Sequential Monte Carlo Sampling Research Articles

A Bayesian Nonparametric Model for Unsupervised Joint Segmentation of a Collection of Images

Jointly segmenting a collection of images with shared classes is expected to yield better results than single-image based methods, due to the use of the shared statistical information across different images. This paper proposes a Bayesian approach for tackling this problem. As a first contribution, the proposed method relies on a new prior distribution for the class labels, which combines a hierarchical Dirichlet process (HDP) with a Potts model. The latter classically favors a spatial dependency, whereas the HDP is a Bayesian nonparametric model that allows the number of classes to be inferred automatically. The HDP also explicitly induces a sharing of classes between the images. The resulting posterior distribution of the labels is not analytically tractable and can be explored using a standard Gibbs sampler. However, such a sampling strategy is known to have poor mixing properties for high-dimensional data. To alleviate this issue, the second contribution reported in this paper consists of an adapted generalized Swendsen-Wang algorithm which is a sampling technique that improves the exploration of the posterior distribution. Finally, since the inferred segmentation depends on the values of the hyperparameters, the third contribution aims at adjusting them while sampling the posterior label distribution by resorting to an original combination of two sequential Monte Carlo samplers. The proposed methods are validated on both simulated and natural images from databases.

The Gibbs sampler with particle efficient importance sampling for state-space models*

We consider Particle Gibbs (PG) for Bayesian analysis of non-linear non-Gaussian state-space models. As a Monte Carlo (MC) approximation of the Gibbs procedure, PG uses sequential MC (SMC) importance sampling inside the Gibbs to update the latent states. We propose to combine PG with the Particle Efficient Importance Sampling (PEIS). By using SMC sampling densities which are approximately globally fully adapted to the targeted density of the states, PEIS can substantially improve the simulation efficiency of the PG relative to existing PG implementations. The efficiency gains are illustrated in PG applications to a non-linear local-level model and stochastic volatility models.

Bayesian estimation of dynamic asset pricing models with informative observations

In dynamic asset pricing models, when the model structure becomes complex and derivatives data are introduced in estimation, traditional MCMC methods converge slowly, are difficult to design efficient proposals for parameters, and have large computational cost. We propose a two-stage sequential Monte Carlo sampler based on common random numbers and a smooth particle filter. This method is robust to potential model misspecification and can deliver almost full-likelihood-based inference at a much smaller computational cost. It is applied to estimate a class of volatility models that take into account price-volatility co-jumps, non-affineness, and self-excitation. An empirical study using S&P 500 index and variance swap rates shows that both non-affineness and self-excitation need to be introduced in modeling volatility dynamics.

Sequential Monte Carlo: Enabling Real-time and High-fidelity Prognostics

Uncertainty quantification and propagation form the foundation of a prognostics and health management (PHM) system. Particle filters have proven to be a valuable tool for this reason but are generally restricted to state-space damage models and lack a natural approach for quantifying model parameter uncertainty. Both of these issues tend to inhibit the real-world application of PHM. While Markov chain Monte Carlo (MCMC) sampling methods avoid some of these restrictions, they are also inherently serial, and, thus, MCMC can become intractable as model fidelity increases. Over the past two decades, sequential Monte Carlo (SMC) methods, of which the particle filter is a special case, have been adapted to sample from a single, static posterior distribution, eliminating the state-space requirement and providing an alternative to MCMC. Additionally, SMC samplers of this type can be run in parallel, resulting in drastic reductions in computation time. In this work, a potential path toward real-time, highfidelity prognostics using a combination of surrogate modeling and a parallel SMC sampler is explored. The use of SMC samplers to enable tractable parameter estimation for full-fidelity (i.e., non-surrogate-assisted) damage models is also discussed. Both of these topics are studied in the context of fatigue crack growth in a geometrically complex, metallic specimen subjected to variable amplitude loading.

Multilevel Sequential2 Monte Carlo for Bayesian inverse problems

The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in P.S. Koutsourelakis (2009) [33] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.

A new moving strategy for the sequential Monte Carlo approach in optimizing the hydrological model parameters

Abstract Sequential Monte Carlo (SMC) samplers have become increasing popular for estimating the posterior parameter distribution with the non-linear dependency structures and multiple modes often present in hydrological models. However, the explorative capabilities and efficiency of the sampler depends strongly on the efficiency in the move step of SMC sampler. In this paper we presented a new SMC sampler entitled the Particle Evolution Metropolis Sequential Monte Carlo (PEM–SMC) algorithm, which is well suited to handle unknown static parameters of hydrologic model. The PEM–SMC sampler is inspired by the works of Liang and Wong (2001) and operates by incorporating the strengths of the genetic algorithm, differential evolution algorithm and Metropolis–Hasting algorithm into the framework of SMC. We also prove that the sampler admits the target distribution to be a stationary distribution. Two case studies including a multi-dimensional bimodal normal distribution and a conceptual rainfall–runoff hydrologic model by only considering parameter uncertainty and simultaneously considering parameter and input uncertainty show that PEM–SMC sampler is generally superior to other popular SMC algorithms in handling the high dimensional problems. The study also indicated that it may be important to account for model structural uncertainty by using multiplier different hydrological models in the SMC framework in future study.

