nils-herrmann/clean_scirepeval huggingface dataset:
Convergence of stochastic approximation for Lyapunov stable dynamics : a proof from rst principles

In this short note we present a proof, aimed at beginners, of the convergence of the stochastic approximation recursion i+1 = i+ i+1h( i)+ i+1 i+1 under the classical 0-level Kushner-Clark noise condition when the underlying dynamic is Lyapunov stable. The technique of proof relies on simple calculus arguments and bypasses the need for the introduction of the associated continuous time ODE. Future work includes the extension of the result to the case where the 0-level Kushner-Clark condition is replaced with a r-level condition for r > 0 . We study the convergence of the recursion i+1 = i + i+1h( i) + i+1 i+1; for a given function h : ! R , a sequence of stepsizes f ig and a noise sequence f ig under simple and veriable conditions. We dene for any T > 0 m(n; T ) := max ( j n : j X
Motivation and Basic Principles of the Monte Carlo Method

The modern history of Monte Carlo techniques dates back from the 1940’s and the Manhattan project. There are earlier descriptions of Monte Carlo experiments, Buffon’s famous needle experiment is one them, but examples have been traced back to Babylonian and old testament times [13]. As we shall see these techniques are particularly useful in scenarios where it is of interest to perform calculations that involve explicitly or implicitly a probability distribution π on a space X (typically X ⊂ Rx for some integer nx), for which closed-form calculations cannot be carried out due to the algebraic complexity of the problem. As we shall see the main principle of Monte Carlo techniques consists of replacing the algebraic representation of π, e.g. 1/ √ 2π exp(−1 2 x 2) with a sample or population representation of π, e.g. a set of samples X1, X2, . . . , XN iid ∼ π(x) = 1/ √ 2π exp(−1 2 x 2). This proves in practice to be extremely powerful as difficult if not impossible exact algebraic calculations are typically replaced with simple calculations in the sample domain. One should however bear in mind that these are random approximations of the true quantity of interest. An important scenario where Monte Carlo methods can be of great help is when one is interested in evaluating expectations of functions, say f , of the type Eπ (f(X)) where π is the probability distributions that defines the expectation. The nature of the approach, where algebraic quantities are approximated by random quantities, requires one to quantify the random fluctuations around the true desired value. As we shall see, the power of Monte Carlo techniques lies in the fact that the rate at which the approximation converges towards the true value of interest is immune to the dimension nx of the space X where π is defined. This is the second interest of Monte Carlo techniques. These numerical techniques have been widely used in physics over the last 50 years, but their interest in the context of Bayesian statistics and more generally statistics was only fully realized in the late eighties early nineties. Although we will here mostly focus on their application in statistics, one should bear in mind that the material presented in this introduction to the topic has applications far beyond statistics. The prerequisites for this introduction are a basic first year undergraduate background in probability and statistics.
Joint space-time trellis code detection and MIMO equalisation via particle filtering

In this contribution we address the problem of MIMO equalisation and transmit diversity. More specifically, we develop a method that jointly decodes space time trellis coded modulation (STTCM) encoded signals and equalises the MIMO channels. The optimal solution for such a setup would require construction of a super-trellis that describes a finite state machine resulting from the combination of MIMO channel and STTCM trellises. However, the number of states in this super-trellis is typically gigantic and as such prohibits practical application. Our solution is based on sequential Monte Carlo technique (aka particle filter), which has gained much interest recently in the communications community. The structure of the STTCM code is incorporated into the proposal distribution of the particle filter. As a results a significant reduction in complexity has been achieved as compared not only to the brute force super-trellis approach, but also when compared to the conventional "turbo" solution.
Reversible Jump Markov Chain Monte Carlo Strategies for Bayesian Model Selection in Autoregressive Processes

This paper addresses the problem of Bayesian inference in autoregressive (AR) processes in the case where the correct model order is unknown. Original hierarchical prior models that allow the stationarity of the model to be enforced are proposed. Obtaining the quantities of interest, such as parameter estimates, predictions of future values of the time series, posterior model‐order probabilities, etc., requires integration with respect to the full posterior distribution, an operation which is analytically intractable. Reversible jump Markov chain Monte Carlo (MCMC) algorithms are developed to perform the required integration implicitly by simulating from the posterior distribution. The methods developed are evaluated in simulation studies on a number of synthetic and real data sets.
Particle smoothing techniques with turbo principle for MIMO systems

