Browsing by Subject "Sequential Monte Carlo"

The modelling and analysis of complex stochastic systems with increasingly large data sets, state-spaces and parameters provides major stimulus to research in Bayesian nonparametric methods and Bayesian computation. This dissertation presents advances in both nonparametric modelling and statistical computation stimulated by challenging problems of analysis in complex spatio-temporal systems and core computational issues in model fitting and model assessment. The first part of the thesis, represented by chapters 2 to 4, concerns novel, nonparametric Bayesian mixture models for spatial point processes, with advances in modelling, computation and applications in biological contexts. Chapter 2 describes and develops models for spatial point processes in which the point outcomes are latent, where indirect observations related to the point outcomes are available, and in which the underlying spatial intensity functions are typically highly heterogenous. Spatial intensities of inhomogeneous Poisson processes are represented via flexible nonparametric Bayesian mixture models. Computational approaches are presented for this new class of spatial point process mixtures and extended to the context of unobserved point process outcomes. Two examples drawn from a central, motivating context, that of immunofluorescence histology analysis in biological studies generating high-resolution imaging data, demonstrate the modelling approach and computational methodology. Chapters 3 and 4 extend this framework to define a class of flexible Bayesian nonparametric models for inhomogeneous spatio-temporal point processes, adding dynamic models for underlying intensity patterns. Dependent Dirichlet process mixture models are introduced as core components of this new time-varying spatial model. Utilizing such nonparametric mixture models for the spatial process intensity functions allows the introduction of time variation via dynamic, state-space models for parameters characterizing the intensities. Bayesian inference and model-fitting is addressed via novel particle filtering ideas and methods. Illustrative simulation examples include studies in problems of extended target tracking and substantive data analysis in cell fluorescent microscopic imaging tracking problems.

The second part of the thesis, consisting of chapters 5 and chapter 6, concerns advances in computational methods for some core and generic Bayesian inferential problems. Chapter 5 develops a novel approach to estimation of upper and lower bounds for marginal likelihoods in Bayesian modelling using refinements of existing variational methods. Traditional variational approaches only provide lower bound estimation; this new lower/upper bound analysis is able to provide accurate and tight bounds in many problems, so facilitates more reliable computation for Bayesian model comparison while also providing a way to assess adequacy of variational densities as approximations to exact, intractable posteriors. The advances also include demonstration of the significant improvements that may be achieved in marginal likelihood estimation by marginalizing some parameters in the model. A distinct contribution to Bayesian computation is covered in Chapter 6. This concerns a generic framework for designing adaptive MCMC algorithms, emphasizing the adaptive Metropolized independence sampler and an effective adaptation strategy using a family of mixture distribution proposals. This work is coupled with development of a novel adaptive approach to computation in nonparametric modelling with large data sets; here a sequential learning approach is defined that iteratively utilizes smaller data subsets. Under the general framework of importance sampling based marginal likelihood computation, the proposed adaptive Monte Carlo method and sequential learning approach can facilitate improved accuracy in marginal likelihood computation. The approaches are exemplified in studies of both synthetic data examples, and in a real data analysis arising in astro-statistics.

Finally, chapter 7 summarizes the dissertation and discusses possible extensions of the specific modelling and computational innovations, as well as potential future work.

This thesis focuses on the development of ABC methods for statistical modeling in complex dynamic systems. Motivated by real applications in biology, I propose computational strategies for Bayesian inference in contexts where standard Monte Carlo methods cannot be directly applied due to the high complexity of the dynamic model and/or data limitations.

Chapter 2 focuses on stochastic bionetwork models applied to data generated from the marginal distribution of a few network nodes at snapshots in time. I present a Bayesian computational strategy, coupled with an approach to summarizing and numerically characterizing biological phenotypes that are represented in terms of the resulting sample distributions of cellular markers. ABC and mixture modeling are used to define the approach to linking mechanistic mathematical models of network dynamics to snapshot data, using a toggle switch example integrating simulated and real data as context.

Chapter 3 focuses on the application of the methodology presented in Chapter 2 to the Myc/Rb/E2F network. This network involves a relatively high number of parameters and stochastic equations in the model specification and, thus, is substantially more complex than the toggle switch example. The analysis of the Myc/Rb/E2F network is performed with simulated and real data. I demonstrate that the proposed method can indicate which parameters can be learned about using the marginal data.

