It has been widely documented that the sampling and resampling steps in particle filters cannot be differentiated. The {\itshape reparameterisation trick} was introduced to allow the sampling step to be reformulated into a differentiable function. We extend the {\itshape reparameterisation trick} to include the stochastic input to resampling therefore limiting the discontinuities in the gradient calculation after this step. Knowing the gradients of the prior and likelihood allows us to run particle Markov Chain Monte Carlo (p-MCMC) and use the No-U-Turn Sampler (NUTS) as the proposal when estimating parameters.
We compare the Metropolis-adjusted Langevin algorithm (MALA), Hamiltonian Monte Carlo with different number of steps and NUTS. We consider two state-space models and show that NUTS improves the mixing of the Markov chain and can produce more accurate results in less computational time.
Stanford（斯坦福大学）；Google AI（谷歌人工智能）；Robotics at Google（谷歌机器人）
Resampling is a key component of sample-based recursive state estimation in particle filters. Recent work explores differentiable particle filters for end-to-end learning. However, resampling remains a challenge in these works, as it is inherently non-differentiable. We address this challenge by replacing traditional resampling with a learned neural network resampler. We present a novel network architecture, the particle transformer, and train it for particle resampling using a likelihood-based loss function over sets of particles. Incorporated into a differentiable particle filter, our model …
Differentiable Particle Filtering using Optimal Placement Resampling
1 Department of Control for Transportation；Vehicle Systems, Faculty of Transportation Engineering；Vehicle Engineering, Budapest University of Technology
Aalto University（阿尔托大学）；University of Oxford（牛津大学）
Particle Filtering (PF) methods are an established class of procedures for performing inference in non-linear state-space models. Resampling is a key ingredient of PF, necessary to obtain low variance likelihood and states estimates. However, traditional resampling methods result in PF-based loss functions being non-differentiable with respect to model and PF parameters. In a variational inference context, resampling also yields high variance gradient estimates of the PF-based evidence lower bound. By leveraging optimal transport ideas, we introduce a principled differentiable particle filter …
Learning Discriminators for Resampling in the Ensemble Gaussian Mixture Filter through a Normalizing Flow Approach
Department of Mathematics, The University of Hawaii at Manoa（夏威夷大学马诺阿分校数学系）；Department of Information and Computer Sciences, The University of Hawaii at Manoa（夏威夷大学马诺阿分校信息与计算机科学系）
We analyze the performance of different resampling strategies for the regularized particle filter regarding parameter estimation. We show in particular, building on analytical insight obtained in the linear Gaussian case, that resampling systematically can prevent the filtered density from converging towards the true posterior distribution. We discuss several means to overcome this limitation, including kernel bandwidth modulation, and provide evidence that the resulting particle filter clearly outperforms traditional bootstrap particle filters. Our results are supported by numerical simulatio…
Among Sequential Monte Carlo (SMC) methods,Sampling Importance Resampling (SIR) algorithms are based on Importance Sampling (IS) and on some resampling-based)rejuvenation algorithm which aims at fighting against weight degeneracy. However %whichever the resampling technique used this mechanism tends to be insufficient when applied to informative or high-dimensional models. In this paper we revisit the rejuvenation mechanism and propose a class of parameterized SIR-based solutions which enable to adjust the tradeoff between computational cost and statistical performances.
We consider particle filters with weakly informative observations (or `potentials') relative to the latent state dynamics. The particular focus of this work is on particle filters to approximate time-discretisations of continuous-time Feynman--Kac path integral models -- a scenario that naturally arises when addressing filtering and smoothing problems in continuous time -- but our findings are indicative about weakly informative settings beyond this context too. We study the performance of different resampling schemes, such as systematic resampling, SSP (Srinivasan sampling process) and strati…
University of British Columbia（不列颠哥伦比亚大学）；Inverted AI；Mila
Particle filters are not compatible with automatic differentiation due to the presence of discrete resampling steps. While known estimators for the score function, based on Fisher's identity, can be computed using particle filters, up to this point they required manual implementation. In this paper we show that such estimators can be computed using automatic differentiation, after introducing a simple correction to the particle weights. This correction utilizes the stop-gradient operator and does not modify the particle filter operation on the forward pass, while also being cheap and easy to c…
Sequential Monte Carlo algorithms, or Particle Filters, are Bayesian filtering algorithms which propagate in time a discrete and random approximation of the a posteriori distribution of interest. Such algorithms are based on Importance Sampling with a bootstrap resampling step which aims at struggling against weights degeneracy. However, in some situations (informative measurements, high dimensional model), the resampling step can prove inefficient. In this paper, we revisit the fundamental resampling mechanism which leads us back to Rubin's static resampling mechanism. We propose an alternati…
Particle filtering is a numerical Bayesian technique that has great potential for solving sequential estimation problems involving non-linear and non-Gaussian models. Since the estimation accuracy achieved by particle filters improves as the number of particles increases, it is natural to consider as many particles as possible. MapReduce is a generic programming model that makes it possible to scale a wide variety of algorithms to Big data. However, despite the application of particle filters across many domains, little attention has been devoted to implementing particle filters using MapReduc…
