arXiv subjects

Sumeetpal S. Singh

Publications and source records attributed to Sumeetpal S. Singh.

At least 19 recordsLinked to original sources

arXiv2026

On the Forgetting of Particle Filters

We study the forgetting properties of the particle filter when its state - the collection of particles - is regarded as a Markov chain. Under a strong mixing assumption on the particle filter's underlying Feynman-Kac model, we find that the particle filter is exponentially mixing, and forgets its initial state in $O(\log N )$ 'time', where $N$ is the number of particles and time refers to the number of particle filter algorithm steps, each comprising a selection (or resampling) and mutation (or prediction) operation. We present an example which shows that this rate is optimal. In contrast to our result, available results to date are extremely conservative, suggesting $O(α^N)$ time steps are needed, for some $α>1$, for the particle filter to forget its initialisation. We also study the conditional particle filter (CPF) and extend our forgetting result to this context. We establish a similar conclusion, namely, CPF is exponentially mixing and forgets its initial state in $O(\log N)$ time. To support this analysis, we establish new time-uniform $L^p$ error estimates for CPF, which can be of independent interest. We also establish new propagation-of-chaos type results using our proof techniques, discuss implications to couplings of particle filters and an application to processing out-of-sequence measurements.

math.PR↗

arXiv2026

Bayesian learning for the stochastic shortest path problem

Sequential decision-making problems are often modelled as a Markov decision process (MDP). We focus on the stochastic shortest path (SSP) problem, which is an infinite-horizon undiscounted MDP with absorbing terminal states. We develop a Bayesian framework to learn the optimal decision strategy through interactions with the decision-making task. Specifically, we learn the optimal action-value function $Q^*$, but unlike many existing Bayesian approaches, we do not rely on unrealistic modelling assumptions and ad-hoc approximations. Our approach is to directly construct the posterior beliefs for $Q^*$ through Bellman's optimality equations. For deterministic rewards, we characterise the posterior as a distribution with a manifold density. To facilitate simpler inference, we relax the likelihood so that a Lebesgue density exists. The flip side is to create unidentifiability issues. Specifically, the relaxed posterior can have significant mass on improper decision rules, while the exact posterior will not. We also calculate the exact posterior probabilities for optimal action selections for the tabular parametrisation of $Q^*$, a Gaussian likelihood relaxation and a Gaussian prior, which is useful in benchmarking studies. Numerical studies on variants of the Deep Sea benchmark verify our findings. We demonstrate that our framework faithfully quantifies uncertainty and, compared to other temporal-difference-based Bayesian methodologies, is more data efficient. We conclude with recommendations for future work.

stat.ML↗

arXiv2025

Mixing time of the conditional backward sampling particle filter

The conditional backward sampling particle filter (CBPF) is a powerful Markov chain Monte Carlo sampler for general state space hidden Markov model (HMM) smoothing. It was proposed as an improvement over the conditional particle filter (CPF), which has an $O(T^2)$ complexity under a general `strong' mixing assumption, where $T$ is the time horizon. Empirical evidence of the superiority of the CBPF over the CPF has never been theoretically quantified. We show that the CBPF has $O(T \log T)$ time complexity under strong mixing: its mixing time is upper bounded by $O(\log T)$, for any sufficiently large number of particles $N$ independent of $T$. This $O(\log T)$ mixing time is optimal. To prove our main result, we introduce a novel coupling of two CBPFs, which employs a maximal coupling of two particle systems at each time instant. The coupling is implementable and we use it to construct unbiased, finite variance, estimates of functionals which have arbitrary dependence on the latent state's path, with a total expected cost of $O(T \log T)$. We use this to construct unbiased estimates of the HMM's score function, and also investigate other couplings which can exhibit improved behaviour. We demonstrate our methods on financial and calcium imaging applications.

stat.CO↗

arXiv2025

Bayesian Deep Learning with Multilevel Trace-class Neural Networks

In this article we consider Bayesian inference associated to deep neural networks (DNNs) and in particular, trace-class neural network (TNN) priors which can be preferable to traditional DNNs as (a) they are identifiable and (b) they possess desirable convergence properties. TNN priors are defined on functions with infinitely many hidden units, and have strongly convergent Karhunen-Loeve-type approximations with finitely many hidden units. A practical hurdle is that the Bayesian solution is computationally demanding, requiring simulation methods, so approaches to drive down the complexity are needed. In this paper, we leverage the strong convergence of TNN in order to apply Multilevel Monte Carlo (MLMC) to these models. In particular, an MLMC method that was introduced is used to approximate posterior expectations of Bayesian TNN models with optimal computational complexity, and this is mathematically proved. The results are verified with several numerical experiments on model problems arising in machine learning, including regression, classification, and reinforcement learning.

