arXiv subjects

Lawrence M. Murray

Publications and source records attributed to Lawrence M. Murray.

14 recordsLinked to original sources

arXiv2026

Resampling in conditional SMC algorithms

Conditional sequential Monte Carlo (CSMC) algorithms arise in particle Markov chain Monte Carlo and a number of related settings. As in standard sequential Monte Carlo (SMC) algorithms, it is possible to employ a number of approaches to resampling within CSMC, but some additional care is required to arrive at a valid algorithm. We present a simple framework for implementing valid SMC and CSMC algorithms which (a) covers most known resampling schemes including those with a complicated dependence structure like systematic resampling, but also adaptive resampling, and even more `exotic' schemes like a version of chopthin resampling; (b) explains how to implement conditional analogues of these and other well known resampling schemes without randomly permuting/shifting the order of the ancestor indices; (c) requires only very weak assumptions which include neither (marginal) "unbiasedness" nor exchangeability.

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arXiv2021

Anytime Monte Carlo

Monte Carlo algorithms simulate some prescribed number of samples, taking some random real time to complete the computations necessary. This work considers the converse: to impose a real-time budget on the computation, which results in the number of samples simulated being random. To complicate matters, the real time taken for each simulation may depend on the sample produced, so that the samples themselves are not independent of their number, and a length bias with respect to compute time is apparent. This is especially problematic when a Markov chain Monte Carlo (MCMC) algorithm is used and the final state of the Markov chain -- rather than an average over all states -- is required, which is the case in parallel tempering implementations of MCMC. The length bias does not diminish with the compute budget in this case. It also occurs in sequential Monte Carlo (SMC) algorithms, which is the focus of this paper. We propose an anytime framework to address the concern, using a continuous-time Markov jump process to study the progress of the computation in real time. We first show that for any MCMC algorithm, the length bias of the final state's distribution due to the imposed real-time computing budget can be eliminated by using a multiple chain construction. The utility of this construction is then demonstrated on a large-scale SMC^2 implementation, using four billion particles distributed across a cluster of 128 graphics processing units on the Amazon EC2 service. The anytime framework imposes a real-time budget on the MCMC move steps within the SMC$^{2}$ algorithm, ensuring that all processors are simultaneously ready for the resampling step, demonstrably reducing idleness to due waiting times and providing substantial control over the total compute budget.

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arXiv2021

Probabilistic programming for birth-death models of evolution using an alive particle filter with delayed sampling

We consider probabilistic programming for birth-death models of evolution and introduce a new widely-applicable inference method that combines an extension of the alive particle filter (APF) with automatic Rao-Blackwellization via delayed sampling. Birth-death models of evolution are an important family of phylogenetic models of the diversification processes that lead to evolutionary trees. Probabilistic programming languages (PPLs) give phylogeneticists a new and exciting tool: their models can be implemented as probabilistic programs with just a basic knowledge of programming. The general inference methods in PPLs reduce the need for external experts, allow quick prototyping and testing, and accelerate the development and deployment of new models. We show how these birth-death models can be implemented as simple programs in existing PPLs, and demonstrate the usefulness of the proposed inference method for such models. For the popular BiSSE model the method yields an increase of the effective sample size and the conditional acceptance rate by a factor of 30 in comparison with a standard bootstrap particle filter. Although concentrating on phylogenetics, the extended APF is a general inference method that shows its strength in situations where particles are often assigned zero weight. In the case when the weights are always positive, the extra cost of using the APF rather than the bootstrap particle filter is negligible, making our method a suitable drop-in replacement for the bootstrap particle filter in probabilistic programming inference.

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arXiv2020

Automated learning with a probabilistic programming language: Birch

This work offers a broad perspective on probabilistic modeling and inference in light of recent advances in probabilistic programming, in which models are formally expressed in Turing-complete programming languages. We consider a typical workflow and how probabilistic programming languages can help to automate this workflow, especially in the matching of models with inference methods. We focus on two properties of a model that are critical in this matching: its structure---the conditional dependencies between random variables---and its form---the precise mathematical definition of those dependencies. While the structure and form of a probabilistic model are often fixed a priori, it is a curiosity of probabilistic programming that they need not be, and may instead vary according to random choices made during program execution. We introduce a formal description of models expressed as programs, and discuss some of the ways in which probabilistic programming languages can reveal the structure and form of these, in order to tailor inference methods. We demonstrate the ideas with a new probabilistic programming language called Birch, with a multiple object tracking example.

