A. Taylan Cemgil

Variational and stochastic inference for Bayesian source separation

We tackle the general linear instantaneous model (possibly underdetermined and noisy) where we model the source prior with a Student t distribution. The conjugate-exponential characterisation of the t distribution as an infinite mixture of scaled Gaussians enables us to do efficient inference. We study two well-known inference methods, Gibbs sampler and variational Bayes for Bayesian source separation. We derive both techniques as local message passing algorithms to highlight their algorithmic similarities and to contrast their different convergence characteristics and computational requirements. Our simulation results suggest that typical posterior distributions in source separation have multiple local maxima. Therefore we propose a hybrid approach where we explore the state space with a Gibbs sampler and then switch to a deterministic algorithm. This approach seems to be able to combine the speed of the variational approach with the robustness of the Gibbs sampler.

Graphical Models for Time-Series

Time-series analysis is central to many problems in signal processing, including acoustics, image processing, vision, tracking, information retrieval, and finance, to name a few. Because of the wide base of application areas, having a common description of the models is useful in transferring ideas between the various communities. Graphical models provide a compact way to represent such models and thereby rapidly transfer ideas. We will discuss briefly how classical timeseries models such as Kalman filters and hidden Markov models (HMMs) can be represented as graphical models and critically how this representation differs from other common graphical representations such as state-transition and block diagrams. We will use this framework to show how one may easily envisage novel models and gain insight into their computational implementation.

Bayesian Time Series Models

Whats going to happen next? Time series data hold the answers, and Bayesian methods represent the cutting edge in learning what they have to say. This ambitious book is the first unified treatment of the emerging knowledge-base in Bayesian time series techniques. Exploiting the unifying framework of probabilistic graphical models, the book covers approximation schemes, both Monte Carlo and deterministic, and introduces switching, multi-object, non-parametric and agent-based models in a variety of application environments. It demonstrates that the basic framework supports the rapid creation of models tailored to specific applications and gives insight into the computational complexity of their implementation. The authors span traditional disciplines such as statistics and engineering and the more recently established areas of machine learning and pattern recognition. Readers with a basic understanding of applied probability, but no experience with time series analysis, are guided from fundamental concepts to the state-of-the-art in research and practice.

Conjugate gamma Markov random fields for modelling nonstationary sources

In modelling nonstationary sources, one possible strategy is to define a latent process of strictly positive variables to model variations in second order statistics of the underlying process. This can be achieved, for example, by passing a Gaussian process through a positive nonlinearity or defining a discrete state Markov chain where each state encodes a certain regime. However, models with such constructs turn out to be either not very flexible or non-conjugate, making inference somewhat harder. In this paper, we introduce a conjugate (inverse-) gamma Markov Random field model that allows random fluctuations on variances which are useful as priors for nonstationary time-frequency energy distributions. The main idea is to introduce auxiliary variables such that full conditional distributions and sufficient statistics are readily available as closed form expressions. This allows straightforward implementation of a Gibbs sampler or a variational algorithm. We illustrate our approach on denoising and single channel source separation.

Unsupervised single-channel source separation using bayesian NMF

We propose a prior structure for single-channel audio source separation using Non-Negative Matrix Factorisation. For the tonal and percussive signals, the model assigns different prior distributions to the corresponding parts of the template and excitation matrices. This partitioning enables not only more realistic modelling, but also a deterministic way to group the components into sources. This also prevents the possibility of not detecting/assigning a component and remove the need for a dataset and training. Our method only needs the number of components of each source to be set, but this does not play a crucial role in the performance. Very promising results can be obtained using the model with too few design decisions and moderate time complexity.

Dynamic bayesian networks for visual surveillance with distributed cameras

This paper presents a surveillance system for tracking multiple people through a wide area with sparsely distributed cameras. The computational core of the system is an adaptive probabilistic model for reasoning about peoples appearances, locations and identities. The system consists of two processing levels. At the low-level, individual persons are detected in the video frames and tracked at a single camera. At the high-level, a probabilistic framework is applied for estimation of identities and camera-to-camera trajectories of people. The system is validated in a real-world office environment with seven color cameras.

Sequential Inference for Factorial Changepoint Models

Conditional Gaussian changepoint models are an interesting subclass of jump-Markov dynamic linear systems, in which, unlike the majority of such intractable hybrid models, exact inference is achievable in polynomial time. However, many applications of interest involve several simultaneously unfolding processes with occasional regime switches and shared observations. In such scenarios, a factorial model, where each process is modelled by a changepoint model is more natural. In this paper, we derive a sequential Monte Carlo algorithm, reminiscent to the Mixture Kalman filter (MKF) [1]. However, unlike MKF, the factorial structure of our model prohibits the computation of the posterior filtering density (the optimal proposal distribution). Even evaluating the likelihood conditioned on a few switch configurations can be time consuming. Therefore, we derive a propagation algorithm (upward-downward) that exploits the factorial structure of the model and facilitates computing Kalman filtering recursions in information form without the need for inverting large matrices. To motivate the utility of the model, we illustrate our approach on a large model for polyphonic pitch tracking.

Gamma Markov Random fields for audio source modelling

Audio processing tasks, such as source separation or denoising, require the construction of realistic models that reflect physical properties of audio signals. In this paper, we modelled the variances of time-frequency coefficients of audio signals with Gamma Markov random fields (GMRFs) so that the dependencies between coefficients are captured. There is positive correlation between consecutive variance variables in this model and the strength of this correlation is determined by the coupling hyperparameters. Inference can be carried out using the Gibbs sampler or variational Bayes because the model is conditionally conjugate. However, the optimisation of the hyperparameters is not straightforward because of the intractable normalising constant. In this work, we used this model in denoising and single-channel source separation problems. The hyperparameters of the model are optimised using contrastive divergence and inference is performed using the Gibbs sampler

Bayesian Time Series Models: Inference and estimation in probabilistic time series models

© Cambridge University Press 2011. The term ‘time series’ refers to data that can be represented as a sequence. This includes for example financial data in which the sequence index indicates time, and genetic data (e.g. ACATGC …) in which the sequence index has no temporal meaning. In this tutorial we give an overview of discrete-time probabilistic models, which are the subject of most chapters in this book, with continuous-time models being discussed separately in Chapters 4, 6, 11 and 17. Throughout our focus is on the basic algorithmic issues underlying time series, rather than on surveying the wide field of applications.

Bayesian methods for multimedia signal processing

In the last years, there have been a significant growth of multimedia information processing applications that employ ideas from statistical machine learning and probabilistic modeling. In this paradigm, multimedia data (music, audio, video, images, text, ...) are viewed as realizations from highly structured stochastic processes. Once a model is constructed, several interesting problems such as transcription, coding, classification, restoration, tracking, source separation or resynthesis etc. can be formulated as Bayesian inference problems. In this context, graphical models provide a language to construct models for quantification of prior knowledge. Unknown parameters in this specification are estimated by probabilistic inference. Often, however, the problem size poses an important challenge and in order to render the approach feasible, specialized inference methods need to be tailored to improve the computational speed and efficiency.n The scope of the proposed tutorial is as follows: First, we will review the fundamentals of probabilistic models, with some focus on music, video and text data. Then, we will discuss the numerical techniques for inference in these models. In particular, we will review exact inference, approximate stochastic inference techniques such as Markov Chain Monte Carlo, Sequential Monte Carlo and deterministic (variational) inference techniques. Our ultimate aim is to provide a basic understanding of probabilistic modeling for multimedia processing, associated computational techniques and a roadmap such that information retrieval researchers new to the Bayesian approach can orient themselves in the relevant literature and understand the current state of the art.