stat.CO↗

arXiv2025

Bayesian learning of the optimal action-value function in a Markov decision process

The Markov Decision Process (MDP) is a popular framework for sequential decision-making problems, and uncertainty quantification is an essential component of it to learn optimal decision-making strategies. In particular, a Bayesian framework is used to maintain beliefs about the optimal decisions and the unknown ingredients of the model, which are also to be learned from the data, such as the rewards and state dynamics. However, many existing Bayesian approaches for learning the optimal decision-making strategy are based on unrealistic modelling assumptions and utilise approximate inference techniques. This raises doubts whether the benefits of Bayesian uncertainty quantification are fully realised or can be relied upon. We focus on infinite-horizon and undiscounted MDPs, with finite state and action spaces, and a terminal state. We provide a full Bayesian framework, from modelling to inference to decision-making. For modelling, we introduce a likelihood function with minimal assumptions for learning the optimal action-value function based on Bellman's optimality equations, analyse its properties, and clarify connections to existing works. For deterministic rewards, the likelihood is degenerate and we introduce artificial observation noise to relax it, in a controlled manner, to facilitate more efficient Monte Carlo-based inference. For inference, we propose an adaptive sequential Monte Carlo algorithm to both sample from and adjust the sequence of relaxed posterior distributions. For decision-making, we choose actions using samples from the posterior distribution over the optimal strategies. While commonly done, we provide new insight that clearly shows that it is a generalisation of Thompson sampling from multi-arm bandit problems. Finally, we evaluate our framework on the Deep Sea benchmark problem and demonstrate the exploration benefits of posterior sampling in MDPs.

stat.ML↗

arXiv2025

Towards a turnkey approach to unbiased Monte Carlo estimation of smooth functions of expectations

Given a smooth function $f$, we develop a general approach to turn Monte Carlo samples with expectation $m$ into an unbiased estimate of $f(m)$. Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of $f$ and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters -- which depend on $m$ -- automatically, with a view to make the whole approach simple to use. We develop our methods for the specific functions $f(x)=\log x$ and $f(x)=1/x$, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.

stat.ME↗

arXiv2023

Conditional particle filters with bridge backward sampling

Conditional particle filters (CPFs) with backward/ancestor sampling are powerful methods for sampling from the posterior distribution of the latent states of a dynamic model such as a hidden Markov model. However, the performance of these methods deteriorates with models involving weakly informative observations and/or slowly mixing dynamics. Both of these complications arise when sampling finely time-discretised continuous-time path integral models, but can occur with hidden Markov models too. Multinomial resampling, which is commonly employed with CPFs, resamples excessively for weakly informative observations and thereby introduces extra variance. Furthermore, slowly mixing dynamics render the backward/ancestor sampling steps ineffective, leading to degeneracy issues. We detail two conditional resampling strategies suitable for the weakly informative regime: the so-called `killing' resampling and the systematic resampling with mean partial order. To avoid the degeneracy issues, we introduce a generalisation of the CPF with backward sampling that involves auxiliary `bridging' CPF steps that are parameterised by a blocking sequence. We present practical tuning strategies for choosing an appropriate blocking. Our experiments demonstrate that the CPF with a suitable resampling and the developed `bridge backward sampling' can lead to substantial efficiency gains in the weakly informative and slow mixing regime.

stat.CO↗

arXiv2022

Quasi-Newton Sequential Monte Carlo

Sequential Monte Carlo samplers represent a compelling approach to posterior inference in Bayesian models, due to being parallelisable and providing an unbiased estimate of the posterior normalising constant. In this work, we significantly accelerate sequential Monte Carlo samplers by adopting the L-BFGS Hessian approximation which represents the state-of-the-art in full-batch optimisation techniques. The L-BFGS Hessian approximation has only linear complexity in the parameter dimension and requires no additional posterior or gradient evaluations. The resulting sequential Monte Carlo algorithm is adaptive, parallelisable and well-suited to high-dimensional and multi-modal settings, which we demonstrate in numerical experiments on challenging posterior distributions.

