arXiv subjects

Ajay Jasra

Publications and source records attributed to Ajay Jasra.

At least 109 records · Page 6Linked to original sources

arXiv2016

Multilevel Sequential Monte Carlo Samplers for Normalizing Constants

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for the example of identifying a parametrized permeability in an elliptic equation given point-wise observations of the pressure.

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arXiv2016

Flexible online multivariate regression with variational Bayes and the matrix-variate Dirichlet process

Flexible regression methods where interest centres on the way that the whole distribution of a response vector changes with covariates are very useful in some applications. A recently developed technique in this regard uses the matrix-variate Dirichlet process as a prior for a mixing distribution on a coefficient in a multivariate linear regression model. The method is attractive, particularly in the multivariate setting, for the convenient way that it allows for borrowing strength across different component regressions and for its computational simplicity and tractability. The purpose of the present article is to develop fast online variational Bayes approaches to fitting this model and to investigate how they perform compared to MCMC and batch variational methods in a number of scenarios.

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arXiv2015

Multilevel particle filter

In this paper the filtering of partially observed diffusions, with discrete-time observations, is considered. It is assumed that only biased approximations of the diffusion can be obtained, for choice of an accuracy parameter indexed by $l$. A multilevel estimator is proposed, consisting of a telescopic sum of increment estimators associated to the successive levels. The work associated to $\mathcal{O}(\varepsilon^2)$ mean-square error between the multilevel estimator and average with respect to the filtering distribution is shown to scale optimally, for example as $\mathcal{O}(\varepsilon^{-2})$ for optimal rates of convergence of the underlying diffusion approximation. The method is illustrated on some toy examples as well as estimation of interest rate based on real S&P 500 stock price data.

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arXiv2015

Bayesian Inference for Duplication-Mutation with Complementarity Network Models

We observe an undirected graph $G$ without multiple edges and self-loops, which is to represent a protein-protein interaction (PPI) network. We assume that $G$ evolved under the duplication-mutation with complementarity (DMC) model from a seed graph, $G_0$, and we also observe the binary forest $Γ$ that represents the duplication history of $G$. A posterior density for the DMC model parameters is established, and we outline a sampling strategy by which one can perform Bayesian inference; that sampling strategy employs a particle marginal Metropolis-Hastings (PMMH) algorithm. We test our methodology on numerical examples to demonstrate a high accuracy and precision in the inference of the DMC model's mutation and homodimerization parameters.

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arXiv2015

Multilevel Sequential Monte Carlo Samplers

In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with the step-size level $h_L$. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels $\infty>h_0>h_1\cdots>h_L$. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context.

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arXiv2015

Biased Online Parameter Inference for State-Space Models

We consider Bayesian online static parameter estimation for state-space models. This is a very important problem, but is very computationally challenging as the state- of-the art methods that are exact, often have a computational cost that grows with the time parameter; perhaps the most successful algorithm is that of SMC2 [9]. We present a version of the SMC2 algorithm which has computational cost that does not grow with the time parameter. In addition, under assumptions, the algorithm is shown to provide consistent estimates of expectations w.r.t. the posterior. However, the cost to achieve this consistency can be exponential in the dimension of the parameter space; if this exponential cost is avoided, typically the algorithm is biased. The bias is investigated from a theoretical perspective and, under assumptions, we find that the bias does not accumulate as the time parameter grows. The algorithm is implemented on several Bayesian statistical models.

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arXiv2015

Variational inference for sparse spectrum Gaussian process regression

We develop a fast variational approximation scheme for Gaussian process (GP) regression, where the spectrum of the covariance function is subjected to a sparse approximation. Our approach enables uncertainty in covariance function hyperparameters to be treated without using Monte Carlo methods and is robust to overfitting. Our article makes three contributions. First, we present a variational Bayes algorithm for fitting sparse spectrum GP regression models that uses nonconjugate variational message passing to derive fast and efficient updates. Second, we propose a novel adaptive neighbourhood technique for obtaining predictive inference that is effective in dealing with nonstationarity. Regression is performed locally at each point to be predicted and the neighbourhood is determined using a measure defined based on lengthscales estimated from an initial fit. Weighting dimensions according to lengthscales, this downweights variables of little relevance, leading to automatic variable selection and improved prediction. Third, we introduce a technique for accelerating convergence in nonconjugate variational message passing by adapting step sizes in the direction of the natural gradient of the lower bound. Our adaptive strategy can be easily implemented and empirical results indicate significant speedups.

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arXiv2014

Sequential Monte Carlo Methods for Bayesian Elliptic Inverse Problems

In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic SMC method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the sequential Monte Carlo (SMC) methods for inverse problems which were introduced in \cite{kantas}; the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology, and its desirable theoretical properties, are demonstrated on numerical examples in both two and three dimensions.

