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Bayesian Inference for Linear Dynamic Models with Dirichlet Process Mixtures

François Caron, Manuel Davy, Arnaud Doucet, Emmanuel Duflos, Philippe Vanheeghe

arXiv:math/0702225v1math.STstat.TH

TL;DR

The paper tackles state estimation in linear dynamic models when noise densities are unknown, using Dirichlet process mixtures and simulation-based inference. It reports efficient inference across synthetic and real applications, while its sequential approach assumes fixed hyperparameters.

  • Problem

    Linear Gaussian Kalman methods do not cover cases where state or observation noise densities are unknown or inadequately modeled as Gaussian.

    Source: §I · p. 2

  • Method

    The paper models state and observation noise with Dirichlet process mixtures and develops Rao-Blackwellized simulation-based algorithms for batch and sequential inference.

    Source: B. Estimation of the state parameters · p. 7, §VIII · p. 23

  • Results

    The authors report efficient applications, including recovery of a two-mode noise density in simulation and posterior jump-probability estimation for real creatinine data.

    Source: 2) Simulation results:  · p. 15, 2) Simulation results · p. 19, §VIII · p. 23

  • Takeaways & Limitations

    State estimation is possible in linear dynamic models even when the dynamic and observation noises have unknown probability densities.

    Source: §VIII · p. 23

    The sequential method assumes a fixed hyperparameter vector; the discussed alternatives have drawbacks or would discard the Polya urn structure.

    Source: D. About the convergence of the… · p. 22

Abstract

from arXiv · show

Using Kalman techniques, it is possible to perform optimal estimation in linear Gaussian state-space models. We address here the case where the noise probability density functions are of unknown functional form. A flexible Bayesian nonparametric noise model based on Dirichlet process mixtures is introduced. Efficient Markov chain Monte Carlo and Sequential Monte Carlo methods are then developed to perform optimal batch and sequential estimation in such contexts. The algorithms are applied to blind deconvolution and change point detection. Experimental results on synthetic and real data demonstrate the efficiency of this approach in various contexts.

1 CNRS/Ecole Centrale de Lille and INRIA-FUTURS SequeL team, Villeneuve d’Ascq, France

The paper addresses state estimation in linear dynamic models when noise distributions are unknown, using Dirichlet process mixtures and Bayesian inference. It situates this approach against Gaussian and fixed-shape alternatives and reviews the density-estimation machinery that supports it.

I. INTRODUCTION

  • Unknown noise distributions can make the usual Gaussian assumption inadequate, while the target remains filtering or smoothing of the hidden state from observations.The paper considers online and offline estimation of the noise densities as well as the state.Source: §I · p. 2

A. Proposed approach

  • Dirichlet process mixtures provide flexible noise-density models, with a Polya urn representation that simplifies latent-variable conditional distributions.The mixture uses a prior over component parameters and a concentration parameter that tunes how many mixture weights are appreciably nonzero without fixing a component count.Source: A. Proposed approach · p. 2

B. Previous works

  • Earlier noise-statistics methods assume Gaussian densities, while other non-Gaussian methods prescribe a parametric density shape; this paper instead uses DPMs for linear dynamic models.The authors state, to their knowledge, that DPMs had not previously been used in this setting.Source: B. Previous works · pp. 2–3

C. Paper organization

  • The paper proceeds from DPM density estimation and dynamic-model specification to offline MCMC, online particle filtering, and applications to blind deconvolution and biomedical change-point detection.It identifies the algorithms as Rao-Blackwellized methods.Source: C. Paper organization · p. 3

A. Density estimation

  • The paper reviews Bayesian nonparametric density estimation, where the density cannot be represented by a finite-dimensional functional expansion and is modeled using a mixing distribution.In the model, the component parameter is latent, and the mixing distribution is assigned a Dirichlet process prior.Source: §II · p. 3, A. Density estimation · p. 4

B. Dirichlet Processes

  • A Dirichlet process yields discrete realizations; its concentration parameter tunes the number of distinct component parameters, from mostly shared values at small α to near-independent draws from G0 at large α.The passage describes this as a way to control clustering without specifying a fixed number of components.Source: B. Dirichlet Processes · pp. 4–5

C. Dirichlet Process Mixtures

The paper models unknown state and observation noise densities with Dirichlet process mixtures and frames inference around latent mixture variables. Its estimation target includes both the state and noise-model quantities, though the state is the quantity of interest.