In Chapter 4, I present an ABC SMC method that uses data-based adaptive weights. This easily implemented and computationally trivial extension of ABC SMC can substantially improve acceptance rates. This is demonstrated through a series of examples with simulated and real data, including the toggle switch example. Theoretical justification is also provided to explain why this method is expected to improve the effectiveness of ABC SMC.

In Chapter 5, I present an integrated Bayesian computational strategy for fitting complex dynamic models to sparse time-series data. This is applied to experimental data from an immunization response study with Indian Rhesus macaques. The computational strategy consists of two stages: first, MCMC is implemented based on simplified sampling steps, and then, the resulting approximate output is used to generate a proposal distribution for the parameters that results in an efficient ABC procedure. The incorporation of ABC as a correction tool improves the model fit, as is demonstrated through predictive posterior analysis on the data sets of the study.

Chapter 6 presents additional discussion and comments on potential future research directions.

Dynamic models, also termed state space models, comprise an extremely rich model class for time series analysis. This dissertation focuses on building state space models for a variety of contexts and computationally efficient methods for Bayesian inference for simultaneous estimation of latent states and unknown fixed parameters.

Chapter 1 introduces state space models and methods of inference in these models. Chapter 2 describes a novel method for jointly sampling the entire latent state vector in a nonlinear Gaussian state space model using a computationally efficient adaptive mixture modeling procedure. This method is embedded in an overall Markov chain Monte Carlo algorithm for estimating fixed parameters as well as states. In Chapter 3 the method of the previous chapter is implemented in a few illustrative

nonlinear models and compared to standard existing methods. This chapter also looks at the effect of the number of mixture components as well as length of the time series on the efficiency of the method. I then turn to an biological application in Chapter 4. I discuss modeling choices as well as derivation of the state space model to be used in this application. Parameter and state estimation are analyzed in these models for both simulated and real data. Chapter 5 extends the methodology introduced in Chapter 2 from nonlinear Gaussian models to general state space models. The method is then applied to a financial

stochastic volatility model on US $ - British £ exchange rates. Bayesian inference in the previous chapter is accomplished through Markov chain Monte Carlo which is suitable for batch analyses, but computationally limiting in sequential analysis. Chapter 6 introduces sequential Monte Carlo. It discusses two methods currently available for simultaneous sequential estimation of latent states and fixed parameters and then introduces a novel algorithm that reduces the key, limiting degeneracy issue while being usable in a wide model class. Chapter 7 implements the novel algorithm in a disease surveillance context modeling influenza epidemics. Finally, Chapter 8 suggests areas for future work in both modeling and Bayesian inference. Several appendices provide detailed technical support material as well as relevant related work.

The advances in three related areas of state-space modeling, sequential Bayesian learning, and decision analysis are addressed, with the statistical challenges of scalability and associated dynamic sparsity. The key theme that ties the three areas is Bayesian model emulation: solving challenging analysis/computational problems using creative model emulators. This idea defines theoretical and applied advances in non-linear, non-Gaussian state-space modeling, dynamic sparsity, decision analysis and statistical computation, across linked contexts of multivariate time series and dynamic networks studies. Examples and applications in financial time series and portfolio analysis, macroeconomics and internet studies from computational advertising demonstrate the utility of the core methodological innovations.

Chapter 1 summarizes the three areas/problems and the key idea of emulating in those areas. Chapter 2 discusses the sequential analysis of latent threshold models with use of emulating models that allows for analytical filtering to enhance the efficiency of posterior sampling. Chapter 3 examines the emulator model in decision analysis, or the synthetic model, that is equivalent to the loss function in the original minimization problem, and shows its performance in the context of sequential portfolio optimization. Chapter 4 describes the method for modeling the steaming data of counts observed on a large network that relies on emulating the whole, dependent network model by independent, conjugate sub-models customized to each set of flow. Chapter 5 reviews those advances and makes the concluding remarks.