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arXiv2014

A Stable Particle Filter in High-Dimensions

We consider the numerical approximation of the filtering problem in high dimensions, that is, when the hidden state lies in $\mathbb{R}^d$ with $d$ large. For low dimensional problems, one of the most popular numerical procedures for consistent inference is the class of approximations termed particle filters or sequential Monte Carlo methods. However, in high dimensions, standard particle filters (e.g. the bootstrap particle filter) can have a cost that is exponential in $d$ for the algorithm to be stable in an appropriate sense. We develop a new particle filter, called the \emph{space-time particle filter}, for a specific family of state-space models in discrete time. This new class of particle filters provide consistent Monte Carlo estimates for any fixed $d$, as do standard particle filters. Moreover, we expect that the state-space particle filter will scale much better with $d$ than the standard filter. We illustrate this analytically for a model of a simple i.i.d. structure and one of a Markovian structure in the $d$-dimensional space-direction, when we show that the algorithm exhibits certain stability properties as $d$ increases at a cost $\mathcal{O}(nNd^2)$, where $n$ is the time parameter and $N$ is the number of Monte Carlo samples, that are fixed and independent of $d$. Similar results are expected to hold, under a more general structure than the i.i.d.~one. independently of the dimension. Our theoretical results are also supported by numerical simulations on practical models of complex structures. The results suggest that it is indeed possible to tackle some high dimensional filtering problems using the space-time particle filter that standard particle filters cannot handle.

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arXiv2014

A Sharp First Order Analysis of Feynman-Kac Particle Models

This article provides a new theory for the analysis of forward and backward particle approximations of Feynman-Kac models. Such formulae are found in a wide variety of applications and their numerical (particle) approximation are required due to their intractability. Under mild assumptions, we provide sharp and non-asymptotic first order expansions of these particle methods, potentially on path space and for possibly unbounded functions. These expansions allows one to consider upper and lower bound bias type estimates for a given time horizon $n$ and particle number $N$; these non-asymptotic estimates are of order $\mathcal{O}(n/N)$. Our approach is extended to tensor products of particle density profiles, leading to new sharp and non-asymptotic propagation of chaos estimates. The resulting upper and lower bound propagation of chaos estimates seems to be the first result of this kind for mean field particle models. As a by-product of our results, we also provide some analysis of the particle Gibbs sampler, providing first order expansions of the kernel and minorization estimates.

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arXiv2014

Theory of Parallel Particle Filters for Hidden Markov Models

The objective of this article is to study the asymptotic behavior of a new particle filtering approach in the context of hidden Markov models (HMMs). In particular, we develop an algorithm where the latent-state sequence is segmented into multiple shorter portions, with an estimation technique based upon a separate particle filter in each portion. The partitioning facilitates the use of parallel processing. Based upon this approach, we introduce new estimators of the latent states and likelihood which have similar or better variance properties compared to estimators derived from standard particle filters. Moreover due to parallelization there is less wall-clock computational time. We show that the likelihood function estimator is unbiased, prove central limit theorem convergences of estimators, and provide consistent in-sample estimation of the asymptotic variances. The theoretical analyses, supported by a numerical study, show that the segmentation reduces the variances in smoothed latent-state estimators, in addition to the savings in wall-clock time.

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arXiv2014

A simulation approach for change-points on phylogenetic trees

We observe $n$ sequences at each of $m$ sites, and assume that they have evolved from an ancestral sequence that forms the root of a binary tree of known topology and branch lengths, but the sequence states at internal nodes are unknown. The topology of the tree and branch lengths are the same for all sites, but the parameters of the evolutionary model can vary over sites. We assume a piecewise constant model for these parameters, with an unknown number of change-points and hence a trans-dimensional parameter space over which we seek to perform Bayesian inference. We propose two novel ideas to deal with the computational challenges of such inference. Firstly, we approximate the model based on the time machine principle: the top nodes of the binary tree (near the root) are replaced by an approximation of the true distribution; as more nodes are removed from the top of the tree, the cost of computing the likelihood is reduced linearly in $n$. The approach introduces a bias, which we investigate empirically. Secondly, we develop a particle marginal Metropolis-Hastings (PMMH) algorithm, that employs a sequential Monte Carlo (SMC) sampler and can use the first idea. Our time-machine PMMH algorithm copes well with one of the bottle-necks of standard computational algorithms: the trans-dimensional nature of the posterior distribution. The algorithm is implemented on simulated and real data examples, and we empirically demonstrate its potential to outperform competing methods based on approximate Bayesian computation (ABC) techniques.

