Markov chain Monte Carlo in the last few decades has become a very popular class of algorithms for sampling from probability distributions based on constructing a Markov chain. A special case of the Markov chain Monte Carlo is the Gibbs sampling algorithm. This algorithm can be used in such a way that it takes into account the prior distribution and likelihood function, carrying a randomly generated variable through the calculation and the simulation. In this thesis, we use the Ising model for the prior of the binary images. Assuming the pixels in binary images are polluted by random noise, we build a Bayesian model for the posterior distribution of the true image data. The posterior distribution enables us to generate the denoised image by designing a Gibbs sampling algorithm.
In this paper, we present a new fast Motion Estimation (ME) algorithm based on Markov Chain Model (MEMCM). Spatial-temporal correlation of video sequence a
While there have been few theoretical contributions on the Markov Chain Monte Carlo (MCMC) methods in the past decade, current understanding and application of MCMC to the solution of inference problems has increased by leaps and bounds. Incorporating changes in theory and highlighting new applications, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition presents a concise, accessible, and comprehensive introduction to the methods of this valuable simulation technique. The second edition includes access to an internet site that provides the code, written in R and WinBUGS, used in many of the previously existing and new examples and exercises. More importantly, the self-explanatory nature of the codes will enable modification of the inputs to the codes and variation on many directions will be available for further exploration. Major changes from the previous edition:. ¿ More examples with discussion of computational details in chapters on Gibbs sampling and ...
The talk will begin by reviewing methods of specifying continuous-time Markov chains and classical limit theorems that arise naturally for chemical network models. Since models arising in molecular biology frequently exhibit multiple state and time scales, analogous limit theorems for these models will be illustrated through simple examples ...
www.MOLUNA.de Linear Algebra, Markov Chains, and Queueing Models [4196258] - Perturbation Theory and Error Analysis.- Error bounds for the computation of null vectors with Applications to Markov Chains.- The influence of nonnormality on matrix computations.- Componentwise error analysis for stationary iterative methods.- The character of a finite Markov chain.- Gaussian elimination, perturbation theory, and Markov chains.- Iterative Methods.- Algorithms for
In this work, we present a novel multiscale texture model, and a related algorithm for the unsupervised segmentation of color images. Elementary textures are characterized by their spatial interactions with neighboring regions along selected directions. Such interactions are modeled in turn by means of a set of Markov chains, one for each direction, whose parameters are collected in a feature vector that synthetically describes the texture. Based on the feature vectors, the texture are then recursively merged, giving rise to larger and more complex textures, which appear at different scales of observation: accordingly, the model is named Hierarchical Multiple Markov Chain (H-MMC). The Texture Fragmentation and Reconstruction (TFR) algorithm, addresses the unsupervised segmen- tation problem based on the H-MMC model. The
In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity. ...
An essential ingredient of the statistical inference theory for hidden Markov models is the nonlinear filter. The asymptotic properties of nonlinear filters have received particular attention in recent years, and their characterization has significant implications for topics such as the convergence of approximate filtering algorithms, maximum likelihood estimation, and stochastic control. Despite much progress in specific models, however, most of the general asymptotic theory of nonlinear filters has suffered from a recently discovered gap in the fundamental work of H. Kunita (1971). In this talk, I will show that this gap can be resolved in the general setting of weakly ergodic signals with nondegenerate observations by exploiting a surprising connection with the theory of Markov chains in random environments. These results hold for both discrete and continuous time models in Polish state spaces, and shed new light on the filter stability problem. In the non-ergodic setting I will argue that a ...