C. Dirichlet Process Mixtures

  • The DPM density model represents a distribution as a mixture whose component parameters are latent and whose mixing distribution has a Dirichlet process prior.The paper notes that Gaussian component densities can accommodate a range of component means and variances.Source: C. Dirichlet Process Mixtures · p. 5

D. Estimation objectives

  • Existing DPM samplers cannot be applied directly because the state-noise and observation-noise sequences are not observed; only their effects through the dynamic model are observed.The paper therefore develops an MCMC algorithm for this model.Source: D. Estimation objectives · p. 6

III. DYNAMIC LINEAR MODEL WITH UNKNOWN NOISE DISTRIBUTION

  • The dynamic model assumes the state and observation noise distributions are DPMs, with Gaussian component densities and normal-inverse-Wishart base distributions.The hyperparameters are treated as known for presentation, while a procedure with priors for unknown hyperparameters is addressed later.Source: §III · p. 6, A. DPM noise models · p. 6

B. Estimation of the state parameters

State estimates are obtained by averaging Kalman-based conditional estimates over posterior samples of latent noise parameters. Offline MCMC and online sequential importance sampling provide those samples.

B. Estimation of the state parameters

  • The state and noise-model quantities are estimated conditional on observations, but the state is the quantity of interest; integrating out the mixing distributions leaves latent component parameters in the posterior.The target may be the online posterior through time or the offline posterior over the full observation sequence.Source: B. Estimation of the state parameters · p. 7
  • Conditional on latent parameters, the state posterior is Gaussian and its means are computed with Kalman filtering or smoothing; Monte Carlo integration over parameter samples yields marginal state estimates.The passage identifies this integration as not available in closed form.Source: B. Estimation of the state parameters · pp. 7–8
  • Offline estimation uses MCMC, whereas online estimation uses sequential importance sampling to generate samples from the latent-parameter posterior.The standard DPM samplers for directly observed data do not apply because the model’s noise sequences are unobserved.Source: B. Estimation of the state parameters · p. 8

IV. MCMC ALGORITHM FOR OFF-LINE STATE ESTIMATION

The offline method samples latent noise parameters with MCMC and then computes state estimates using Kalman smoothing. A backward-forward likelihood recursion reduces the stated full-sample computational complexity from quadratic to linear in sequence length.

A. Fixed and known hyperparameters

  • For fixed, known hyperparameters, the offline posterior over latent parameters is sampled with a Gibbs procedure whose conditional calculations use forward Kalman and backward recursions.The algorithm’s full posterior-sampling procedure is described as accounting for a factorized likelihood.Source: A. Fixed and known hyperparameters · pp. 8–9
  • Each latent-parameter update uses a Metropolis-Hastings step with the conditional prior as proposal and an acceptance probability requiring the data likelihood.After burn-in, retained samples are used to compute MMSE estimates of latent parameters and states.Source: A. Fixed and known hyperparameters · pp. 8–10
  • A backward-forward likelihood recursion reduces the offline algorithm’s complexity from O(T^2) to O(T).Repeated O(T) Kalman likelihood evaluations across time would otherwise produce the quadratic cost.Source: A. Fixed and known hyperparameters · p. 9
  • For each retained latent-parameter sample, state means are computed with a Kalman smoother to produce offline state estimates.The sampled Markov chain targets the latent-parameter posterior given the full observation sequence.Source: A. Fixed and known hyperparameters · p. 10

B. Unknown hyperparameters

The unknown hyperparameters are incorporated into inference through prior distributions and conditional posterior updates. The resulting updates use Metropolis–Hastings when direct sampling is unavailable.