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arXiv2014

On the Convergence of Adaptive Sequential Monte Carlo Methods

In several implementations of Sequential Monte Carlo (SMC) methods it is natural, and important in terms of algorithmic efficiency, to exploit the information of the history of the samples to optimally tune their subsequent propagations. In this article we provide a carefully formulated asymptotic theory for a class of such \emph{adaptive} SMC methods. The theoretical framework developed here will cover, under assumptions, several commonly used SMC algorithms. There are only limited results about the theoretical underpinning of such adaptive methods: we will bridge this gap by providing a weak law of large numbers (WLLN) and a central limit theorem (CLT) for some of these algorithms. The latter seems to be the first result of its kind in the literature and provides a formal justification of algorithms used in many real data context. We establish that for a general class of adaptive SMC algorithms the asymptotic variance of the estimators from the adaptive SMC method is \emph{identical} to a so-called `perfect' SMC algorithm which uses ideal proposal kernels. Our results are supported by application on a complex high-dimensional posterior distribution associated with the Navier-Stokes model, where adapting high-dimensional parameters of the proposal kernels is critical for the efficiency of the algorithm.

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arXiv2014

Approximate Bayesian Computation for a Class of Time Series Models

In the following article we consider approximate Bayesian computation (ABC) for certain classes of time series models. In particular, we focus upon scenarios where the likelihoods of the observations and parameter are intractable, by which we mean that one cannot evaluate the likelihood even up-to a positive unbiased estimate. This paper reviews and develops a class of approximation procedures based upon the idea of ABC, but, specifically maintains the probabilistic structure of the original statistical model. This idea is useful, in that it can facilitate an analysis of the bias of the approximation and the adaptation of established computational methods for parameter inference. Several existing results in the literature are surveyed and novel developments with regards to computation are given.

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arXiv2013

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae under Verifiable Conditions

In the following article we consider the time-stability associated to the sequential Monte Carlo (SMC) estimate of the backward interpretation of Feynman-Kac Formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models (HMMs). We prove a central limit theorem (CLT) under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression, for additive functionals grows at most linearly in time, under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some state-space models.

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arXiv2013

Parameter Estimation in Hidden Markov Models with Intractable Likelihoods Using Sequential Monte Carlo

We propose sequential Monte Carlo based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable likelihood using ideas from approximate Bayesian computation. The static parameter estimation algorithms are gradient based and cover both offline and online estimation. We demonstrate their performance by estimating the parameters of three intractable models, namely the alpha-stable distribution, g-and-k distribution, and the stochastic volatility model with alpha-stable returns, using both real and synthetic data.

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arXiv2013

Twisting the Alive Particle Filter

This work focuses on sampling from hidden Markov models (Cappe et al, 2005) whose observations have intractable density functions. We develop a new sequential Monte Carlo (Doucet et al, 2000 and Gordon et al, 1993) algorithm and a new particle marginal Metropolis-Hastings (Andrieu et al, 2010) algorithm for these purposes. We build from Jasra, et al (2013) and Whiteley, et al (2013) to construct the sequential Monte Carlo (SMC) algorithm (which we call the alive twisted particle filter). Like the alive particle filter of Jasra, et al (2013), our new SMC algorithm adopts an approximate Bayesian computation (Tavare et al, 1997) estimate of the HMM. Our alive twisted particle filter also uses a twisted proposal as in Whiteley, et al (2013) to obtain a low-variance estimate of the HMM normalising constant. We demonstrate via numerical examples that, in some scenarios, this estimate has a much lower variance than that of the estimate obtained via the alive particle filter. The low variance of this normalising constant estimate encourages the implementation of our SMC algorithm within a particle marginal Metropolis-Hastings (PMMH) scheme, and we call the resulting methodology ``alive twisted PMMH''. We numerically demonstrate on a stochastic volatility model how our alive twisted PMMH can converge faster than the standard alive PMMH of Jasra, et al (2013).

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arXiv2013

Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A case study for the Navier-Stokes equations

We consider the inverse problem of estimating the initial condition of a partial differential equation, which is only observed through noisy measurements at discrete time intervals. In particular, we focus on the case where Eulerian measurements are obtained from the time and space evolving vector field, whose evolution obeys the two-dimensional Navier-Stokes equations defined on a torus. This context is particularly relevant to the area of numerical weather forecasting and data assimilation. We will adopt a Bayesian formulation resulting from a particular regularization that ensures the problem is well posed. In the context of Monte Carlo based inference, it is a challenging task to obtain samples from the resulting high dimensional posterior on the initial condition. In real data assimilation applications it is common for computational methods to invoke the use of heuristics and Gaussian approximations. The resulting inferences are biased and not well-justified in the presence of non-linear dynamics and observations. On the other hand, Monte Carlo methods can be used to assimilate data in a principled manner, but are often perceived as inefficient in this context due to the high-dimensionality of the problem. In this work we will propose a generic Sequential Monte Carlo (SMC) sampling approach for high dimensional inverse problems that overcomes these difficulties. The method builds upon Markov chain Monte Carlo (MCMC) techniques, which are currently considered as benchmarks for evaluating data assimilation algorithms used in practice. In our numerical examples, the proposed SMC approach achieves the same accuracy as MCMC but in a much more efficient manner.

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