Particle Filter with Missing Data
Sequential Monte Carlo Particle Filter for State-Space Models with Missing Observations · Also known as: SMC with missing data, bootstrap filter with missing observations, sequential Monte Carlo missing data, particle filtering incomplete data
A particle filter adapted for state-space models in which some observations are absent. The algorithm tracks a hidden state over time using a cloud of weighted random samples (particles); when a time step has no observed value, the weight-update step is simply skipped, so the particles propagate forward using only the transition model until new data arrives.
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When to use it
Use a particle filter with missing data when analysing a non-linear or non-Gaussian dynamic system observed sequentially, and when some time steps yield no measurement due to sensor failure, dropout, or censoring. It is preferable to the Kalman filter when the model is non-linear or the noise is non-Gaussian, and to batch methods when real-time or online inference is needed. Do not use it when the state space is very high-dimensional (particle degeneracy becomes severe), when missingness is informative and unmodelled, or when a linear-Gaussian Kalman filter with missing data suffices — that case has an exact, cheaper closed-form solution.
Strengths & limitations
- Handles non-linear and non-Gaussian state-space models exactly (in the Monte Carlo sense), unlike the Kalman filter.
- Missing observations are handled with no structural change — simply skip the weight update step at those time steps.
- Provides a full posterior distribution over the hidden state at each time point, not just a point estimate.
- Online (sequential) update: inference can proceed in real time as new data arrive.
- Flexible: any tractable transition and emission model can be used.
- Particle degeneracy in high-dimensional state spaces: exponentially many particles are needed for reliable coverage.
- Resampling introduces Monte Carlo variance and can cause loss of diversity (particle impoverishment).
- Computationally expensive compared with analytic filters; cost scales linearly with the number of particles.
- Long stretches of missing data cause weight distributions to flatten, reducing effective sample size and increasing uncertainty substantially.
- Tuning the proposal distribution and resampling threshold requires expertise.
Frequently asked
How exactly does the particle filter skip a missing observation?
At any time step where y_t is missing, the likelihood p(y_t | x_t) cannot be evaluated. The weight update formula therefore reduces to multiplying each particle's weight by the transition density ratio only — effectively leaving weights unchanged when the bootstrap filter is used. Particles continue to propagate forward according to the transition model, so the posterior broadens over missing intervals.
Do I need to do anything special if many consecutive observations are missing?
Long gaps amplify uncertainty: particles spread according to the transition model and effective sample size may drop. You may need more particles to maintain adequate coverage, and you should monitor ESS across the sequence. If missingness is predictable, you can use a targeted proposal that accounts for the impending gap.
When should I use the Kalman filter instead?
If your state-space model is linear and all noise is Gaussian, the Kalman filter handles missing observations exactly and cheaply by simply not performing the measurement-update step at missing time points. The particle filter is only needed when non-linearity or non-Gaussian noise makes the Kalman filter approximate.
What if the data are not missing at random?
The standard approach assumes missingness is ignorable (missing at random or completely at random). If whether an observation is missing depends on the unobserved state — for instance, a sensor fails precisely when the signal is extreme — you must model the missingness mechanism jointly with the state, adding a separate emission model for the indicator r_t.
How many particles do I need?
There is no universal rule: it depends on the dimensionality of the state space and the quality of the proposal. Common starting points range from a few hundred to several thousand. Monitor the effective sample size (ESS = 1 / sum of squared normalised weights) over time; if ESS routinely falls far below N/2, increase the particle count or improve the proposal.
Sources
- Doucet, A., de Freitas, N. & Gordon, N. J. (Eds.) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York. ISBN: 978-0387951461
- Doucet, A., Godsill, S. & Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10(3), 197-208. DOI: 10.1023/A:1008935410038 ↗
How to cite this page
ScholarGate. (2026, June 3). Sequential Monte Carlo Particle Filter for State-Space Models with Missing Observations. ScholarGate. https://scholargate.app/en/bayesian/particle-filter-with-missing-data
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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- Dynamic Particle FilterBayesian↔ compare
- Kalman Filter with Missing DataBayesian↔ compare
- MCMC with missing dataBayesian↔ compare
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- Sequential Monte CarloBayesian↔ compare