The evolutionary algorithm stochastic process is well-known to be Markovian. These have been under investigation in much of the theoretical evolutionary computing research. When mutation rate is positive, the Markov chain modeling an evolutionary algorithm is irreducible and, therefore, has a unique stationary distribution, yet, rather little is known about the stationary distribution. On the other hand, knowing the stationary distribution may provide some information about the expected times to hit optimum, assessment of the biases due to recombination and is of importance in population genetics to assess whats called a ``genetic load (see the introduction for more details). In this talk I will show how the quotient construction method can be exploited to derive rather explicit bounds on the ratios of the stationary distribution values of various subsets of the state space. In fact, some of the bounds obtained in the current work are expressed in terms of the parameters involved in all the ...
Earlier this week, my company, Lander Analytics, organized our first public Bayesian short course, taught by Andrew Gelman, Bob Carpenter and Daniel Lee. Needless to say the class sold out very quickly and left a long wait list. So we will schedule another public training (exactly when tbd) and will make the same course available for private training.. This was the first time we utilized three instructors (as opposed to a main instructor and assistants which we often use for large classes) and it led to an amazing dynamic. Bob laid the theoretical foundation for Markov chain Monte Carlo (MCMC), explaining both with math and geometry, and discussed the computational considerations of performing simulation draws. Daniel led the participants through hands-on examples with Stan, covering everything from how to describe a model, to efficient computation to debugging. Andrew gave his usual, crowd dazzling performance use previous work as case studies of when and how to use Bayesian methods.. It was an ...
Philosophers have been trying to understand in any branches of Artificial Intelligence since thousand years ago. Al in Bioscience was born combined with Bioinformatic and produces many kinds of research areas especially in computational problem. Hidden Markov Model is a powerful statistical tool for describing an event within hidden states (unknown condition), such as predictor for exon section at Deoxyribonucleic Acid (DNA) sequence. Number of states, transition probabilities and emission distributional oribabilities are 3 major elements of HMM. Hidden Markov Model used Forward-Backward algorithm and Viterbi algorithm for implementing the HMM basic problems and solutions, includes evaluation, training and testing. Whereas, all this functions has been planted in Single Board Computer as the embedded platform. The reason of choosing SBC was influenced by Open Source Software (OSS) development area ...
Abstract: This talk presents suffcient conditions for the existence of stationary optimal policies for average-cost Markov Decision Processes with Borel state and action sets and with weakly continuous transition probabilities. The one-step cost functions may be unbounded, and the action sets may be noncompact. The main contributions of this paper are: (i) general sufficient conditions for the existence of stationary discount-optimal and average-cost optimal policies and descriptions of properties of value functions and sets of optimal actions, (ii) a sufficient condition for the average-cost optimality of a stationary policy in the form of optimality inequalities, and (iii) approximations of average-cost optimal actions by discount-optimal actions ...
Chromosome Classification Using Continuous Hidden Markov Models - Up-to-date results on the application of Markov models to chromosome analysis are presented. On the one hand, this means using continuous Hidden Markov Models (HMMs) instead of discrete models. On the other hand, this also means to conduct empirical tests on the same large chromosome datasets that are currently used to evaluate state-ofthe-art classifiers. It is shown that the use of continuous HMMs allows to obtain error rates that are very close to those provided by the most accurate classifiers.
Key concepts Markov chains Hidden Markov models Computing the probability of a sequence Estimating parameters of a Markov model Hidden Markov models States Emission and transition probabilities Parameter estimation Forward and backward algorithm Viterbi algorithm
AS Jean-Luc Jannink and Rohan L. Fernando (Jannink and Fernando 2004) nicely illustrated, when applying Markov chain Monte Carlo methods in a form where the dimension [the number of quantitative trait loci (QTL)] is not fixed, it can sometimes be hard to establish the correct form of the acceptance ratio for the proposals that are made. Therefore, as a safety precaution, the correct performance of the sampler should be checked also (under the prior model) without data.. We recently learned that Patrick Gaffney (Gaffney 2001) in his Ph.D. thesis had made essentially the same observation as Jannink and Fernando, correcting our mistake in Sillanpää and Arjas (1998). Somewhat earlier Vogl and Xu (2000) had expressed similar kinds of thoughts. As Gaffney (2001) explained, the acceptance ratio given in our article would correspond to an analysis, where an accelerated truncated Poisson prior (with a square term in the denominator) was assumed for the number of QTL, instead of an "ordinary" truncated ...