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arXiv2020

Particle filter with rejection control and unbiased estimator of the marginal likelihood

We consider the combined use of resampling and partial rejection control in sequential Monte Carlo methods, also known as particle filters. While the variance reducing properties of rejection control are known, there has not been (to the best of our knowledge) any work on unbiased estimation of the marginal likelihood (also known as the model evidence or the normalizing constant) in this type of particle filter. Being able to estimate the marginal likelihood without bias is highly relevant for model comparison, computation of interpretable and reliable confidence intervals, and in exact approximation methods, such as particle Markov chain Monte Carlo. In the paper we present a particle filter with rejection control that enables unbiased estimation of the marginal likelihood.

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arXiv2020

Lazy object copy as a platform for population-based probabilistic programming

This work considers dynamic memory management for population-based probabilistic programs, such as those using particle methods for inference. Such programs exhibit a pattern of allocating, copying, potentially mutating, and deallocating collections of similar objects through successive generations. These objects may assemble data structures such as stacks, queues, lists, ragged arrays, and trees, which may be of random, and possibly unbounded, size. For the simple case of $N$ particles, $T$ generations, $D$ objects, and resampling at each generation, dense representation requires $O(DNT)$ memory, while sparse representation requires only $O(DT+DN\log DN)$ memory, based on existing theoretical results. This work describes an object copy-on-write platform to automate this saving for the programmer. The core idea is formalized using labeled directed multigraphs, where vertices represent objects, edges the pointers between them, and labels the necessary bookkeeping. A specific labeling scheme is proposed for high performance under the motivating pattern. The platform is implemented for the Birch probabilistic programming language, using smart pointers, hash tables, and reference-counting garbage collection. It is tested empirically on a number of realistic probabilistic programs, and shown to significantly reduce memory use and execution time in a manner consistent with theoretical expectations. This enables copy-on-write for the imperative programmer, lazy deep copies for the object-oriented programmer, and in-place write optimizations for the functional programmer.

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arXiv2018

Automatic Alignment of Sequential Monte Carlo Inference in Higher-Order Probabilistic Programs

Probabilistic programming is a programming paradigm for expressing flexible probabilistic models. Implementations of probabilistic programming languages employ a variety of inference algorithms, where sequential Monte Carlo methods are commonly used. A problem with current state-of-the-art implementations using sequential Monte Carlo inference is the alignment of program synchronization points. We propose a new static analysis approach based on the 0-CFA algorithm for automatically aligning higher-order probabilistic programs. We evaluate the automatic alignment on a phylogenetic model, showing a significant decrease in runtime and increase in accuracy.

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arXiv2018

Delayed Sampling and Automatic Rao-Blackwellization of Probabilistic Programs

We introduce a dynamic mechanism for the solution of analytically-tractable substructure in probabilistic programs, using conjugate priors and affine transformations to reduce variance in Monte Carlo estimators. For inference with Sequential Monte Carlo, this automatically yields improvements such as locally-optimal proposals and Rao-Blackwellization. The mechanism maintains a directed graph alongside the running program that evolves dynamically as operations are triggered upon it. Nodes of the graph represent random variables, edges the analytically-tractable relationships between them. Random variables remain in the graph for as long as possible, to be sampled only when they are used by the program in a way that cannot be resolved analytically. In the meantime, they are conditioned on as many observations as possible. We demonstrate the mechanism with a few pedagogical examples, as well as a linear-nonlinear state-space model with simulated data, and an epidemiological model with real data of a dengue outbreak in Micronesia. In all cases one or more variables are automatically marginalized out to significantly reduce variance in estimates of the marginal likelihood, in the final case facilitating a random-weight or pseudo-marginal-type importance sampler for parameter estimation. We have implemented the approach in Anglican and a new probabilistic programming language called Birch.

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arXiv2017

Better together? Statistical learning in models made of modules

In modern applications, statisticians are faced with integrating heterogeneous data modalities relevant for an inference, prediction, or decision problem. In such circumstances, it is convenient to use a graphical model to represent the statistical dependencies, via a set of connected "modules", each relating to a specific data modality, and drawing on specific domain expertise in their development. In principle, given data, the conventional statistical update then allows for coherent uncertainty quantification and information propagation through and across the modules. However, misspecification of any module can contaminate the estimate and update of others, often in unpredictable ways. In various settings, particularly when certain modules are trusted more than others, practitioners have preferred to avoid learning with the full model in favor of approaches that restrict the information propagation between modules, for example by restricting propagation to only particular directions along the edges of the graph. In this article, we investigate why these modular approaches might be preferable to the full model in misspecified settings. We propose principled criteria to choose between modular and full-model approaches. The question arises in many applied settings, including large stochastic dynamical systems, meta-analysis, epidemiological models, air pollution models, pharmacokinetics-pharmacodynamics, and causal inference with propensity scores.

