We therefore define a family of continuous mixing measures such that the resulting mixture of Dirichlet or Pitman-Yor processes produces an approximately flat prior over H. We explore the theoretical properties of the resulting estimators and show that they perform well on data sampled from both exponential and power-law tailed distributions.

We develop a novel Bayesian nonparametric model for random bipartite graphs. The model is based on the theory of completely random measures and is able to handle a potentially infinite number of nodes. We show that the model has appealing properties and in particular it may exhibit a power-law behavior. We derive a posterior characterization, a generative process for network growth, and a simple Gibbs sampler for posterior simulation. Our model is shown to be well fitted to several real-world social networks.

Accurate and detailed models of neurodegenerative disease progression are crucially important for reliable early diagnosis and the determination of effective treatments. We introduce the ALPACA (Alzheimer's disease Probabilistic Cascades) model, a generative model linking latent Alzheimer's progression dynamics to observable biomarker data. In contrast with previous works which model disease progression as a fixed event ordering, we explicitly model the variability over such orderings among patients which is more realistic, particularly for highly detailed progression models. We describe efficient learning algorithms for ALPACA and discuss promising experimental results on a real cohort of Alzheimer's patients from the Alzheimer's Disease Neuroimaging Initiative.

Overview: The authors propose the Gibbs error criterion for active learning; seeking the samples that maximize the expected Gibbs error under the current posterior. They propose a greedy algorithm that maximises this criterion (Max-GEC). The objective reduces to maximising a specific instance of the Tsallis entropy of the predictive distribution which is very similar to Maximum Entropy Sampling (MES) which uses the Shannon entropy of the predictive distribution. They consider the non-adaptive, adaptive and batch settings separately, and in each setting they prove using submodularity results that the greedy approach achieves near-maximal performance compared to optimal policy. They show how to implement their fully adaptive policy (approximately) in CRFs with application to named entity recognition, and implement the batch algorithm with a Naive Bayes classifier, with application to a text classification task.

We introduce a new objective function for pool-based Bayesian active learning with probabilistic hypotheses. This objective function, called the policy Gibbs error, is the expected error rate of a random classifier drawn from the prior distribution on the examples adaptively selected by the active learning policy. Exact maximization of the policy Gibbs error is hard, so we propose a greedy strategy that maximizes the Gibbs error at each iteration, where the Gibbs error on an instance is the expected error of a random classifier selected from the posterior label distribution on that instance. We apply this maximum Gibbs error criterion to three active learning scenarios: non-adaptive, adaptive, and batch active learning. In each scenario, we prove that the criterion achieves near-maximal policy Gibbs error when constrained to a fixed budget. For practical implementations, we provide approximations to the maximum Gibbs error criterion for Bayesian conditional random fields and transductive Naive Bayes. Our experimental results on a named entity recognition task and a text classification task show that the maximum Gibbs error criterion is an effective active learning criterion for noisy models.

The idea of computer vision as the Bayesian inverse problem to computer graphics has a long history and an appealing elegance, but it has proved difficult to directly implement. Instead, most vision tasks are approached via complex bottom-up processing pipelines. Here we show that it is possible to write short, simple probabilistic graphics programs that define flexible generative models and to automatically invert them to interpret real-world images. Generative probabilistic graphics programs (GPGP) consist of a stochastic scene generator, a renderer based on graphics software, a stochastic likelihood model linking the renderer's output and the data, and latent variables that adjust the fidelity of the renderer and the tolerance of the likelihood. Representations and algorithms from computer graphics are used as the deterministic backbone for highly approximate and stochastic generative models. This formulation combines probabilistic programming, computer graphics, and approximate Bayesian computation, and depends only on generalpurpose, automatic inference techniques. We describe two applications: reading sequences of degraded and adversarially obscured characters, and inferring 3D road models from vehicle-mounted camera images. Each of the probabilistic graphics programs we present relies on under 20 lines of probabilistic code, and yields accurate, approximately Bayesian inferences about real-world images.

Shannon's entropy is a basic quantity in information theory, and a fundamental building block for the analysis of neural codes. Estimating the entropy of a discrete distribution from samples is an important and difficult problem that has received considerable attention in statistics and theoretical neuroscience. However, neural responses have characteristic statistical structure that generic entropy estimators fail to exploit. For example, existing Bayesian entropy estimators make the naive assumption that all spike words are equally likely a priori, which makes for an inefficient allocation of prior probability mass in cases where spikes are sparse. Here we develop Bayesian estimators for the entropy of binary spike trains using priors designed to flexibly exploit the statistical structure of simultaneouslyrecorded spike responses.

Causal identification in causal Bayes nets (CBNs) is an important tool in causal inference allowing the derivation of interventional distributions from observational distributions where this is possible in principle. However, most existing formulations of causal identification using techniques such as d-separation and do-calculus are expressed within the mathematical language of classical probability theory on CBNs. However, there are many causal settings where probability theory and hence current causal identification techniques are inapplicable such as relational databases, dataflow programs such as hardware description languages, distributed systems and most modern machine learning algorithms. We show that this restriction can be lifted by replacing the use of classical probability theory with the alternative axiomatic foundation of symmetric monoidal categories. In this alternative axiomatization, we show how an unambiguous and clean distinction can be drawn between the general syntax of causal models and any specific semantic implementation of that causal model. This allows a purely syntactic algorithmic description of general causal identification by a translation of recent formulations of the general ID algorithm through fixing. Our description is given entirely in terms of the non-parametric ADMG structure specifying a causal model and the algebraic signature of the corresponding monoidal category, to which a sequence of manipulations is then applied so as to arrive at a modified monoidal category in which the desired, purely syntactic interventional causal model, is obtained. We use this idea to derive purely syntactic analogues of classical back-door and front-door causal adjustment, and illustrate an application to a more complex causal model.