We formulate tempo tracking in a Bayesian framework where a tempo tracker is modeled as a stochastic dynamical system. The tempo is modeled as a hidden state variable of the system and is estimated from a MIDI performance by Kalman filtering and smoothing. We also introduce the Tempogram representation, a wavelet-like multiscale expansion of a real performance, on which the Kalman filter operates.

We present a probabilistic generative model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables denote the tempo. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. Exact computation of posterior features such as the MAP state is intractable in this model class, so we introduce Monte Carlo methods for integration and optimization. We compare Markov Chain Monte Carlo (MCMC) methods (such as Gibbs sampling, simulated annealing and iterative improvement) and sequential Monte Carlo methods (particle filters). Our simulation results suggest better results with sequential methods. The methods can be applied in both online and batch scenarios such as tempo tracking and transcription and are thus potentially useful in a number of music applications such as adaptive automatic accompaniment, score typesetting and music information retrieval.

In this article we consider the issue of optimal control in collaborative multi-agent systems with stochastic dynamics. The agents have a joint task in which they have to reach a number of target states. The dynamics of the agents contains additive control and additive noise, and the autonomous part factorizes over the agents. Full observation of the global state is assumed. The goal is to minimize the accumulated joint cost, which consists of integrated instantaneous costs and a joint end cost. The joint end cost expresses the joint task of the agents. The instantaneous costs are quadratic in the control and factorize over the agents. The optimal control is given as a weighted linear combination of single-agent to single-target controls. The single-agent to single-target controls are expressed in terms of diffusion processes. These controls, when not closed form expressions, are formulated in terms of path integrals, which are calculated approximately by Metropolis-Hastings sampling. The weights in the control are interpreted as marginals of a joint distribution over agent to target assignments. The structure of the latter is represented by a graphical model, and the marginals are obtained by graphical model inference. Exact inference of the graphical model will break down in large systems, and so approximate inference methods are needed. We use naive mean field approximation and belief propagation to approximate the optimal control in systems with linear dynamics. We compare the approximate inference methods with the exact solution, and we show that they can accurately compute the optimal control. Finally, we demonstrate the control method in multi-agent systems with nonlinear dynamics consisting of up to 80 agents that have to reach an equal number of target states.

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We give an error potential for selforganising learning rules. The gradient of this error potential leads to the well-known learning rule of Kohonen, except for the determination of the "winning" unit. The existence of an error potential facilitates a global description of the learning process. A one-dimensional topological map is treated as an example.

An attempt is made to study learning in neural networks with local minima. For small learning parameters g, the transition time from one mimimum to another is asymptotica11y given by exp(g/q), with g, a constant independent of g, called the reference learning parameter. A general scheme to calculate the reference learning parameter is presented. This scheme is valid for a large class of learning rules.

Bayesian network are widely accepted as models for reasoning with uncertainty. In this chapter we focus on models that are created using domain expertise only. After a short review of Bayesian networks models and common Bayesian network modeling approaches, we will discuss in more detail three applications of Bayesian networks. With these applications, we aim to illustrate the modeling power and flexibility of the Bayesian networks that goes beyond the standard textbook applications. The first network is applied in a system for medical diagnostic decision support. A distinguishing feature of this network is the large amount of variables in the model. The second one involves an application for petrophysical decision support to determine the mineral content of a well based on borehole measurements. This model differs from standard Bayesian networks by its continuous variables and nonlinear relations. Finally, we will discuss an application for victim identification by kinship analysis based on DNA profiles. The distinguishing feature in this application is that Bayesian networks are generated and computed on-the-fly based on case information.