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arXiv2015

Sequential Monte Carlo with Highly Informative Observations

We propose sequential Monte Carlo (SMC) methods for sampling the posterior distribution of state-space models under highly informative observation regimes, a situation in which standard SMC methods can perform poorly. A special case is simulating bridges between given initial and final values. The basic idea is to introduce a schedule of intermediate weighting and resampling times between observation times, which guide particles towards the final state. This can always be done for continuous-time models, and may be done for discrete-time models under sparse observation regimes; our main focus is on continuous-time diffusion processes. The methods are broadly applicable in that they support multivariate models with partial observation, do not require simulation of the backward transition (which is often unavailable), and, where possible, avoid pointwise evaluation of the forward transition. When simulating bridges, the last cannot be avoided entirely without concessions, and we suggest an epsilon-ball approach (reminiscent of Approximate Bayesian Computation) as a workaround. Compared to the bootstrap particle filter, the new methods deliver substantially reduced mean squared error in normalising constant estimates, even after accounting for execution time. The methods are demonstrated for state estimation with two toy examples, and for parameter estimation (within a particle marginal Metropolis--Hastings sampler) with three applied examples in econometrics, epidemiology and marine biogeochemistry.

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arXiv2015

Parallel resampling in the particle filter

Modern parallel computing devices, such as the graphics processing unit (GPU), have gained significant traction in scientific and statistical computing. They are particularly well-suited to data-parallel algorithms such as the particle filter, or more generally Sequential Monte Carlo (SMC), which are increasingly used in statistical inference. SMC methods carry a set of weighted particles through repeated propagation, weighting and resampling steps. The propagation and weighting steps are straightforward to parallelise, as they require only independent operations on each particle. The resampling step is more difficult, as standard schemes require a collective operation, such as a sum, across particle weights. Focusing on this resampling step, we analyse two alternative schemes that do not involve a collective operation (Metropolis and rejection resamplers), and compare them to standard schemes (multinomial, stratified and systematic resamplers). We find that, in certain circumstances, the alternative resamplers can perform significantly faster on a GPU, and to a lesser extent on a CPU, than the standard approaches. Moreover, in single precision, the standard approaches are numerically biased for upwards of hundreds of thousands of particles, while the alternatives are not. This is particularly important given greater single- than double-precision throughput on modern devices, and the consequent temptation to use single precision with a greater number of particles. Finally, we provide auxiliary functions useful for implementation, such as for the permutation of ancestry vectors to enable in-place propagation.

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arXiv2013

On Disturbance State-Space Models and the Particle Marginal Metropolis-Hastings Sampler

We investigate nonlinear state-space models without a closed-form transition density, and propose reformulating such models over their latent noise variables rather than their latent state variables. In doing so the tractable noise density emerges in place of the intractable transition density. For importance sampling methods such as the auxiliary particle filter, this enables importance weights to be computed where they could not be otherwise. As case studies we take two multivariate marine biogeochemical models and perform state and parameter estimation using the particle marginal Metropolis-Hastings sampler. For the particle filter within this sampler, we compare several proposal strategies over noise variables, all based on lookaheads with the unscented Kalman filter. These strategies are compared using conventional means for assessing Metropolis-Hastings efficiency, as well as with a novel metric called the conditional acceptance rate for assessing the consequences of using an estimated, and not exact, likelihood. Results indicate the utility of reformulating the model over noise variables, particularly for fast-mixing process models.

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arXiv2013

Bayesian State-Space Modelling on High-Performance Hardware Using LibBi

LibBi is a software package for state-space modelling and Bayesian inference on modern computer hardware, including multi-core central processing units (CPUs), many-core graphics processing units (GPUs) and distributed-memory clusters of such devices. The software parses a domain-specific language for model specification, then optimises, generates, compiles and runs code for the given model, inference method and hardware platform. In presenting the software, this work serves as an introduction to state-space models and the specialised methods developed for Bayesian inference with them. The focus is on sequential Monte Carlo (SMC) methods such as the particle filter for state estimation, and the particle Markov chain Monte Carlo (PMCMC) and SMC^2 methods for parameter estimation. All are well-suited to current computer hardware. Two examples are given and developed throughout, one a linear three-element windkessel model of the human arterial system, the other a nonlinear Lorenz '96 model. These are specified in the prescribed modelling language, and LibBi demonstrated by performing inference with them. Empirical results are presented, including a performance comparison of the software with different hardware configurations.

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arXiv2011

High-Performance Pseudo-Random Number Generation on Graphics Processing Units

This work considers the deployment of pseudo-random number generators (PRNGs) on graphics processing units (GPUs), developing an approach based on the xorgens generator to rapidly produce pseudo-random numbers of high statistical quality. The chosen algorithm has configurable state size and period, making it ideal for tuning to the GPU architecture. We present a comparison of both speed and statistical quality with other common parallel, GPU-based PRNGs, demonstrating favourable performance of the xorgens-based approach.

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