When system identification methods are used to construct mathematical models of real systems, it is important to collect data that reveal useful information about the systems dynamics. Experimental data are always corrupted by noise and this causes uncertainty in the model estimate. Therefore, design of input signals that guarantee a certain model accuracy is an important issue in system identification.. This thesis studies input design problems for system identification where time domain constraints have to be considered. A finite Markov chain is used to model the input of the system. This allows to directly include input amplitude constraints into the input model, by properly choosing the state space of the Markov chain. The state space is defined so that the model generates a binary signal. The probability distribution of the Markov chain is shaped in order to minimize an objective function defined in the input design problem.. Two identification issues are considered in this thesis: ...
03/14/19 - We consider the recently proposed reinforcement learning (RL) framework of Contextual Markov Decision Processes (CMDP), where the ...
In the runup to PyconUK 2014, I made the following ill-advised statement in an IRC channel: I feel like I should find something to talk about at PyconUK. I wish I had something interesting to talk about. Nine seconds later someone replied create a markov chain to generate a talk from the names of the talks at pycon and europython, then talk about how you did that, using a title it generates as the title of the talk.. Challenge accepted. This is that talk, admittedly one year late.. In this talk I will briefly describe Markov Chains as a means to simulate conversations and graph databases as a means to store Markov Chains. After this, I will discuss various considerations for creating interesting candidate responses in conversations, along with the challenges of too little and too much data. Finally, I will demonstrate my implementation and generate the title of this talk.. ...
Atomistic simulations have the potential to elucidate the molecular basis of biological processes such as protein misfolding in Alzheimers disease or the conformational changes that drive transcription or translation. However, most simulations can only capture the nanosecond to microsecond timescale, whereas most biological processes of interest occur on millisecond and longer timescales. Also, even with an infinitely fast computer, extracting meaningful insight from simulations is difficult because of the complexity of the underlying free energy landscapes. Fortunately, Markov State Models (MSMs) can help overcome these limitations.. MSMs may be used to model any random process where the next state depends solely on the current state. For example, imagine exploring New York City by rolling a die to randomly select which direction to go in each time you came to an intersection. Such a process could be described by an MSM with a state for each intersection. Each state might have a probability of ...
Read the book An Introduction To Markov State Models And Their Application To Long Timescale Molecular Simulation by Gregory R. Bowman ; Vijay S. Pande ; Frank Noé, Ed online or Preview the book. Please wait while, the book is loading ...
In this article, we present a modification of the popular Bayesian clustering program STRUCTURE (Pritchard et al. 2000) for inferring population substructure and self-fertilization simultaneously. Using extensive simulations with four distinct demographic models (K = 1, 2, 3, 6), we demonstrate that our method can accurately estimate selfing rates in the presence of population structure in the data. Additionally it can classify individuals into their appropriate subpopulations without the assumption of Hardy-Weinberg equilibrium within subpopulations.. It is important to note that the accuracy of selfing rate estimation is influenced by multiple factors, including sample size and number of loci, with decreased precision when they are small, as is illustrated in Table 2. Likewise, we find that the complexity of the true demographic history underlying data (e.g., the number of subpopulations derived from a common ancestral population) also influences accuracy. In general, more complicated models ...
Hi. For a project I am using a Markov Chain model with 17 states. I have used data to estimate transition probabilities. From these transition
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the Markov order). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions ...