Resampling Filter Design for Multirate Neural Audio Effect Processing
Acoustics and Audio, University of Edinburgh（声学与音频，爱丁堡大学）；Acoustics Lab, Department of Information and Communications Engineering, Aalto University（声学实验室，信息与通信工程系，阿尔托大学）；Scottish Graduate School of Arts and Humanities (SGSAH)（苏格兰艺术与人文研究生学院）
When the weights in a particle filter are not available analytically, standard resampling methods cannot be employed. To circumvent this problem state-of-the-art algorithms replace the true weights with non-negative unbiased estimates. This algorithm is still valid but at the cost of higher variance of the resulting filtering estimates in comparison to a particle filter using the true weights. We propose here a novel algorithm that allows for resampling according to the true intractable weights when only an unbiased estimator of the weights is available. We demonstrate our algorithm on several…
Data assimilation is concerned with sequentially estimating a temporally-evolving state. This task, which arises in a wide range of scientific and engineering applications, is particularly challenging when the state is high-dimensional and the state-space dynamics are unknown. This paper introduces a machine learning framework for learning dynamical systems in data assimilation. Our auto-differentiable ensemble Kalman filters (AD-EnKFs) blend ensemble Kalman filters for state recovery with machine learning tools for learning the dynamics. In doing so, AD-EnKFs leverage the ability of ensemble …
Robotics and Biology Laboratory, Technische Universität Berlin, Germany（德国柏林工业大学机器人与生物实验室）
We present differentiable particle filters (DPFs): a differentiable implementation of the particle filter algorithm with learnable motion and measurement models. Since DPFs are end-to-end differentiable, we can efficiently train their models by optimizing end-to-end state estimation performance, rather than proxy objectives such as model accuracy. DPFs encode the structure of recursive state estimation with prediction and measurement update that operate on a probability distribution over states. This structure represents an algorithmic prior that improves learning performance in state estimati…
We propose a method for inference on moderately high-dimensional, nonlinear, non-Gaussian, partially observed Markov process models for which the transition density is not analytically tractable. Markov processes with intractable transition densities arise in models defined implicitly by simulation algorithms. Widely used particle filter methods are applicable to nonlinear, non-Gaussian models but suffer from the curse of dimensionality. Improved scalability is provided by ensemble Kalman filter methods, but these are inappropriate for highly nonlinear and non-Gaussian models. We propose a par…
We introduce a new version of particle filter in which the number of "children" of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that descendants of different particles can evolve independently. It makes easy to parallelize computations. Moreover, particle filter with Poisson resampling is readily adapted to the case when a hidden process is a continuous time, piecewise deterministic semi-Markov process. We show that the basic techniques of particle MCMC, namely particle independent Met…
Multi-sample, importance-weighted variational autoencoders (IWAE) give tighter bounds and more accurate uncertainty estimates than variational autoencoders (VAE) trained with a standard single-sample objective. However, IWAEs scale poorly: as the latent dimensionality grows, they require exponentially many samples to retain the benefits of importance weighting. While sequential Monte-Carlo (SMC) can address this problem, it is prohibitively slow because the resampling step imposes sequential structure which cannot be parallelised, and moreover, resampling is non-differentiable which is problem…
Particle filters provide Monte Carlo approximations of intractable quantities such as point-wise evaluations of the likelihood in state space models. In many scenarios, the interest lies in the comparison of these quantities as some parameter or input varies. To facilitate such comparisons, we introduce and study methods to couple two particle filters in such a way that the correlation between the two underlying particle systems is increased. The motivation stems from the classic variance reduction technique of positively correlating two estimators. The key challenge in constructing such a cou…
Resampling is a standard step in particle filters and more generally sequential Monte Carlo methods. We present an algorithm, called chopthin, for resampling weighted particles. In contrast to standard resampling methods the algorithm does not produce a set of equally weighted particles; instead it merely enforces an upper bound on the ratio between the weights. Simulation studies show that the chopthin algorithm consistently outperforms standard resampling methods. The algorithms chops up particles with large weight and thins out particles with low weight, hence its name. It implicitly guaran…
Tsinghua University（清华大学）；Stanford University（斯坦福大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Institute for Advanced Study（高等研究院）；Massachusetts Institute of Technology（麻省理工学院）
A major difficulty of solving continuous POMDPs is to infer the multi-modal distribution of the unobserved true states and to make the planning algorithm dependent on the perceived uncertainty. We cast POMDP filtering and planning problems as two closely related Sequential Monte Carlo (SMC) processes, one over the real states and the other over the future optimal trajectories, and combine the merits of these two parts in a new model named the DualSMC network. In particular, we first introduce an adversarial particle filter that leverages the adversarial relationship between its internal compon…