B. Unknown hyperparameters

  • Hyperparameters affecting the DPMs are treated as unknowns with prior distributions in the inference.The passage identifies the hyperparameters in φ as influencing estimation of Fv and Fw.Source: B. Unknown hyperparameters · p. 10
  • The conditional posterior for αv reduces to p(αv|Mv,T), and it can be sampled with a Metropolis–Hastings step using a Gamma prior.Direct-sampling alternatives are also cited and.Source: B. Unknown hyperparameters · pp. 10–11
  • When direct sampling is unavailable, ψv is sampled with a Metropolis–Hastings step using its prior Gamma density as the proposal.Source: B. Unknown hyperparameters · p. 11

V. RAO-BLACKWELLIZED PARTICLE FILTER ALGORITHM FOR ONLINE STATE ESTIMATION

The online method uses a Rao–Blackwellized particle filter to sample noise-cluster variables while computing state posteriors with Kalman techniques. Its application section then presents blind deconvolution and biomedical time-series change-point detection with unknown state-noise statistics modeled by a DPM.

V. RAO-BLACKWELLIZED PARTICLE FILTER ALGORITHM FOR ONLINE STATE ESTIMATION

  • The particle filter samples θ1:t sequentially because Kalman techniques compute p(x0:t|θ1:t,z1:t), extending the Rao–Blackwellized particle filter to DPMs.The method targets online posterior distributions as observations arrive; the hyperparameter vector φ is assumed known here.Source: §V · p. 11
  • For each particle, Kalman filtering recursively computes the state estimate and posterior covariance used in the particle approximation.The algorithm updates particle weights and resamples when the effective sample size falls below a threshold.Source: §V · pp. 11–12
  • The conditionally optimal importance distribution is not used because its importance weights lack a closed-form expression; the implementation uses the evolution density instead.Computing the optimal weights requires integration over θt; cluster means can be integrated analytically, but covariances cannot.Source: §V · p. 12

VI. APPLICATIONS

  • The applications are blind deconvolution and biomedical time-series change-point detection, with unknown state-noise statistics modeled as a DPM.Source: §VI · p. 13

A. Blind deconvolution of impulse processes

The blind-deconvolution application models the observed signal as a filtered latent sequence with additive white Gaussian noise. Its state noise can be zero or drawn from a Gaussian DPM, and the joint posterior is approximated by MCMC.

A. Blind deconvolution of impulse processes

  • Blind deconvolution is presented as an application for impulse-process restoration, following the model of.The passage cites image de-blurring, spectroscopic analysis, and audio restoration as application areas.Source: A. Blind deconvolution of impulse… · p. 13

1) Statistical Model:

  • The observation model treats zt as the convolution of xt with a finite impulse response filter H, plus additive white Gaussian noise wt.Source: 1) Statistical Model · p. 13
  • The state-transition noise is either zero or distributed according to a DPM of Gaussians, with latent Bernoulli variables representing the zero-versus-mixture choice.The mixture probability λ has a beta prior.Source: 1) Statistical Model · pp. 13–14
  • MCMC approximates the joint posterior of v1:T, ϕ1:T, and φ given z1:T, using the particle algorithm for cluster variables and conditional sampling for the others.The simulation smoother has O(T) complexity.Source: 1) Statistical Model · p. 14

2) Simulation results: 

In the synthetic simulation, the proposed model estimates the hidden signal and its unknown noise density, and its state MSE is close to that of a model with the true density fixed. The experiments also compare alternative noise assumptions and describe a biomedical change-point setting.