Designers often search for new solutions by iteratively adapting a current design. By engaging in this search, designers not only improve solution quality but also begin to learn what operational patterns might improve the solution in future iterations. Previous work in psychology has demonstrated that humans can fluently and adeptly learn short operational sequences that aid problem-solving. This paper explores how designers learn and employ sequences within the realm of engineering design. Specifically, this work analyzes behavioral patterns in two human studies in which participants solved configuration design problems. Behavioral data from the two studies is first analyzed using Markov chains to determine how much representation complexity is necessary to quantify the sequential patterns that designers employ during solving. It is discovered that first-order Markov chains are capable of accurately representing designers sequences. Next, the ability to learn first-order sequences is ...
0040] FIG. 1 shows a schematic diagram of a sequence generator 100 according to an embodiment. In particular, FIG. 1 shows the details of a processing part 10 of the sequence generator 100 (e.g. a processor or other suitable processing part). The sequence generator 100 creates a non-homogenous Markov process M that generates sequences, wherein each sequence has a finite length L, comprises items from a set of a specific number n of items, and satisfies one or more control constraints specifying one or more requirements on the sequence. As an example, at least one of the control constraints can require a specific item to be at a specific position within the sequence, or can require a specific transition between two positions within the sequence. Each sequence can for example comprise items of music notes, text components or drawings, or any other suitable type of items. The sequence generator 100 comprises a Markov process unit 11 adapted to provide data defining an initial Markov process M of a ...
We present a discriminative learning method for pattern discovery of binding sites in nucleic acid sequences based on hidden Markov models. Sets of positive and negative example sequences are mined for sequence motifs whose occurrence frequency varies between the sets. The method offers several objective functions, but we concentrate on mutual information of condition and motif occurrence. We perform a systematic comparison of our method and numerous published motif-finding tools. Our method achieves the highest motif discovery performance, while being faster than most published methods. We present case studies of data from various technologies, including ChIP-Seq, RIP-Chip and PAR-CLIP, of embryonic stem cell transcription factors and of RNA-binding proteins, demonstrating practicality and utility of the method. For the alternative splicing factor RBM10, our analysis finds motifs known to be splicing-relevant. The motif discovery method is implemented in the free software package Discrover. It ...
... ,The Markov Chain Algorithm 1.2 is A classic algorithm which can produce entertaining output, given a sufficiently
Markov Chains, part I December 8, Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X
Linear Models and Markov Chain MBA Assignment Help, Online MBA Assignment Writing Service and Homework Help Linear Models and Markov Chain Assignment Help Linear models explain a constant reaction variable as a function of several predictor variables. They can he
Artificial Intelligence has made tremendous progress in industry in terms of problem solving pattern recognition. Mirror neuron systems (MNS), a new branch in intention recognition, has been successful in human robot interface, but with some limitations. First, it is a cognitive function in relation to the basic research limited. Second, it lacks an experimental paradigm. Therefore MNS requires a firm mathematical modeling. If we design engineering modeling based on mathematical, we will be able to apply mirror neuron system to brain-computer interface. This paper proposes a hybrid model-based classification of the action for brain-computer interface, a combination of Hidden Markov Model and Gaussian Mixture Model. Both models are possible to collect specific information. This hybrid model has been compared with Hidden Markov Model-based classification. The recognition rates achieved by Hidden Markov Model were 76.62% and the proposed model showed 84.38 ...
