Pierre Del Moral

Approximate Bayesian Computation for Smoothing

We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter ε > 0. We provide theoretical results which quantify, in terms of ε, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is , where n is the number of time steps over which smoothing is performed. For numerical implementation, we adopt the forward-only sequential Monte Carlo (SMC) scheme of [14] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors, with a finite number of data and simulated samples.

An island particle algorithm for rare event analysis

Estimating rare event probability with accuracy is of great interest for safety and reliability applications. In this paper, we focus on rare events which can be modeled by a threshold exceedance of a deterministic input–output function with random inputs. Some parameters of this function or density parameters of input random variables may be fixed by an experimenter for simplicity reasons. From a risk analysis point of view, it is not only interesting to evaluate the probability of a critical event but it is also important to determine the impact of such tuning of parameters on the realization of a critical event, because a bad estimation of these parameters can strongly modify rare event probability estimations. In the present paper, we present an example of island particle algorithm referred to as sequential Monte Carlo square (SMC2). This algorithm gives an estimate of the law of random phenomena that leads to critical events. The principles of this statistical technique are described throughout this article and its results are analysed on different realistic aerospace test cases.

The Alive Particle Filter and Its Use in Particle Markov Chain Monte Carlo

In the following article, we investigate a particle filter for approximating Feynman–Kac models with indicator potentials and we use this algorithm within Markov chain Monte Carlo (MCMC) to learn static parameters of the model. Examples of such models include approximate Bayesian computation (ABC) posteriors associated with hidden Markov models (HMMs) or rare-event problems. Such models require the use of advanced particle filter or MCMC algorithms to perform estimation. One of the drawbacks of existing particle filters is that they may “collapse,” in that the algorithm may terminate early, due to the indicator potentials. In this article, using a newly developed special case of the locally adaptive particle filter, we use an algorithm that can deal with this latter problem, while introducing a random cost per-time step. In particular, we show how this algorithm can be used within MCMC, using particle MCMC. It is established that, when not taking into account computational time, when the new MCMC algorithm is applied to a simplified model it has a lower asymptotic variance in comparison to a standard particle MCMC algorithm. Numerical examples are presented for ABC approximations of HMMs.

Particle Association Measures and Multiple Target Tracking

In the last decade, the area of multiple target tracking has witnessed the introduction of important concepts and methods, aiming at establishing principled approaches for dealing with the estimation of multiple objects in an efficient way. One of the most successful classes of multi-object filters that have been derived out of these new grounds includes all the variants of the Probability Hypothesis Density (phd) filter. In spite of the attention that these methods have attracted, their theoretical performances are still not fully understood. In this chapter, we first focus on the different ways of establishing the equations of the phd filter, using a consistent set of notations. The objective is then to introduce the idea of observation path, upon which association measures are defined. We will see how these concepts highlight the structure of the first moment of the multi-object distributions in time, and how they allow for devising solutions to practical estimation problems.

A sequential Monte Carlo approximation of the HISP filter

A formulation of the hypothesised filter for independent stochastic populations (hisp) is proposed, based on the concept of association measure, which is a measure on the set of observation histories. Using this formulation, a particle approximation is introduced at the level of the association measure for handling the exponential growth in the number of underlying hypotheses. This approximation is combined with a sequential Monte Carlo implementation for the underlying single-object distributions to form a mixed particle association model. Finally, the performance of this approach is compared against a Kalman filter implementation on simulated data based on a finite-resolution sensor.

A Note on Random Walks with Absorbing barriers and Sequential Monte Carlo Methods

ABSTRACT In this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.

Probabilistic Safety Analysis of the Collision Between a Space Debris and a Satellite with an Island Particle Algorithm

Collision between satellites and space debris seldom happens, but the loss of a satellite by collision may have catastrophic consequences both for the satellite mission and for the space environment. To support the decision to trigger off a collision avoidance manoeuver, an adapted tool is the determination of the collision probability between debris and satellite. This probability estimation can be performed with rare event simulation techniques when Monte Carlo techniques are not enough accurate. In this chapter, we focus on analyzing the influence of different simulation parameters (such as the drag coefficient) that are set for to simplify the simulation, on the collision probability estimation. A bad estimation of these simulation parameters can strongly modify rare event probability estimations. We design here a new island particle Markov chain Monte Carlo algorithm to determine the parameters that, in case of bad estimation, tend to increase the collision probability value. This algorithm also gives an estimate of the collision probability maximum taking into account the likelihood of the parameters. The principles of this statistical technique are described throughout this chapter.