This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons.

We present a probabilistic generarive 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. Ex- act 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.

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori information with new information obtained by measuring some observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.). When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have a large number of model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available. The most well known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.

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This paper shows how state-of-the-art state estimation techniques can be used to provide efficient solutions to the difficult problem of real-time diagnosis in mobile robots. The power of the adopted estimation techniques resides in our ability to combine particle filters with classical algorithms, such as Kalman filters. We demonstrate these techniques in two scenarios: a mobile waiter robot and planetary rovers designed by NASA for Mars exploration. Keywords—Diagnosis, Rao–Blackwellized particle filtering, robotics, state estimation. I.

Autonomous vehicles need to be able to plan trajectories to a specified goal that avoid obstacles, and are robust to the inherent uncertainty in the problem. This uncertainty arises due to uncertain state estimation, disturbances and modeling errors. Previous solutions to the robust path planning problem solved this problem using a finite horizon optimal stochastic control approach. This approach finds the optimal path subject to chance constraints, which ensure that the probability of collision with obstacles is below a given threshold. This approach is limited to problems where all uncertain distributions are Gaussian, and typically result in highly conservative plans. In many cases, however, the Gaussian assumption is invalid; for example in the case of localization, the belief state about a vehicle’s position can consist of highly non-Gaussian, even multimodal, distributions. In this paper we present a novel method for finite horizon stochastic control of dynamic systems subject to chance constraints. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore the approximation becomes exact as the number of particles tends to infinity. For a general class of chance constrained problems with linear system dynamics, we show that the approximate problem can be solved using efficient Mixed-Integer Linear Programming techniques. We apply the new method to aircraft control in turbulence, and show simulation results that demonstrate the efficacy of the approach. I.

We develop a Bayesian procedure for estimation and inference for spatial models of roll call voting. This approach is extremely flexible, applicable to any legislative setting, irrespective of size, the extremism of the legislators ’ voting histories, or the number of roll calls available for analysis. The model is easily extended to let other sources of information inform the analysis of roll call data, such as the number and nature of the underlying dimensions, the presence of party whipping, the determinants of legislator preferences, and the evolution of the legislative agenda; this is especially helpful since generally it is inappropriate to use estimates of extant methods (usually generated under assumptions of sincere voting) to test models embodying alternate assumptions (e.g., log-rolling, party discipline). A Bayesian approach also provides a coherent framework for estimation and inference with roll call data that eludes extant methods; moreover, via Bayesian simulation methods, it is straightforward to generate uncertainty assessments or hypothesis tests concerning any auxiliary quantity of interest or to formally compare models. In a series of examples we show how our method is easily extended to accommodate theoretically interesting models of legislative behavior. Our goal is to provide a statistical framework for combining the measurement of legislative preferences with tests of models of legislative behavior. Modern studies of legislative behavior focus

This paper shows that the probability of extreme default losses on portfolios of U.S. corporate debt is much greater than would be estimated under the standard assumption that default correlation arises only from exposure to observable risk factors. At the high confidence levels at which bank loan portfolio and CDO default losses are typically measured for economic-capital and rating purposes, our empirical results indicate that conventionally based estimates are downward biased by a full order of magnitude on test portfolios. Our estimates are based on U.S. public non-financial firms existing between 1979 and 2004. We find strong evidence for the presence of common latent factors, even when controlling for observable factors that provide the most accurate available model of firm-by-firm default probabilities. ∗ We are grateful for financial support from Moody’s Corporation and Morgan Stanley, and for research assistance from Sabri Oncu and Vineet Bhagwat. We are also grateful for remarks from Torben Andersen, André Lucas, Richard Cantor, Stav Gaon, Tyler Shumway, and especially Michael Johannes. This revision is much improved because of suggestions by a referee, an associate editor, and Campbell Harvey. We are thankful to Moodys and to Ed Altman for generous assistance with data. Duffie is at The Graduate School of Business, Stanford University. Eckner and Horel are at Merrill Lynch. Saita is at Lehman

The Monte Carlo method is rapidly becoming the model of choice for simulating light transport in tissue. This paper provides all the details necessary for implementation of a Monte Carlo program. Variance reduction schemes that improve the efficiency of the Monte Carlo method are discussed. Analytic expressions facilitating convolution calculations for finite flat and Gaussian beams are included. Useful validation benchmarks are presented. 1

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