The channel capacity of a deterministic system with confidential data is an upper bound on the amount of bits of data an attacker can learn from the system. We encode all possible attacks to a system using a probabilistic specification, an Interval Markov Chain. Then the channel capacity computation reduces to finding a model of a specification with highest entropy. Entropy maximization for probabilistic process specifications has not been studied before, even though it is well known in Bayesian inference for discrete distributions. We give a characterization of global entropy of a process as a reward function, a polynomial algorithm to verify the existence of a system maximizing entropy among those respecting a specification, a procedure for the maximization of reward functions over Interval Markov Chains and its application to synthesize an implementation maximizing entropy. We show how to use Interval Markov Chains to model abstractions of deterministic systems with confidential data, and use the
Background:Natural history models of breast cancer progression provide an opportunity to evaluate and identify optimal screening scenarios. This paper describes a detailed Markov model characterising breast cancer tumour progression.Methods:Breast cancer is modelled by a 13-state continuous-time Markov model. The model differentiates between indolent and aggressive ductal carcinomas in situ tumours, and aggressive tumours of different sizes. We compared such aggressive cancers, that is, which are non-indolent, to those which are non-growing and regressing. Model input parameters and structure were informed by the 1978-1984 Ostergotland county breast screening randomised controlled trial. Overlaid on the natural history model is the effect of screening on diagnosis. Parameters were estimated using Bayesian methods. Markov chain Monte Carlo integration was used to sample the resulting posterior distribution.Results:The breast cancer incidence rate in the Ostergotland population was 21 (95% CI: ...
Title: Approximate conditional independence of separated subtrees and phylogenetic inference Abstract: Bayesian methods to reconstruct evolutionary trees from aligned DNA sequence data from different species depend on Markov chain Monte Carlo sampling of phylogenetic trees from a posterior distribution. The probabilities of tree topologies are typically estimated with the simple relative frequencies of the trees in the sample. When the posterior distribution is spread thinly over a very large number of trees, the simple relative frequencies from finite samples are often inaccurate estimates of the posterior probabilities for many trees. We present a new method for estimating the posterior distribution on the space of trees from samples based on the approximation of conditional independence between subtrees given their separation by an edge in the tree. This approximation procedure effectively spreads the estimated posterior distribution from the sampled trees to the larger set of trees that ...
Title: Approximate conditional independence of separated subtrees and phylogenetic inference Abstract: Bayesian methods to reconstruct evolutionary trees from aligned DNA sequence data from different species depend on Markov chain Monte Carlo sampling of phylogenetic trees from a posterior distribution. The probabilities of tree topologies are typically estimated with the simple relative frequencies of the trees in the sample. When the posterior distribution is spread thinly over a very large number of trees, the simple relative frequencies from finite samples are often inaccurate estimates of the posterior probabilities for many trees. We present a new method for estimating the posterior distribution on the space of trees from samples based on the approximation of conditional independence between subtrees given their separation by an edge in the tree. This approximation procedure effectively spreads the estimated posterior distribution from the sampled trees to the larger set of trees that ...
Parameters for computing emission probabilities for a 6-state HMM, including starting values for the mean and standard deviations for log R ratios (assumed to be Gaussian) and B allele frequencies (truncated Gaussian), and initial state probabilities. Constructor for EmissionParam class This function is exported primarily for internal use by other BioC packages.
A recent paper answers the following question: for which of this kind of Markov chain problems can a so-called filtered estimator be found in combination with a Markov importance measure under which this estimator has variance zero. Adaptive importance sampling algorithms aim to approach this zero variance measure on-the-fly and already two special cases were known for which this works: the resulting sequence of estimates converges at an exponential rate. For a while I thought that finding a general convergence proof would be impossible, but in recent months I have made some progress with this. In the talk I will describe the proof including the part where the conditions are not weak enough to my liking---maybe you have an idea.... More details on the seminars website: ...
The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and ... More. The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution ...
Abstract: In many applications, the failure rate function may present a bathtub shape curve. In this paper, an expectation maximization algorithm is proposed to construct a suitable continuous-time Markov chain which models the failure time data by the first time reaching the absorbing state. Assume that a system is described by methods of supplementary variables, the device of stage, and so on. Given a data set, the maximum likelihood estimators of the initial distribution and the infinitesimal transition rates of the Markov chain can be obtained by our novel algorithm. Suppose that there are transient states in the system and that there are failure time data. The devised algorithm only needs to compute the exponential of upper triangular matrices for times in each iteration. Finally, the algorithm is applied to two real data sets, which indicates the practicality and efficiency of our algorithm. 1. Introduction Among many quantitative analysis methods of system reliability, the state space ...
