Concept Architecture
Concept
Theoretically, Sequential Monte Carlo is a class of simulation methods for approximating a sequence of probability distributions using weighted random samples known as particles. The method updates the particle population as new data or model stages become available, combining importance sampling with resampling to control weight degeneracy. In health economics, sequential Monte Carlo is used in dynamic Bayesian models, state-space models, infectious disease transmission models and other settings in which latent states or parameters must be estimated recursively.
Mathematically, sequential Monte Carlo represents each target distribution �?(x?:?) by an empirical distribution formed from particles x?:???? and normalised weights W????. Particles are propagated through a proposal distribution, reweighted according to the ratio of the target and proposal densities, and resampled when the concentration of weights becomes excessive. The resulting weighted sample is used to approximate expectations, marginal distributions and other quantities associated with the evolving target distribution.
In practice, sequential Monte Carlo is implemented by selecting an initial particle distribution, propagating particles through the model, calculating and normalising importance weights, assessing weight degeneracy and resampling when required. Health economic applications include estimating unobserved epidemic states, calibrating dynamic transmission models, updating disease progression estimates as surveillance data accrue and propagating posterior uncertainty into cost-effectiveness analyses.
Purpose
Used to approximate evolving probability distributions, estimate latent states and parameters sequentially, and update dynamic health economic models as new evidence becomes available.
Mathematical Formulae
Primary Formula
The sequential importance weight is
w???? = w?????? ? �?(x?:????) / [�???(x?:??????) q?(x???? | x?:??????)]
where:
- w???? = unnormalised weight of particle i at time t
- �?(x?:?) = target distribution at time t
- q?(x? | x?:???) = proposal distribution
- x?:???? = trajectory represented by particle i
Supporting Formulae
Normalised particle weight:
W???? = w???? / �???? w????
Empirical approximation:
�?(x?:?) � �???? W???? �??:????(x?:?)
Monte Carlo expectation:
E?�??[f(X)] � �???? W???? f(x?:????)
Effective sample size:
N?ff = 1 / �???? (W????)�
Related Mathematical Methods
- Importance sampling
- Sequential importance sampling
- Particle filtering
- Resampling
- Bayesian filtering
- State-space modelling
- Markov Chain Monte Carlo
Example
A dynamic influenza transmission model uses 20,000 particles to estimate the unobserved number of infectious individuals each week. After the particles are propagated through the transmission model, their likelihoods are calculated using newly reported hospital admissions.
Suppose the normalised weights for four illustrative particles are 0.10, 0.20, 0.30 and 0.40, with corresponding predicted weekly hospital costs of �80,000, �90,000, �110,000 and �130,000. The posterior expected cost is
E(C) = (0.10 ? �80,000) + (0.20 ? �90,000) + (0.30 ? �110,000) + (0.40 ? �130,000)
E(C) = �111,000
The weighted particle population is then resampled if the effective sample size falls below the modelled threshold, and the updated state estimates are used to project costs and QALYs under alternative vaccination strategies.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =SUM(C2:C20001) | Calculate the total unnormalised particle weight |
| Division | =C2/SUM($C$2:$C$20001) | Normalise each particle weight |
| SUMPRODUCT | =SUMPRODUCT(B2:B20001,D2:D20001) | Calculate a weighted posterior estimate of costs, cases or QALYs |
| SUMSQ | =1/SUMSQ(D2:D20001) | Calculate the effective sample size from normalised weights |
| RAND | =RAND() | Generate random numbers for particle propagation or resampling |
| XLOOKUP | =XLOOKUP(F2,$E$2:$E$20001,$A$2:$A$20001,,1) | Select a particle using cumulative resampling probabilities |
VBA (Optional)
Automate particle propagation, weight normalisation, effective sample size assessment, resampling and export of posterior model estimates.
Sources
- Doucet A, de Freitas N, Gordon NJ, eds. Sequential Monte Carlo Methods in Practice. Springer; 2001.
- Del Moral P. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer; 2004.
- Capp� O, Moulines E, Ryd�n T. Inference in Hidden Markov Models. Springer; 2005.
- Arulampalam MS, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for online nonlinear/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.
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 sequential Monte Carlo?
A class of algorithms, including particle filtering, estimating a sequence of unknown quantities over time by updating weighted samples as new data arrive.
Source: Gordon, Salmond & Smith 1993
How does sequential Monte Carlo update its estimate over time?
Sequential Monte Carlo represents the current estimate of an unknown quantity by a set of weighted samples and revises them each time new data arrive, rather than reworking the whole calculation from scratch. As each observation comes in, the samples are reweighted to reflect how well they fit it, and periodically resampled to concentrate on the more plausible values. This step-by-step updating suits problems where data accumulate over time. Particle filtering is the best-known instance. Doucet and colleagues (2001) describe the family.
Source: Doucet et al. 2001
How does sequential Monte Carlo work?
Sequential Monte Carlo works by representing the distribution of the unknown quantities with weighted samples, then, as each new observation arrives, propagating the samples according to the system's dynamics, reweighting them by how well they match the observation, and resampling to focus on the more probable samples. This process repeats over the sequence of observations, so the weighted samples track the changing distribution over time. The method builds up the estimate sequentially, updating rather than recomputing as data accumulate.
Source: Gordon, Salmond & Smith 1993
How does sequential Monte Carlo relate to particle filtering?
Particle filtering is the best-known form of sequential Monte Carlo, applied to estimating the state of a dynamic system over time. Sequential Monte Carlo is the broader class of algorithms that estimate sequences of quantities by updating weighted samples, of which particle filters are the application to state-space models. The terms are often used together, with sequential Monte Carlo naming the general methodology and particle filtering its application to filtering problems, both sharing the propagate, weight, and resample structure.
Source: Gordon, Salmond & Smith 1993
When is sequential Monte Carlo used?
Sequential Monte Carlo is used for problems where quantities must be estimated over time as data arrive, particularly dynamic systems that are non-linear or non-Gaussian, where methods such as the Kalman filter do not apply. It suits tracking, filtering, and sequential inference where the distribution of interest evolves and observations come in sequence. Its ability to handle complex, changing distributions makes it valuable for estimating the state of systems, such as evolving epidemics or signals, from sequential, noisy data.
Source: Metropolis & Ulam 1949
What are the limitations of sequential Monte Carlo?
Sequential Monte Carlo is computationally demanding, needing many samples to represent distributions well, and this cost grows with the dimension of the problem, limiting its use in very high dimensions. It can suffer sample degeneracy, where few samples carry most of the weight, reducing accuracy unless resampling and sufficient samples are used, and its estimates carry Monte Carlo error. These limitations mean sequential Monte Carlo trades generality for computational cost, so it is applied where its flexibility for dynamic, complex problems is needed.
Source: Gordon, Salmond & Smith 1993
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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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