We address the problem of space-time communications in the context where the turbo principle is used in order to improve the robustness of the transmission. Recovering the sequence of emitted symbols in the wide-band context is known to lead to a computationally very challenging problem. We propose an adaptive numerical approximation of the BCJR algorithm, based on particle filtering/smoothing techniques. When combined with classical estimation techniques for the channel and variance of the observation noise, this algorithm is very efficient, and scales very favourably with the dimension of the problem. This is demonstrated through numerical simulations.
Joint blind and semi-blind detection and channel estimation for space-time trellis coded systems

The paper considers a multiple-input multiple-output (MIMO) communication system, which uses space-time trellis coding (STTC). A novel method of decoding STTC without a need to transmit training sequences is developed. The technique uses only a single channel estimate to acquire a complete set of the channels' estimates while performing STTC detection. The method is akin to blind trellis search techniques (per-survivor processing - PSP) and adaptive Viterbi. Our solution consists of the deployment of a bank of Kalman filters. The bank of Kalman filters is coupled with Viterbi type decoders, which produce tentative decisions based on Kalman channel predictions. In return, the Kalman filters use the tentative decisions to update and track the MIMO channels corresponding to a number of tracked hypotheses. The proposed technique is particularly applicable to space-time systems operating in rapidly fading environments, where STTC can be decoded and MIMO channels efficiently tracked without relying on periodic pilot sequences.
Efficient particle filtering for jump Markov systems. Application to time-varying autoregressions

We present an efficient particle filtering method to perform optimal estimation in jump Markov (nonlinear) systems (JMSs). Such processes consist of a mixture of heterogeneous models and possess a natural hierarchical structure. We take advantage of these specificities in order to develop a generic filtering methodology for these models. The method relies on an original and nontrivial combination of techniques that have been presented recently in the filtering literature, namely, the auxiliary particle filter and the unscented transform. This algorithm is applied to the complex problem of time-varying autoregressive estimation with an unknown time-varying model order. More precisely, we develop an attractive and original probabilistic model that relies on a flexible pole representation that easily lends itself to interpretations. We show that this problem can be formulated as a JMS and that the associated filtering problem can be efficiently addressed using the generic methodology developed in this paper. Simulations demonstrate the performance of our method compared to standard particle filtering techniques.
Monte Carlo Methods for Absolute Beginners

The modern history of Monte Carlo techniques dates back from the 1940’s and the Manhattan project. There are earlier descriptions of Monte Carlo experiments, Buffon’s famous needle experiment is one them, but examples have been traced back to Babylonian and old testament times [13]. As we shall see these techniques are particularly useful in scenarios where it is of interest to perform calculations that involve – explicitly or implicitly – a probability distribution π on a space X (typically \(\mathsf{X} \subset \mathbb{R}^{n_x}\) for some integer n x ), for which closed-form calculations cannot be carried out due to the algebraic complexity of the problem. As we shall see the main principle of Monte Carlo techniques consists of replacing the algebraic representation of π, e.g. \(1/\sqrt{2\pi}\exp(\frac{-1}{2}x^{2}) \) with a sample or population representation of π, e.g. a set of samples \(X_{1},X_{2},\ldots,X_{N}\overset{iid}{\sim}\pi(x)=1/\sqrt{2\pi}\exp(\frac{-1}{2}x^{2})\). This proves in practice to be extremely powerful as difficult – if not impossible -exact algebraic calculations are typically replaced with simple calculations in the sample domain. One should however bear in mind that these are randomapproximations of the true quantity of interest. An important scenario where Monte Carlo methods can be of great help is when one is interested in evaluating expectations of functions, say f, of the type \(\mathbb{E}_{\pi}\left( f(X)\right) \) where π is the probability distributions that defines the expectation. The nature of the approach, where algebraic quantities are approximated by random quantities, requires one to quantify the random fluctuations around the true desired value. As we shall see, the power of Monte Carlo techniques lies in the fact that the rate at which the approximation converges towards the true value of interest is immune to the dimension n x of the space X where π is defined. This is the second interest of Monte Carlo techniques.
Semi-blind identification of wideband MIMO channels via stochastic sampling

In this paper we address the problem of wide-band multiple-input multiple-output (MIMO) channel (multidimensional time invariant FIR filter) identification using Markov chains Monte Carlo methods. Towards this end we develop a novel stochastic sampling technique that produces a sequence of multidimensional channel samples. The method is semi-blind in the sense that it uses a very short training sequence. In such a framework the problem is no longer analytically tractable; hence we resort to stochastic sampling technique. The developed technique samples the channel, the variance of the noise and the symbols in order to build an ergodic Markov chain whose equilibrium distribution is the distribution of interest. The estimates of the MIMO channel and the noise variance are inferred from marginal posterior distributions, which are by-products of the output of the algorithm.
Bayesian deconvolution in nuclear spectroscopy using RJMCMC