Abstract: Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint, strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this ...
Abstract: Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint, strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this ...
At the end of the nineties, Jordan, Kinderlehrer, and Otto discovered a new interpretation of the heat equation in R^n, as the gradient flow of the entropy in the Wasserstein space of probability measures. In this talk, I will present a discrete counterpart to this result: given a reversible Markov kernel on a finite set, there exists a Riemannian metric on the space of probability densities, for which the law of the continuous time Markov chain evolves as the gradient flow of the entropy ...
Cattle supply an important source of nutrition for humans in the world. CpG islands (CGIs) are very important and useful, as they carry functionally relevant epigenetic loci for whole genome studies. As a matter of fact, there have been no formal analyses of CGIs at the DNA sequence level in cattle genomes and therefore this study was carried out to fill the gap. We used hidden markov model algorithm to detect CGIs. The total number of predicted CGIs for cattle was 90668. The number of detected CGIs and CGI densities downwardly varied across chromosomes. Chromosome 25 had the largest number of CGIs (4556) and the highest CGI density (106.20 CGIs/Mb).A significant positive correlation observed among CGI densities with guanine-cytosine (GC) content, ObsCpG/ExpCpG, recombination rate and gene density. When the size of chromosomes increased, the CGI densities decreased and a trend of higher CGI densities in the telomeric regions observed. This feature may be the reason of a positive correlation between CGI
Empirical codon models (ECMs) estimated from a large number of globular protein families outperformed mechanistic codon models in their description of the general process of protein evolution. Among other factors, ECMs implicitly model the influence of amino acid properties and multiple nucleotide substitutions (MNS). However, the estimation of ECMs requires large quantities of data, and until recently, only few suitable data sets were available. Here, we take advantage of several new Drosophila species genomes to estimate codon models from genome-wide data. The availability of large numbers of genomes over varying phylogenetic depths in the Drosophila genus allows us to explore various divergence levels. In consequence, we can use these data to determine the appropriate level of divergence for the estimation of ECMs, avoiding overestimation of MNS rates caused by saturation. To account for variation in evolutionary rates along the genome, we develop new empirical codon hidden Markov models ...
Leos-Barajas, V., Gangloff, E. J., Adam, T., Langrock, R., van Beest, F. M., Nabe-Nielsen, J., & Morales, J. M. (2017). Multi-scale Modeling of Animal Movement and General Behavior Data Using Hidden Markov Models with Hierarchical Structures. Journal of Agricultural, Biological and Environmental Statistics, 22(3), 232-248. doi:10.1007/s13253-017-0282- ...
This article talks about Hidden Markov Model, which are especially known for their application in temporal pattern recognition such as speech, handwriting,
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As explained in the accompanying paper, if we were to start with all possible opinions initially and perform enough sweeps until all of them agreed, we would obtain perfect sample from the posterior distribution. This explanation glosses over sometechnical details, including the fact that the perfect sample occurs right before the next successful coalescence of opinions, and that we must throw away the first of these. In the applet, every possible opinion could be drawn besides the blue line we see, but this would be very confusing. Instead, we sandwich the other opinions between a yellow and an orange region. Once these two meet, we know that all possible initial opinions about the allocations have finally agreed. We budget 100 sweeps by default for each coalescence. At the end of these, we either have a coalescence or we dont, but we start again with all possible opinions. The bottom panel shows some statistics about the perfect simulation method, and also draws a color coded histogram of the ...
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Downloadable! Continuous-time stochastic volatility models are becoming a more and more popular way to describe moderate and high-frequency financial data. Recently, Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein-Uhlenbeck process, driven by a positive Levy process without Gaussian component. They also consider superpositions of such processes and we extend that to the inclusion of an uncorrelated component. Our aim is to design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo methods and we use a series representation of Levy processes. Inference for such models is complicated by the fact that parameter changes will often induce a change of dimension in the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the