Adapting the Number of Particles in Sequential Monte Carlo Methods through an Online Scheme for Convergence Assessment

Particle filters are broadly used to approximate posterior distributions of hidden states in state-space models by means of sets of weighted particles. While the convergence of the filter is guaranteed when the number of particles tends to infinity, the quality of the approximation is usually unknown but strongly dependent on the number of particles. In this paper, we propose a novel method for assessing the convergence of particle filters online manner, as well as a simple scheme for the online adaptation of the number of particles based on the convergence assessment. The method is based on a sequential comparison between the actual observations and their predictive probability distributions approximated by the filter. We provide a rigorous theoretical analysis of the proposed methodology and, as an example of its practical use, we present simulations of a simple algorithm for the dynamic and online adaption of the number of particles during the operation of a particle filter on a stochastic version of the Lorenz system

Rethinking the Effective Sample Size

The effective sample size (ESS) is widely used in sample-based simulation methods for assessing the quality of a Monte Carlo approximation of a given distribution and of related integrals. In this paper, we revisit and complete the approximation of the ESS in the specific context of importance sampling (IS). The derivation of this approximation, that we will denote as $\widehat{\text{ESS}}$, is only partially available in Kong [1992]. This approximation has been widely used in the last 25 years due to its simplicity as a practical rule of thumb in a wide variety of importance sampling methods. However, we show that the multiple assumptions and approximations in the derivation of $\widehat{\text{ESS}}$, makes it difficult to be considered even as a reasonable approximation of the ESS. We extend the discussion of the ESS in the multiple importance sampling (MIS) setting, and we display numerical examples. This paper does not cover the use of ESS for MCMC algorithms

Latent variable analysis of causal interactions in atrial fibrillation electrograms

Multi-channel intracardiac electrocardiograms of atrial fibrillation (AF) patients are acquired at the electrophysiology laboratory in order to guide radiofrequency (RF) ablation surgery. Unfortunately, the success rate of RF ablation is still moderate, since the mechanisms underlying AF initiation and maintenance are still not precisely known. In this paper, we use an advanced machine learning model, the Gaussian process latent force model (GP-LFM), to infer the relationship between the observed signals and the unknown (latent or exogenous) sources causing them. The resulting GP-LFM provides valuable information about signal generation and propagation inside the heart, and can then be used to perform causal analysis. Results on realistic synthetic signals, generated using the FitzHugh-Nagumo model, are used to showcase the potential of the proposed approach

A principled stopping rule for importance sampling

Importance sampling (IS) is a Monte Carlo technique that relies on weighted samples, simulated from a proposal distribution, to estimate intractable integrals. The quality of the estimators improves with the number of samples. However, for achieving a desired quality of estimation, the required number of samples is unknown and depends on the quantity of interest, the estimator, and the chosen proposal. We present a sequential stopping rule that terminates simulation when the overall variability in estimation is relatively small. The proposed methodology closely connects to the idea of an effective sample size in IS and overcomes crucial shortcomings of existing metrics, e.g., it acknowledges multivariate estimation problems. Our stopping rule retains asymptotic guarantees and provides users a clear guideline on when to stop the simulation in IS

An efficient sampling scheme for the eigenvalues of dual wishart matrices

Despite the numerous results in the literature about the eigenvalue distributions of Wishart matrices, the existing closed-form probability density function (pdf) expressions do not allow for efficient sampling schemes from such densities. In this letter, we present a stochastic representation for the eigenvalues of 2×2 complex central uncorrelated Wishart matrices with an arbitrary number of degrees of freedom (referred to as dual Wishart matrices). The draws from the joint pdf of the eigenvalues are generated by means of a simple transformation of a chi-squared random variable and an independent beta random variable. Moreover, this stochastic representation allows a simple derivation, alternative to those already existing in the literature, of some eigenvalue function distributions such as the condition number or the ratio of the maximum eigenvalue to the trace of the matrix. The proposed sampling scheme may be of interest in wireless communications and multivariate statistical analysis, where Wishart matrices play a central role.The work of Ignacio Santamaria was supported in part by the Ministerio de Ciencia e Innovación and AEI/10.13039/501100011033, under Grant PID2019-104958RB-C43 (ADELE). The work of Víctor Elvira was supported in part by the Agence Nationale de la Recherche of France PISCES under Grant ANR-17-CE40-0031-01, and in part by the ARL/ARO under Grant W911NF-20-1-0126