Sequential Monte Carlo (SMC) samplers have received attention as an alternative to Markov chain Monte Carlo for Bayesian inference problems due to their strong empirical performance on difficult multimodal problems, natural synergy with parallel computing environments, and accuracy when estimating ratios of normalizing constants. However, while these properties have been demonstrated empirically, the extent of these advantages remain unexplored theoretically. Typical convergence results for SMC are limited to root N results; they obscure the relationship between the algorithmic factors (weights, Markov kernels, target distribution) and the error of the resulting estimator. This limitation makes it difficult to compare SMC to other estimation methods and challenging to design efficient SMC algorithms from a theoretical perspective.

In this thesis, we provide conditions under which SMC provides a randomized approximation scheme, showing how to choose the number of of particles and Markov kernel transitions at each SMC step in order to ensure an accurate approximation with bounded error. These conditions rely on the sequence of SMC interpolating distributions and the warm mixing times of the Markov kernels, explicitly relating the algorithmic choices to the error of the SMC estimate. This allows us to provide finite-sample complexity bounds for SMC in a variety of settings, including finite state-spaces, product spaces, and log-concave target distributions.

A key advantage of this approach is that the bounds provide insight into the selection of efficient sequences of SMC distributions. When the target distribution is spherical Gaussian or log-concave, we show that judicious selection of interpolating distributions results in an SMC algorithm with a smaller complexity bound than MCMC. These results are used to motivate the use of a well known SMC algorithm that adaptively chooses interpolating distributions. We provide conditions under which the adaptive algorithm gives a randomized approximation scheme, providing theoretical validation for the automatic selection of SMC distributions.

Selecting efficient sequences of distributions is a problem that also arises in the estimation of normalizing constants using path sampling. In the final chapter of this thesis, we develop automatic methods for choosing sequences of distributions that provide low-variance path sampling estimators. These approaches are motived by properties of the theoretically optimal, lowest-variance path, which is given by the geodesic of the Riemann manifold associated with the path sampling family. For one dimensional paths we provide a greedy approach to step size selection that has good empirical performance. For multidimensional paths, we present an approach using Gaussian process emulation that efficiently finds low variance paths in this more complicated setting.

This dissertation aims to introduce a new sequential Monte Carlo (SMC) based estimation framework for structural models used in macroeconomics and industrial organization. Current Markov chain Monte Carlo (MCMC) estimation methods for structural models suffer from slow Markov chain convergence, which means parameter and state spaces of interest might not be properly explored unless huge numbers of samples are simulated. This could lead to insurmountable computational burdens for the estimation of those structural models that are expensive to solve. In contrast, SMC methods rely on the principle of sequential importance sampling to jointly evolve simulated particles, thus bypassing the dependence on Markov chain convergence altogether. This dissertation will explore the feasibility and the potential benefits to estimating structural models using SMC based methods.

Chapter 1 casts the structural estimation problem in the form of inference of hidden Markov models and demonstrates with a simple growth model.

Chapter 2 presents the key ingredients, both conceptual and theoretical, to successful SMC parameter estimation strategies in the context of structural economic models.

Chapter 3, based on Chen, Petralia and Lopes (2010), develops SMC estimation methods for dynamic stochastic general equilibrium (DSGE) models. SMC algorithms allow a simultaneous filtering of time-varying state vectors and estimation of fixed parameters. We first establish empirical feasibility of the full SMC approach by comparing estimation results from both MCMC batch estimation and SMC on-line estimation on a simple neoclassical growth model. We then estimate a large scale DSGE model for the Euro area developed in Smets and Wouters (2003) with a full SMC approach, and revisit the on-going debate between the merits of reduced form and structural models in the macroeconomics context by performing sequential model assessment between the DSGE model and various VAR/BVAR models.

Chapter 4 proposes an SMC estimation procedure and show that it readily applies to the estimation of dynamic discrete games with serially correlated endogenous state variables. I apply this estimation procedure to a dynamic oligopolistic game of entry using data from the generic pharmaceutical industry and demonstrate that the proposed SMC method can potentially better explore the parameter posterior space while being more computationally efficient than MCMC estimation. In addition, I show how the unobserved endogenous cost paths could be recovered using particle smoothing, both with and without parameter uncertainty. Parameter estimates obtained using this SMC based method largely concur with earlier findings that spillover effect from market entry is significant and plays an important role in the generic drug industry, but that it might not be as high as previously thought when full model uncertainty is taken into account during estimation.