Online Bayesian Phylogenetic Inference: Theoretical Foundations via Sequential Monte Carlo.

Phylogenetics, the inference of evolutionary trees from molecular sequence data such as DNA, is an enterprise that yields valuable evolutionary understanding of many biological systems. Bayesian phylogenetic algorithms, which approximate a posterior distribution on trees, have become a popular if computationally expensive means of doing phylogenetics. Modern data collection technologies are quickly adding new sequences to already substantial databases. With all current techniques for Bayesian phylogenetics, computation must start anew each time a sequence becomes available, making it costly to maintain an up-to-date estimate of a phylogenetic posterior. These considerations highlight the need for an online Bayesian phylogenetic method which can update an existing posterior with new sequences. Here, we provide theoretical results on the consistency and stability of methods for online Bayesian phylogenetic inference based on Sequential Monte Carlo (SMC) and Markov chain Monte Carlo. We first show a consistency result, demonstrating that the method samples from the correct distribution in the limit of a large number of particles. Next, we derive the first reported set of bounds on how phylogenetic likelihood surfaces change when new sequences are added. These bounds enable us to characterize the theoretical performance of sampling algorithms by bounding the effective sample size (ESS) with a given number of particles from below. We show that the ESS is guaranteed to grow linearly as the number of particles in an SMC sampler grows. Surprisingly, this result holds even though the dimensions of the phylogenetic model grow with each new added sequence.

Estimation for the Capacity Value of PV and Wind Plants Considering Output Correlation

There is urgent necessity for estimating the capacity value of renewable energy sources (RES) in the planning of modern power systems to avoid excessive investment at generation side and RES curtailment. Firstly in this paper, the models of conventional units and the load curve are introduced along with the output model of RES, which takes the correlation between PV and wind plants into account using copulas. On the basis of these models, the sequential Monte Carlo sampling (SMCS) method is employed to compute the reliability index of generating system, such as loss of load expectation (LOLE) and expected energy not supplied (EENS). After that, in view of the concept of effective load carrying capacity (ELCC), the capacity value of RES is estimated by adopting secant method to maintain the reliability indices before and after RES integration unchanged. Finally, a study case is designed with the actual data of a certain western province in China to prove the correctness and feasibility of this proposed estimation method.

Bayesian estimation of scaled mutation rate under the coalescent: a sequential Monte Carlo approach

BackgroundSamples of molecular sequence data of a locus obtained from random individuals in a population are often related by an unknown genealogy. More importantly, population genetics parameters, for instance, the scaled population mutation rate Θ=4Neμ for diploids or Θ=2Neμ for haploids (where Ne is the effective population size and μ is the mutation rate per site per generation), which explains some of the evolutionary history and past qualities of the population that the samples are obtained from, is of significant interest.ResultsIn this paper, we present the evolution of sequence data in a Bayesian framework and the approximation of the posterior distributions of the unknown parameters of the model, which include Θ via the sequential Monte Carlo (SMC) samplers for static models. Specifically, we approximate the posterior distributions of the unknown parameters with a set of weighted samples i.e., the set of highly probable genealogies out of the infinite set of possible genealogies that describe the sampled sequences. The proposed SMC algorithm is evaluated on simulated DNA sequence datasets under different mutational models and real biological sequences. In terms of the accuracy of the estimates, the proposed SMC method shows a comparable and sometimes, better performance than the state-of-the-art MCMC algorithms.ConclusionsWe showed that the SMC algorithm for static model is a promising alternative to the state-of-the-art approach for simulating from the posterior distributions of population genetics parameters.

Multiresolution alignment for multiple unsynchronized audio sequences using sequential Monte Carlo samplers

It is increasingly more common that an occasion is recorded by multiple individuals with the proliferation of recording devices such as smart phones. When properly aligned, these recordings may provide several audio and visual perspectives to a scene which leads to several applications in restoring, remastering and remixing frameworks in various fields. In this work, we propose a multiresolution alignment algorithm for aligning multiple unsynchronized audio sequences using Sequential Monte Carlo samplers. We employ a model based approach and a score function analogous to similarity based methods. The optimum alignments are obtained in a course to fine structure with multiresolution sampling and a heuristic sequential search method. The proposed method is evaluated with a real-life dataset from Jiku Mobile Video Datasets. The simulation results suggest that our method is competitive with the baseline methods in terms of accuracy with suitable choice of parameters.

