Concept Architecture
Concept
Theoretically, Particle Filter is a sequential Monte Carlo method used to estimate the evolving state of a dynamic system when observations are received over time and the underlying process is stochastic or non-linear. The method represents the unknown state distribution using a finite set of weighted random samples, known as particles, which are updated recursively as new observations become available. In health economics, particle filters are applied to state-space models, infectious disease transmission models, and dynamic decision models where latent disease states cannot be observed directly.
Mathematically, the particle filter approximates the posterior distribution of the hidden state by propagating particles through the system model and updating their weights according to the likelihood of the observed data. A resampling step is performed periodically to eliminate particles with negligible weights and concentrate computational effort on particles that better represent the posterior distribution. As the number of particles increases, the empirical approximation converges to the true filtering distribution under standard regularity conditions.
In practice, particle filters are implemented within Bayesian state-space modelling frameworks to estimate time-varying epidemiological parameters, latent disease prevalence, transmission dynamics, or treatment response trajectories. They are particularly valuable when conventional Kalman filtering assumptions of linearity and Gaussian errors are violated. Estimated posterior state distributions can subsequently inform health economic decision models and probabilistic analyses.
Purpose
Used to estimate evolving hidden states and time-varying parameters in dynamic stochastic models by sequentially updating probability distributions as new observational data become available.
Mathematical Formulae
Primary Formula
The filtering distribution is approximated by
p(x? | y?:?) � �???? w???? �(x? ? x????)
where:
- x???? = particle i
- w???? = normalised particle weight
- N = number of particles
- �(�) = Dirac delta function
Supporting Formulae
Weight update:
w???? ? w?????? p(y? | x????)
Normalisation:
w???? = w???? / �???? w????
Effective sample size:
N?ff = 1 / �???? (w????)�
Related Mathematical Methods
- Sequential Monte Carlo
- Bayesian filtering
- State-space modelling
- Hidden Markov models
- Markov Chain Monte Carlo
- Kalman filtering
- Importance sampling
- Bayesian parameter estimation
Example
A dynamic influenza transmission model estimates the unobserved number of infectious individuals each week.
The model propagates 10,000 particles through the transmission equations. Weekly surveillance data are used to update particle weights according to the likelihood of the observed case counts. After weighting, particles are resampled when the effective sample size falls below a predefined threshold. The weighted particle distribution provides an estimate of the posterior number of infectious individuals, which is subsequently used to project healthcare costs and quality-adjusted life years under alternative vaccination strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generate random particles during simulation |
| NORM.INV | =NORM.INV(RAND(),Mean,SD) | Sample from proposal or transition distributions |
| SUM | =SUM(C2:C10001) | Calculate the total particle weight before normalisation |
| COUNT | =COUNT(A2:A10001) | Count simulated particles |
| SUMPRODUCT | =SUMPRODUCT(B2:B10001,C2:C10001) | Calculate weighted posterior estimates from particle values |
VBA (Optional)
Automate sequential particle propagation, weight updating, resampling, and export of posterior state estimates for dynamic health economic models.
Sources
- Doucet A, de Freitas N, Gordon N, eds. Sequential Monte Carlo Methods in Practice. Springer; 2001.
- Capp� O, Godsill S, Moulines E. An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo. Proceedings of the IEEE. 2007;95(5):899-924.
- Arulampalam MS, Maskell S, Gordon N, Clapp T. A Tutorial on Particle Filters for Online Non-linear/Non-Gaussian Bayesian Tracking. IEEE Transactions on Signal Processing. 2002;50(2):174-188.
- Briggs AH, Claxton K, Sculpher MJ. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
Related Concepts (3)
Library
Publications
1
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →
Frequently Asked Questions (6)
What is a particle filter?
A Monte Carlo technique estimating a dynamic system's state over time using weighted sample points, called particles, updated as new data arrive.
Source: Gordon, Salmond & Smith 1993
How does a particle filter represent a system's state?
A particle filter represents what is known about a system's state by a cloud of many sample points, each a possible state carrying a weight reflecting how well it fits the data so far. As new measurements arrive, the weights are updated and the points resampled, so the cloud concentrates where the true state is most likely. Because it uses a spread of samples rather than a single formula, it can handle systems whose behaviour is not neatly linear. Doucet and colleagues (2001) describe the method.
Source: Doucet et al. 2001
How does a particle filter work?
A particle filter works by representing the distribution of the system's state with many weighted particles, each a candidate state. At each step, the particles are propagated forward using the system's dynamics, then weighted according to how well they agree with the new observation, and resampled to concentrate on the more probable states. This predict, weight, and resample cycle updates the particle set as data arrive, so the weighted particles approximate the evolving distribution of the state over time.
Source: Gordon, Salmond & Smith 1993
How does a particle filter differ from a Kalman filter?
A particle filter and a Kalman filter both estimate a system's state over time from noisy observations, but they differ in scope. The Kalman filter is exact and efficient for linear systems with Gaussian noise, representing the state by a mean and covariance. The particle filter uses many weighted samples to approximate the state distribution, so it can handle non-linear dynamics and non-Gaussian noise that the Kalman filter cannot, at greater computational cost. Particle filters thus extend sequential estimation to more general systems.
Source: Kalman 1960
When is a particle filter used?
A particle filter is used for estimating the state of dynamic systems that are non-linear or have non-Gaussian noise, where the Kalman filter's assumptions do not hold, and where the state distribution may be complex or multimodal. It suits tracking and estimation problems in which observations arrive sequentially and the system evolves in ways not well approximated by linear-Gaussian models. Its flexibility makes it applicable across fields, including estimating the evolving state of epidemics or other systems from imperfect, sequential data.
Source: Gordon, Salmond & Smith 1993
What are the limitations of particle filters?
Particle filters are computationally intensive, requiring many particles to represent the state distribution well, and this demand grows with the dimension of the state, limiting their use for very high-dimensional problems. They can also suffer particle degeneracy, where few particles carry most of the weight, reducing accuracy unless resampling and enough particles are used. Their estimates carry Monte Carlo error. These limitations mean particle filters trade generality for computational cost, so they are applied where their flexibility justifies the effort.
Source: Gordon, Salmond & Smith 1993
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 10 Oct 2025
Content version: 1.0.0
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- Term code
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