stat.ME↗

arXiv2022

Trace-class Gaussian priors for Bayesian learning of neural networks with MCMC

This paper introduces a new neural network based prior for real valued functions on $\mathbb R^d$ which, by construction, is more easily and cheaply scaled up in the domain dimension $d$ compared to the usual Karhunen-Loève function space prior. The new prior is a Gaussian neural network prior, where each weight and bias has an independent Gaussian prior, but with the key difference that the variances decrease in the width of the network in such a way that the resulting function is \emph{almost surely} well defined in the limit of an infinite width network. We show that in a Bayesian treatment of inferring unknown functions, the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is popular, e.g. in the Bayesian Inverse Problems literature, because it is stable under \emph{mesh refinement}, i.e. the acceptance probability does not shrink to $0$ as more parameters of the function's prior are introduced, even \emph{ad infinitum}. In numerical examples we demonstrate these stated competitive advantages over other function space priors. We also implement examples in Bayesian Reinforcement Learning to automate tasks from data and demonstrate, for the first time, stability of MCMC to mesh refinement for these type of problems.

stat.ME↗

arXiv2022

On resampling schemes for particle filters with weakly informative observations

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 stratified resampling, as the time-discretisation becomes finer and also identify their continuous-time limit, which is expressed as a suitably defined `infinitesimal generator.' By contrasting these generators, we find that (certain modifications of) systematic and SSP resampling `dominate' stratified and independent `killing' resampling in terms of their limiting overall resampling rate. The reduced intensity of resampling manifests itself in lower variance in our numerical experiment. This efficiency result, through an ordering of the resampling rate, is new to the literature. The second major contribution of this work concerns the analysis of the limiting behaviour of the entire population of particles of the particle filter as the time discretisation becomes finer. We provide the first proof, under general conditions, that the particle approximation of the discretised continuous-time Feynman--Kac path integral models converges to a (uniformly weighted) continuous-time particle system.

stat.CO↗

arXiv2022

De-biasing particle filtering for a continuous time hidden Markov model with a Cox process observation model

We develop a (nearly) unbiased particle filtering algorithm for a specific class of continuous-time state-space models, such that (a) the latent process $X_t$ is a linear Gaussian diffusion; and (b) the observations arise from a Poisson process with intensity $λ(X_t)$. The likelihood of the posterior probability density function of the latent process includes an intractable path integral. Our algorithm relies on Poisson estimates which approximate unbiasedly this integral. We show how we can tune these Poisson estimates to ensure that, with large probability, all but a few of the estimates generated by the algorithm are positive. Then replacing the negative estimates by zero leads to a much smaller bias than what would obtain through discretisation. We quantify the probability of negative estimates for certain special cases and show that our particle filter is effectively unbiased. We apply our method to a challenging 3D single molecule tracking example with a Born and Wolf observation model.

stat.CO↗

arXiv2022

Ensemble Kalman Inversion for General Likelihoods

In this letter we generalise Ensemble Kalman inversion techniques to general Bayesian models where previously they were restricted to additive Gaussian likelihoods - all in the difficult setting where the likelihood can be sampled from, but its density not necessarily evaluated.

stat.ME↗

arXiv2021

Online Particle Smoothing with Application to Map-matching

We introduce a novel method for online smoothing in state-space models that utilises a fixed-lag approximation to overcome the well known issue of path degeneracy. Unlike classical fixed-lag techniques that only approximate certain marginals, we introduce an online resampling algorithm, called particle stitching, that converts these marginal samples into a full posterior approximation. We demonstrate the utility of our method in the context of map-matching, the task of inferring a vehicle's trajectory given a road network and noisy GPS observations. We develop a new state-space model for the difficult task of map-matching on dense, urban road networks.

stat.ME↗

arXiv2020

Asymptotic Analysis of Model Selection Criteria for General Hidden Markov Models

The paper obtains analytical results for the asymptotic properties of Model Selection Criteria -- widely used in practice -- for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical i.i.d.-like model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is not weakly consistent. Numerical experiments support our theoretical results.