Sequential Monte Carlo (SMC), also known as particle filters, has been widely accepted as a powerful computational tool for making inference with dynamical systems. A key step in SMC is resampling, which plays the role of steering the algorithm towards the future dynamics. Several strategies have been proposed and used in practice, including multinomial resampling, residual resampling (Liu and Chen 1998), optimal resampling (Fearnhead and Clifford 2003), stratified resampling (Kitagawa 1996), and optimal transport resampling (Reich 2013). We show that, in the one dimensional case, optimal tran…
Filtering in spatially-extended dynamical systems is a challenging problem with significant practical applications such as numerical weather prediction. Particle filters allow asymptotically consistent inference but require infeasibly large ensemble sizes for accurate estimates in complex spatial models. Localisation approaches, which perform local state updates by exploiting low dependence between variables at distant points, have been suggested as a potential resolution to this issue. Naively applying the resampling step of the particle filter locally however produces implausible spatially d…
Sequential Monte Carlo (SMC) algorithms were originally designed for estimating intractable conditional expectations within state-space models, but are now routinely used to generate approximate samples in the context of general-purpose Bayesian inference. In particular, SMC algorithms are often used as subroutines within larger Monte Carlo schemes, and in this context, the demands placed on SMC are different: control of mean-squared error is insufficient---one needs to control the divergence from the target distribution directly. Towards this goal, we introduce the conditional adaptive resamp…
We develop a novel parallel resampling algorithm for fully parallelized particle filters, which is designed with GPUs (graphics processing units) or similar parallel computing devices in mind. With our new algorithm, a full cycle of particle filtering (computing the value of the likelihood for each particle, constructing the cumulative distribution function (CDF) for resampling, resampling the particles with the CDF, and propagating new particles for the next cycle) can be executed in a massively and completely parallel manner. One of the advantages of our algorithm is that every single numeri…
We present a variational inference (VI) framework that unifies and leverages sequential Monte-Carlo (particle filtering) with \emph{approximate} rejection sampling to construct a flexible family of variational distributions. Furthermore, we augment this approach with a resampling step via Bernoulli race, a generalization of a Bernoulli factory, to obtain a low-variance estimator of the marginal likelihood. Our framework, Variational Rejection Particle Filtering (VRPF), leads to novel variational bounds on the marginal likelihood, which can be optimized efficiently with respect to the variation…
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 approx…
It is crucially important to estimate unknown parameters in earth system models by integrating observation and numerical simulation. For many applications in earth system sciences, an optimization method which allows parameters to temporally change is required. In the present paper, an efficient and practical method to estimate the time-varying parameters of relatively low dimensional models is presented. In the newly proposed method, called Hybrid Offline Online Parameter Estimation with Particle Filtering (HOOPE-PF), an inflation method to maintain the spread of ensemble members in a samplin…
The particle filter is a popular Bayesian filtering algorithm for use in cases where the state-space model is nonlinear and/or the random terms (initial state or noises) are non-Gaussian distributed. We study the behavior of the error in the particle filter algorithm as the number of particles gets large. After a decomposition of the error into two terms, we show that the difference between the estimator and the conditional mean is asymptotically normal when the resampling is done at every step in the filtering process. Two nonlinear/non-Gaussian examples are tested to verify this conclusion.
This paper concerns numerical assessment of Monte Carlo error in particle filters. We show that by keeping track of certain key features of the genealogical structure arising from resampling operations, it is possible to estimate variances of a number of standard Monte Carlo approximations which particle filters deliver. All our estimators can be computed from a single run of a particle filter with no further simulation. We establish that as the number of particles grows, our estimators are weakly consistent for asymptotic variances of the Monte Carlo approximations and some of them are also n…
We present simple conditions under which the limiting genealogical process associated with a class of interacting particle systems with non-neutral selection mechanisms, as the number of particles grows, is a time-rescaled Kingman coalescent. Sequential Monte Carlo algorithms are popular methods for approximating integrals in problems such as non-linear filtering and smoothing which employ this type of particle system. Their performance depends strongly on the properties of the induced genealogical process. We verify the conditions of our main result for standard sequential Monte Carlo algorit…
Autonomous systems such as Unmanned Aerial Vehicles (UAVs) need to be able to recognise and track crowds of people, e.g. for rescuing and surveillance purposes. Large groups generate multiple measurements with uncertain origin. Additionally, often the sensor noise characteristics are unknown but measurements are bounded within certain intervals. In this work we propose two solutions to the crowds tracking problem - with a box particle filtering approach and with a convolution particle filtering approach. The developed filters can cope with the measurement origin uncertainty in an elegant way, …
