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  • 1.
    Jacob, Pierre
    et al.
    Harvard University, USA.
    Lindsten, Fredrik
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Retracted article: Smoothing with Couplings of Conditional Particle Filters2018In: Journal of the American Statistical Association, ISSN 0162-1459, E-ISSN 1537-274XArticle in journal (Refereed)
    Abstract [en]

    In state space models, smoothing refers to the task of estimating a latent stochastic process given noisy measurements related to the process. We propose an unbiased estimator of smoothing expectations. The lack-of-bias property has methodological benefits: independent estimators can be generated in parallel, and confidence intervals can be constructed from the central limit theorem to quantify the approximation error. To design unbiased estimators, we combine a generic debiasing technique for Markov chains, with a Markov chain Monte Carlo algorithm for smoothing. The resulting procedure is widely applicable and we show in numerical experiments that the removal of the bias comes at a manageable increase in variance. We establish the validity of the proposed estimators under mild assumptions. Numerical experiments are provided on toy models, including a setting of highly-informative observations, and for a realistic Lotka-Volterra model with an intractable transition density.

  • 2.
    Lindsten, Fredrik
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Bunch, Pete
    Department of Engineering, University of Cambridge, Cambridge, UK.
    Särkkä, Simo
    Department of Electrical Engineering and Automation, Aalto University, Aalto, Finland.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Godsill, Simon J.
    Department of Engineering, University of Cambridge, Cambridge, UK.
    Rao–Blackwellized particle smoothers for conditionally linear Gaussian models2016In: IEEE Journal on Selected Topics in Signal Processing, ISSN 1932-4553, E-ISSN 1941-0484, Vol. 10, no 2, p. 353-365Article in journal (Refereed)
    Abstract [en]

    Sequential Monte Carlo (SMC) methods, such as the particle filter, are by now one of the standard computational techniques for addressing the filtering problem in general state-space models. However, many applications require post-processing of data offline. In such scenarios the smoothing problem-in which all the available data is used to compute state estimates-is of central interest. We consider the smoothing problem for a class of conditionally linear Gaussian models. We present a forward-backward-type Rao-Blackwellized particle smoother (RBPS) that is able to exploit the tractable substructure present in these models. Akin to the well known Rao-Blackwellized particle filter, the proposed RBPS marginalizes out a conditionally tractable subset of state variables, effectively making use of SMC only for the “intractable part” of the model. Compared to existing RBPS, two key features of the proposed method are: 1) it does not require structural approximations of the model, and 2) the aforementioned marginalization is done both in the forward direction and in the backward direction.

  • 3.
    Schön, Thomas B.
    et al.
    Uppsala universitet, Reglerteknik, Sweden.
    Svensson, Andreas
    Uppsala universitet, Reglerteknik, Sweden.
    Murray, Lawrence
    Uppsala universitet, Reglerteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Probabilistic learning of nonlinear dynamical systems using sequential Monte Carlo2018In: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 104, p. 866-883Article in journal (Refereed)
    Abstract [en]

    Probabilistic modeling provides the capability to represent and manipulate uncertainty in data, models, predictions and decisions. We are concerned with the problem of learning probabilistic models of dynamical systems from measured data. Specifically, we consider learning of probabilistic nonlinear state-space models. There is no closed-form solution available for this problem, implying that we are forced to use approximations. In this tutorial we will provide a self-contained introduction to one of the state-of-the-art methods the particle Metropolis-Hastings algorithm which has proven to offer a practical approximation. This is a Monte Carlo based method, where the particle filter is used to guide a Markov chain Monte Carlo method through the parameter space. One of the key merits of the particle Metropolis-Hastings algorithm is that it is guaranteed to converge to the "true solution" under mild assumptions, despite being based on a particle filter with only a finite number of particles. We will also provide a motivating numerical example illustrating the method using a modeling language tailored for sequential Monte Carlo methods. The intention of modeling languages of this kind is to open up the power of sophisticated Monte Carlo methods including particle Metropolis-Hastings to a large group of users without requiring them to know all the underlying mathematical details.

  • 4.
    Svensson, Andreas
    et al.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Avdelningen för systemteknik, Sweden.
    Learning nonlinear state-space models using smooth particle-filter-based likelihood approximations2018In: 18th IFAC Symposium on System IdentificationSYSID 2018 Proceedings, Elsevier, 2018, p. 652-657Conference paper (Refereed)
    Abstract [en]

    When classical particle filtering algorithms are used for maximum likelihood parameter estimation in nonlinear state-space models, a key challenge is that estimates of the likelihood function and its derivatives are inherently noisy. The key idea in this paper is to run a particle filter based on a current parameter estimate, but then use the output from this particle filter to re-evaluate the likelihood function approximation also for other parameter values. This results in a (local) deterministic approximation of the likelihood and any standard optimization routine can be applied to find the maximum of this approximation. By iterating this procedure we eventually arrive at a final parameter estimate.

  • 5.
    Svensson, Andreas
    et al.
    Uppsala universitet, Reglerteknik, Sweden.
    Schön, Thomas B.
    Uppsala universitet, Reglerteknik, Sweden.
    Lindsten, Fredrik
    Uppsala universitet, Reglerteknik, Sweden.
    Learning of state-space models with highly informative observations: A tempered sequential Monte Carlo solution2018In: Mechanical systems and signal processing, ISSN 0888-3270, E-ISSN 1096-1216, Vol. 104, p. 915-928Article in journal (Refereed)
    Abstract [en]

    Probabilistic (or Bayesian) modeling and learning offers interesting possibilities for systematic representation of uncertainty using probability theory. However, probabilistic learning often leads to computationally challenging problems. Some problems of this type that were previously intractable can now be solved on standard personal computers thanks to recent advances in Monte Carlo methods. In particular, for learning of unknown parameters in nonlinear state-space models, methods based on the particle filter (a Monte Carlo method) have proven very useful. A notoriously challenging problem, however, still occurs when the observations in the state-space model are highly informative, i.e. when there is very little or no measurement noise present, relative to the amount of process noise. The particle filter will then struggle in estimating one of the basic components for probabilistic learning, namely the likelihood p(datalparameters). To this end we suggest an algorithm which initially assumes that there is substantial amount of artificial measurement noise present. The variance of this noise is sequentially decreased in an adaptive fashion such that we, in the end, recover the original problem or possibly a very close approximation of it. The main component in our algorithm is a sequential Monte Carlo (SMC) sampler, which gives our proposed method a clear resemblance to the SMC2 method. Another natural link is also made to the ideas underlying the approximate Bayesian computation (ABC). We illustrate it with numerical examples, and in particular show promising results for a challenging Wiener-Hammerstein benchmark problem.

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