math.ST↗

arXiv2019

Coupled conditional backward sampling particle filter

The conditional particle filter (CPF) is a promising algorithm for general hidden Markov model smoothing. Empirical evidence suggests that the variant of CPF with backward sampling (CBPF) performs well even with long time series. Previous theoretical results have not been able to demonstrate the improvement brought by backward sampling, whereas we provide rates showing that CBPF can remain effective with a fixed number of particles independent of the time horizon. Our result is based on analysis of a new coupling of two CBPFs, the coupled conditional backward sampling particle filter (CCBPF). We show that CCBPF has good stability properties in the sense that with fixed number of particles, the coupling time in terms of iterations increases only linearly with respect to the time horizon under a general (strong mixing) condition. The CCBPF is useful not only as a theoretical tool, but also as a practical method that allows for unbiased estimation of smoothing expectations, following the recent developments by Jacob et al. (to appear). Unbiased estimation has many advantages, such as enabling the construction of asymptotically exact confidence intervals and straightforward parallelisation.

stat.CO↗

arXiv2018

Stability of Conditional Sequential Monte Carlo

The particle Gibbs (PG) sampler is a Markov Chain Monte Carlo (MCMC) algorithm, which uses an interacting particle system to perform the Gibbs steps. Each Gibbs step consists of simulating a particle system conditioned on one particle path. It relies on a conditional Sequential Monte Carlo (cSMC) method to create the particle system. We propose a novel interpretation of the cSMC algorithm as a perturbed Sequential Monte Carlo (SMC) method and apply telescopic decompositions developed for the analysis of SMC algorithms \cite{delmoral2004} to derive a bound for the distance between the expected sampled path from cSMC and the target distribution of the MCMC algorithm. This can be used to get a uniform ergodicity result. In particular, we can show that the mixing rate of cSMC can be kept constant by increasing the number of particles linearly with the number of observations. Based on our decomposition, we also prove a central limit theorem for the cSMC Algorithm, which cannot be done using the approaches in \cite{Andrieu2013} and \cite{Lindsten2014}.

stat.CO↗

arXiv2018

Multilevel Monte Carlo for Smoothing via Transport Methods

In this article we consider recursive approximations of the smoothing distribution associated to partially observed stochastic differential equations (SDEs), which are observed discretely in time. Such models appear in a wide variety of applications including econometrics, finance and engineering. This problem is notoriously challenging, as the smoother is not available analytically and hence require numerical approximation. This usually consists by applying a time-discretization to the SDE, for instance the Euler method, and then applying a numerical (e.g. Monte Carlo) method to approximate the smoother. This has lead to a vast literature on methodology for solving such problems, perhaps the most popular of which is based upon the particle filter (PF) e.g. [9]. In the context of filtering for this class of problems, it is well-known that the particle filter can be improved upon in terms of cost to achieve a given mean squared error (MSE) for estimates. This in the sense that the computational effort can be reduced to achieve this target MSE, by using multilevel (ML) methods [12, 13, 18], via the multilevel particle filter (MLPF) [16, 20, 21]. For instance, to obtain a MSE of $\mathcal{O}(ε^2)$ for some $ε> 0$ when approximating filtering distributions associated with Euler-discretized diffusions with constant diffusion coefficients, the cost of the PF is $\mathcal{O}(ε^{-3})$ while the cost of the MLPF is $\mathcal{O}(ε^{-2}\log(ε)^2)$. In this article we consider a new approach to replace the particle filter, using transport methods in [27]. In the context of filtering, one expects that the proposed method improves upon the MLPF by yielding, under assumptions, a MSE of $\mathcal{O}(ε^2)$ for a cost of $\mathcal{O}(ε^{-2})$. This is established theoretically in an "ideal" example and numerically in numerous examples.

stat.ME↗

arXiv2018

On Large Lag Smoothing for Hidden Markov Models

In this article we consider the smoothing problem for hidden Markov models (HMM). Given a hidden Markov chain $\{X_n\}_{n\geq 0}$ and observations $\{Y_n\}_{n\geq 0}$, our objective is to compute $\mathbb{E}[φ(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued, integrable functional $φ$ and $k$ fixed, $k \ll n$ and for some realisation $(y_0,\dots,y_n)$ of $(Y_0,\dots,Y_n)$. We introduce a novel application of the multilevel Monte Carlo (MLMC) method with a coupling based on the Knothe-Rosenblatt rearrangement. We prove that this method can approximate the afore-mentioned quantity with a mean square error (MSE) of $\mathcal{O}(ε^2)$, for arbitrary $ε>0$ with a cost of $\mathcal{O}(ε^{-2})$. This is in contrast to the same direct Monte Carlo method, which requires a cost of $\mathcal{O}(nε^{-2})$ for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of \cite{span} is used, which directly approximates our strategy. We show that our theoretical improvements are achieved, even under approximation, in several numerical examples.

stat.ME↗