Multiparameter estimation along quantum trajectories with sequential Monte Carlo methods

This paper proposes an efficient method for the simultaneous estimation of the state of a quantum system and the classical parameters that govern its evolution. This hybrid approach benefits from efficient numerical methods for the integration of stochastic master equations for the quantum system, and efficient parameter estimation methods from classical signal processing. The classical techniques use Sequential Monte Carlo (SMC) methods, which aim to optimize the selection of points within the parameter space, conditioned by the measurement data obtained. We illustrate these methods using a specific example, an SMC sampler applied to a nonlinear system, the Duffing oscillator, where the evolution of the quantum state of the oscillator and three Hamiltonian parameters are estimated simultaneously.

A sequential Monte Carlo approach to gene expression deconvolution.

High-throughput gene expression data are often obtained from pure or complex (heterogeneous) biological samples. In the latter case, data obtained are a mixture of different cell types and the heterogeneity imposes some difficulties in the analysis of such data. In order to make conclusions on gene expresssion data obtained from heterogeneous samples, methods such as microdissection and flow cytometry have been employed to physically separate the constituting cell types. However, these manual approaches are time consuming when measuring the responses of multiple cell types simultaneously. In addition, exposed samples, on many occasions, end up being contaminated with external perturbations and this may result in an altered yield of molecular content. In this paper, we model the heterogeneous gene expression data using a Bayesian framework, treating the cell type proportions and the cell-type specific expressions as the parameters of the model. Specifically, we present a novel sequential Monte Carlo (SMC) sampler for estimating the model parameters by approximating their posterior distributions with a set of weighted samples. The SMC framework is a robust and efficient approach where we construct a sequence of artificial target (posterior) distributions on spaces of increasing dimensions which admit the distributions of interest as marginals. The proposed algorithm is evaluated on simulated datasets and publicly available real datasets, including Affymetrix oligonucleotide arrays and national center for biotechnology information (NCBI) gene expression omnibus (GEO), with varying number of cell types. The results obtained on all datasets show a superior performance with an improved accuracy in the estimation of cell type proportions and the cell-type specific expressions, and in addition, more accurate identification of differentially expressed genes when compared to other widely known methods for blind decomposition of heterogeneous gene expression data such as Dsection and the nonnegative matrix factorization (NMF) algorithms. MATLAB implementation of the proposed SMC algorithm is available to download at https://github.com/moyanre/smcgenedeconv.git.

Learning of state-space models with highly informative observations: A tempered sequential Monte Carlo solution

Probabilistic (or Bayesian) modeling and learning offers interesting possibilities for systematic representation of uncertainty using probability theory. However, probabilistic learning often leads to computationally challenging problems. Some problems of this type that were previously intractable can now be solved on standard personal computers thanks to recent advances in Monte Carlo methods. In particular, for learning of unknown parameters in nonlinear state-space models, methods based on the particle filter (a Monte Carlo method) have proven very useful. A notoriously challenging problem, however, still occurs when the observations in the state-space model are highly informative, i.e. when there is very little or no measurement noise present, relative to the amount of process noise. The particle filter will then struggle in estimating one of the basic components for probabilistic learning, namely the likelihood $p($data$|$parameters$)$. To this end we suggest an algorithm which initially assumes that there is substantial amount of artificial measurement noise present. The variance of this noise is sequentially decreased in an adaptive fashion such that we, in the end, recover the original problem or possibly a very close approximation of it. The main component in our algorithm is a sequential Monte Carlo (SMC) sampler, which gives our proposed method a clear resemblance to the SMC^2 method. Another natural link is also made to the ideas underlying the approximate Bayesian computation (ABC). We illustrate it with numerical examples, and in particular show promising results for a challenging Wiener-Hammerstein benchmark problem.

Simulation from quasi-stationary distributions on reducible state spaces

AbstractQuasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of sequential Monte Carlo samplers is utilised to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally, an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter.