This paper addresses the general problem of estimating parameters in nuclear spectroscopy. We present a unified Bayesian formulation to tackle the various aspects of this problem. This includes deconvolution and modelling of both the peaks and background. The peaks are modelled with Gaussian or Lorentzian type functions and the background with cubic B-splines. The number of peaks and spline knots are treated as unknowns and as such are also estimated together with the model parameters. The Bayesian model allows us to define a posterior probability on the parameter space upon which all subsequent Bayesian inference is based. Direct evaluation of this distribution or its derived features such as the conditional expectation is, unfortunately, not possible on account of the need to evaluate high-dimension integrals. As such we resort to a stochastic numerical Bayesian technique, the reversible-jump Markov-chain Monte Carlo(RJMCMC) method.
Bayesian curve fitting using MCMC with applications to signal segmentation

We propose some Bayesian methods to address the problem of fitting a signal modeled by a sequence of piecewise constant linear (in the parameters) regression models, for example, autoregressive or Volterra models. A joint prior distribution is set up over the number of the changepoints/knots, their positions, and over the orders of the linear regression models within each segment if these are unknown. Hierarchical priors are developed and, as the resulting posterior probability distributions and Bayesian estimators do not admit closed-form analytical expressions, reversible jump Markov chain Monte Carlo (MCMC) methods are derived to estimate these quantities. Results are obtained for standard denoising and segmentation of speech data problems that have already been examined in the literature. These results demonstrate the performance of our methods.
Particle filtering for partially observed Gaussian state space models

Solving Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data has many applications for dynamic models. A large number of algorithms based on particle filtering methods, also known as sequential Monte Carlo algorithms, have recently been proposed to solve these problems. We propose a special particle filtering method which uses random mixtures of normal distributions to represent the posterior distributions of partially observed Gaussian state space models. This algorithm is based on a marginalization idea for improving efficiency and can lead to substantial gains over standard algorithms. It differs from previous algorithms which were only applicable to conditionally linear Gaussian state space models. Computer simulations are carried out to evaluate the performance of the proposed algorithm for dynamic tobit and probit models.
Particle methods for Bayesian modeling and enhancement of speech signals

This paper applies time-varying autoregressive (TVAR) models with stochastically evolving parameters to the problem of speech modeling and enhancement. The stochastic evolution models for the TVAR parameters are Markovian diffusion processes. The main aim of the paper is to perform on-line estimation of the clean speech and model parameters and to determine the adequacy of the chosen statistical models. Efficient particle methods are developed to solve the optimal filtering and fixed-lag smoothing problems. The algorithms combine sequential importance sampling (SIS), a selection step and Markov chain Monte Carlo (MCMC) methods. They employ several variance reduction strategies to make the best use of the statistical structure of the model. It is also shown how model adequacy may be determined by combining the particle filter with frequentist methods. The modeling and enhancement performance of the models and estimation algorithms are evaluated in simulation studies on both synthetic and real speech data sets.
Iterative algorithms for state estimation of jump Markov linear systems

Jump Markov linear systems (JMLSs) are linear systems whose parameters evolve with time according to a finite state Markov chain. Given a set of observations, our aim is to estimate the states of the finite state Markov chain and the continuous (in space) states of the linear system. In this paper, we present original deterministic and stochastic iterative algorithms for optimal state estimation of JMLSs. The first stochastic algorithm yields minimum mean square error (MMSE) estimates of the finite state space Markov chain and of the continuous state of the JMLS. A deterministic and a stochastic algorithm are given to obtain the marginal maximum a posteriori (MMAP) sequence estimate of the finite state Markov chain. Finally, a deterministic and a stochastic algorithm are derived to obtain the MMAP sequence estimate of the continuous state of the JMLS. Computer simulations are carried out to evaluate the performance of the proposed algorithms. The problem of deconvolution of Bernoulli-Gaussian (BG) processes and the problem of tracking a maneuvering target are addressed.
Controlled MCMC for Optimal Sampling

In this paper we develop an original and general framework for automatically optimizing the statistical properties of Markov chain Monte Carlo (MCMC) samples, which are typically used to evaluate complex integrals. The Metropolis-Hastings algorithm is the basic building block of classical MCMC methods and requires the choice of a proposal distribution, which usually belongs to a parametric family. The correlation properties together with the exploratory ability of the Markov chain heavily depend on the choice of the proposal distribution. By monitoring the simulated path, our approach allows us to learn “on the fly” the optimal parameters of the proposal distribution for several statistical criteria.
Bayesian Blind and Semi-Blind Equalization of Channels with Markov Inputs