2) Simulation results: 

  • With 120 observations and 10,000 MCMC iterations, the estimated hidden signal is almost correct and the estimated noise density matches both modes of the true mixture.The runs used 7,500 burn-in iterations; multiple starting values gave results described as insensitive to initialization.Source: 2) Simulation results:  · p. 15, Figure 2 · p. 16
  • The estimated vector h converges toward its true value, [−1.5, 0.5, −0.2].Source: 2) Simulation results:  · p. 17
  • M1’s state MSE is only 10% higher than M2’s, despite M1 not estimating the noise pdf exactly.M2 uses the true pdf as fixed.Source: Table I · p. 17, 2) Simulation results:  · p. 17
  • M3’s MSE is 17% higher than the proposed approach’s; fixing α_v also raises MSE, especially at α_v = 0.1, where cluster proposals are rare and posterior convergence is slow.Source: 2) Simulation results:  · p. 18

B. Change-point problems in biomedical time series

1) Statistical model:

The creatinine model uses a Gaussian Dirichlet process mixture to represent state-noise jumps without fixing their levels in advance. At a 0.5 threshold, MCMC detects three jumps, while the Rao-Blackwellized particle filter detects two.

1) Statistical model:

  • The model uses a Gaussian Dirichlet process mixture for state-noise jumps, estimating their levels and number instead of fixing jump levels as in.The measurement-noise model accounts for errors such as transcription mistakes, equipment malfunction, or blood contamination.Source: 1) Statistical model · p. 18

2) Simulation results:

A. About Dirichlet Process-based modeling

The section weighs Dirichlet process mixtures’ flexible modeling and implementation advantages against inference limitations, then discusses the behavior and scope of the proposed algorithms. It concludes that state estimation is possible with unknown dynamic and observation noise pdfs.

A. About Dirichlet Process-based modeling

  • DPMs simplify posterior sampling through the Polya urn scheme and allow straightforward component-count estimation, but can favor imbalanced groups and too many groups.The authors describe this as a trade-off between versatile modeling properties and implementation advantages.Source: A. About Dirichlet Process-based… · pp. 20–21

B. About MCMC algorithms for DPMs

  • Single-site marginal MCMC can become stuck because separating noises assigned to one cluster requires passage through a rare, low-probability intermediate state.Group-sampling alternatives avoid this issue at increased computational cost; the authors report their scheme efficient in the presented applications.Source: B. About MCMC algorithms for DPMs · p. 21

C. About the hyperparameter estimation in the MCMC algorithm

D. About the convergence of the Rao-Blackwellized particle filter

  • The particle filter accumulates errors because static DPM noise distributions prevent cluster values from moving after initialization, though this is less critical when the static component is prior G rather than cluster θ_t.The hyperparameter vector φ is fixed; artificial dynamics are unsuitable because they would lose the Polya urn structure.Source: D. About the convergence of the… · pp. 21–22

E. About related approaches

  • Unlike common jump linear systems with a known number of known models and known prior probabilities, this model estimates the number of models, their parameters, and their prior probabilities.Source: E. About related approaches · p. 22

F. About observability

VIII. CONCLUSION

  • The proposed Rao-Blackwellized simulation algorithms are reported efficient in applications, including state estimation despite unknown dynamic and observation noise pdfs.The authors identify nonlinear dynamic models and time-varying DPMs for evolving noise statistics as extensions under investigation.Source: §VIII · p. 23

APPENDIX

The appendix describes a backward-forward recursion and related computational quantities. It notes that a singular filtered covariance can be factored using singular value decomposition.

A. Notations µ

  • The appendix states that Σ is sampled from the Normal inverse Wishart distribution G0, parameterized in part by µ0 and κ0.Source: A. Notations µ · p. 23

B. Backward forward recursion

  • When the filtered covariance is nonzero, the appendix factors it into Q and Π matrices obtained by singular value decomposition.Source: B. Backward forward recursion · p. 24
  • The one-step filtered state estimate and covariance, innovation, and innovation covariance are supplied by the Kalman filter conditional on the linear-Gaussian system.Source: B. Backward forward recursion · p. 24
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