A rare event approach to high-dimensional approximate Bayesian computation

Approximate Bayesian computation (ABC) methods permit approximate inference for intractable likelihoods when it is possible to simulate from the model. However, they perform poorly for high-dimensional data and in practice must usually be used in conjunction with dimension reduction methods, resulting in a loss of accuracy which is hard to quantify or control. We propose a new ABC method for high-dimensional data based on rare event methods which we refer to as RE-ABC. This uses a latent variable representation of the model. For a given parameter value, we estimate the probability of the rare event that the latent variables correspond to data roughly consistent with the observations. This is performed using sequential Monte Carlo and slice sampling to systematically search the space of latent variables. In contrast, standard ABC can be viewed as using a more naive Monte Carlo estimate. We use our rare event probability estimator as a likelihood estimate within the pseudo-marginal Metropolis–Hastings algorithm for parameter inference. We provide asymptotics showing that RE-ABC has a lower computational cost for high-dimensional data than standard ABC methods. We also illustrate our approach empirically, on a Gaussian distribution and an application in infectious disease modelling.

Posterior exploration based sequential Monte Carlo for global optimization

We propose a global optimization algorithm based on the Sequential Monte Carlo (SMC) sampling framework. In this framework, the objective function is normalized to be a probabilistic density function (pdf), based on which a sequence of annealed target pdfs is designed to asymptotically converge on the set of global optima. A sequential importance sampling (SIS) procedure is performed to simulate the resulting targets, and the maxima of the objective function is assessed from the yielded samples. The disturbing issue lies in the design of the importance sampling (IS) pdf, which crucially influences the IS efficiency. We propose an approach to design the IS pdf online by embedding a posterior exploration (PE) procedure into each iteration of the SMC framework. The PE procedure can also explore the important regions of the parameter space supported by the target pdf. A byproduct of the PE procedure is an adaptive mechanism to design the annealing temperature schedule online. We compare the performance of the proposed algorithm with those of several existing related alternatives by applying them to over a dozen standard benchmark functions. The result demonstrates the appealing properties of our algorithm.

Effcient Sequential Monte-Carlo Samplers for Bayesian Inference

In many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such inference problems. However, in the presence of a high-dimensional and/or multi-modal posterior distribution, it is widely documented that standard Monte-Carlo techniques could lead to poor performance. In this paper, the study is focused on a Sequential Monte-Carlo (SMC) sampler framework, a more robust and efficient Monte Carlo algorithm. Although this approach presents many advantages over traditional Monte-Carlo methods, the potential of this emergent technique is however largely under-exploited in signal processing. In this work, we aim at proposing some novel strategies that will improve the efficiency and facilitate practical implementation of the SMC sampler specifically for signal processing applications. Firstly, we propose an automatic and adaptive strategy that selects the sequence of distributions within the SMC sampler that minimizes the asymptotic variance of the estimator of the posterior normalization constant. This is critical for performing model selection in modelling applications in Bayesian signal processing. The second original contribution we present improves the global efficiency of the SMC sampler by introducing a novel correction mechanism that allows the use of the particles generated through all the iterations of the algorithm (instead of only particles from the last iteration). This is a significant contribution as it removes the need to discard a large portion of the samples obtained, as is standard in standard SMC methods. This will improve estimation performance in practical settings where computational budget is important to consider.

Multiple-Input Multiple-Output OFDM With Index Modulation: Low-Complexity Detector Design

Multiple-input multiple-output orthogonal frequency division multiplexing with index modulation (MIMO-OFDM-IM), which provides a flexible trade-off between spectral efficiency and error performance, is recently proposed as a promising transmission technique for energy-efficient 5G wireless communications systems. However, due to the dependence of subcarrier symbols within each subblock and the strong interchannel interference, it is challenging to detect the transmitted data effectively while imposing low computational burden to the receiver. In this paper, we propose two types of low-complexity detectors based on the sequential Monte Carlo (SMC) theory for the detection of MIMO-OFDM-IM signals. The first detector draws samples independently at the subblock level, while the second detector draws samples at the subcarrier level with further reduced complexity. To meet the constraint of the subcarrier combinations within each subblock, the second detector is further coupled with a carefully designed legality examination method. Attributed to the effectiveness of legality examination and deterministic SMC sampling, both proposed detectors achieve near-optimal error performance for the MIMO-OFDM-IM system.

Divide-and-Conquer With Sequential Monte Carlo

ABSTRACTWe propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved subproblems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed divide-and-conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging subproblems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem. Supplementary materials including proofs and additional numerical results are available online.

Estimating the parameters of dynamical systems from Big Data using Sequential Monte Carlo samplers

In this paper the authors present a method which facilitates computationally efficient parameter estimation of dynamical systems from a continuously growing set of measurement data. It is shown that the proposed method, which utilises Sequential Monte Carlo samplers, is guaranteed to be fully parallelisable (in contrast to Markov chain Monte Carlo methods) and can be applied to a wide variety of scenarios within structural dynamics. Its ability to allow convergence of one’s parameter estimates, as more data is analysed, sets it apart from other sequential methods (such as the particle filter).