An original full Bayesian approach is developed for blind and semi-blind equalisation of fading channels with Markov inputs. The sequence of discrete symbols is estimated according to a marginal maximum a posteriori criterion; the other unknown parameters are regarded as random nuisance parameters and are integrated out analytically. A batch algorithm is proposed to maximise the marginal posterior distribution. Simulation results are presented to demonstrate the effectiveness of the method.
Improved auxiliary particle filtering: applications to time-varying spectral analysis

This paper addresses optimal estimation for time-varying autoregressive (TVAR) models. First, we propose a statistical model on the time evolution of the frequencies, moduli and real poles instead of a standard model on the AR coefficients, as it makes more sense from a physical viewpoint. Second, optimal estimation involves solving a complex optimal filtering problem which does not admit any closed-form solution. We propose a new particle filtering scheme which is an improvement over the so-called auxiliary particle filter. The hyperparameters timing the evolution of the model parameters are also estimated on-line to make the model robust. Simulations demonstrate the efficiency of both our model and algorithm.
Bayesian deconvolution of noisy filtered point processes

The detection and estimation of filtered point processes using noisy data is an essential requirement in many seismic, ultrasonic, and nuclear applications. We address this joint detection/estimation problem using a Bayesian approach, which allows us to easily include any relevant prior information. Performing Bayesian inference for such a complex model is a challenging computational problem as it requires the evaluation of intricate high-dimensional integrals. We develop here an efficient stochastic procedure based on a reversible jump Markov chain Monte Carlo method to solve this problem and prove the geometric convergence of the algorithm. The proposed model and algorithm are demonstrated on an application arising in nuclear science.
Recursive Monte Carlo algorithms for parameter estimation in general state space models

We present new algorithms that aim at estimating the "static" parameters of a latent variable process in an on-line manner. This new class of on-line algorithms is inspired by Monte Carlo Markov chain (MCMC) methods whose use has been mainly restricted to static problems, i.e., for which the set of observations is fixed. The main interest of this new class of algorithms is that it combines MCMC and particle filtering techniques, for which extensive know-how and literature are now available.
Rao-Blackwellised Particle Filtering Data Augmentation VIa

Arnaud Doucet EE Engineering University of Melbourne Parkville, Victoria 3052 Australia doucet@ee.mu.oz.au In this paper, we extend the Rao-Blackwellised particle filtering method to more complex hybrid models consisting of Gaussian latent variables and discrete observations. This is accomplished by augmenting the models with artificial variables that enable us to apply Rao-Blackwellisation. Other improvements include the design of an optimal importance proposal distribution and being able to swap the sampling an selection steps to handle outliers. We focus on sequential binary classifiers that consist of linear combinations of basis functions , whose coefficients evolve according to a Gaussian smoothness prior. Our results show significant improvements.
An Introduction to Monte Carlo Methods for Bayesian Data Analysis

Often it is natural to describe a signal processing or dynamical modeling problem in terms of probability distributions, and in particular tin Bayesian terms, where the unknown parameters are taken to be random variables and their distributions are updated by applying Bayes’ theorem to gave the distributions of the parameters conditional on the data. In the past, it was not possible to handle many non-trivial problems in this way because the distributions seldom took tractable forms. Considerable progress has been made in recent years in applying Monte Carla methods to overcome this, and in this chapter we describe some of the new results that have made a full Bayesian approach to signal processing tractable as well as powerful.
Optimal Estimation of Amplitude and Phase Modulated Signals

In this paper, we address the problem of on-line optimal estimation of signals modelled äs the sum. of sinusoids with time-varying amplitudes and phases. This is a complex optimal filtering problem. We present an efficient Sequential Monte Carlo (SMC) method, i.e. particle filtering method, to solve it. It combines several variance reduction methods and Markov chain Monte Carlo (MCMC) steps.
Particle filtering for multiuser detection in fading CDMA channels

We address the problem of multiuser CDMA detection under fading conditions. The optimal detection problem can be reformulated as an optimal filtering problem for jump Markov linear systems, i.e., linear Gaussian state space models switching according to an unobserved finite state space Markov chain. Several approaches based on particle filtering techniques are reviewed to perform optimal filtering in this framework. A brief simulation study is carried out.
On sequential Monte Carlo sampling methods for Bayesian filtering

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literature; these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.
Stochastic algorithms for marginal MAP retrieval of sinusoids in non-Gaussian noise

In this paper we propose a method to estimate the frequencies of sinusoids embedded in non-Gaussian noise. We model the noise using mixtures of Gaussians and propose two original, efficient algorithms that allow for the marginal maximum a posteriori (MAP) estimation of the sinusoid parameters to be estimated. Outline of the proof of convergence of the algorithms is also given and simulation results are presented.