Particle Filter

Particle Filter

A particle filter is a sequential Monte Carlo method for estimating an evolving, unobserved state as new observations arrive. It represents the filtering distribution with many simulated state values, called particles, and weights them according to how well each explains the latest data. It is particularly useful when dynamics or observations are nonlinear or non-Gaussian, provided a credible state-space model and computational resources are available.

State-space problem

Let $X_t$ be a latent state at time $t$ and $Y_t$ the observation. A basic state-space model supplies an initial distribution $p(x_0)$, transition density $p(x_t\mid x_{t-1})$, and observation likelihood $p(y_t\mid x_t)$. The usual filtering assumption is that the next state depends on the past through the preceding state, and the current observation depends on the history through the current state, conditional on any stated model inputs. These are modelling assumptions, not properties supplied by the algorithm.

The goal is $p(x_t\mid y_{1:t})$, the distribution of the current state given observations through $t$. Bayesian filtering alternates prediction and updating:

$$p(x_t\mid y_{1:t-1})=\int p(x_t\mid x_{t-1})p(x_{t-1}\mid y_{1:t-1})\,dx_{t-1},$$

$$p(x_t\mid y_{1:t})\propto p(y_t\mid x_t)p(x_t\mid y_{1:t-1}).$$

The first step propagates uncertainty under the transition model; the second rewards states that make the new observation more likely. A particle filter approximates these distributions numerically. Filtering uses observations available now; smoothing, which estimates an earlier state using later observations, is a distinct task and generally needs a suitable backward or trajectory method.

Bootstrap particle filter step by step

Represent the filtered distribution at $t-1$ by particles $x_{t-1}^{(i)}$ and nonnegative weights $w_{t-1}^{(i)}$ summing to one, for $i=1,\ldots,N$. The bootstrap filter draws each proposed next state from the transition model, evaluates the observation likelihood, normalises weights, and resamples when appropriate.

1. Initialise: Draw $N$ states from the initial distribution, or start from a justified approximation to the initial filtering distribution, with weights $1/N$.
2. Propagate: Draw $x_t^{(i)}\sim p(x_t\mid x_{t-1}^{(i)})$ from the transition model. If the previous particles had nonuniform weights and no resampling was done, retain those weights in the next weighting step.
3. Weight: Compute unnormalised $\widetilde w_t^{(i)}=w_{t-1}^{(i)}p(y_t\mid x_t^{(i)})$ when propagating without prior resampling, or $\widetilde w_t^{(i)}\propto p(y_t\mid x_t^{(i)})$ after resampling to equal weights. Set $w_t^{(i)}=\widetilde w_t^{(i)}/\sum_j\widetilde w_t^{(j)}$.
4. Assess and resample: Calculate effective sample size $N_{\mathrm{eff}}=1/\sum_i(w_t^{(i)})^2$. If it falls below a chosen threshold, sample particle indices with probabilities $w_t^{(i)}$ and assign the descendants equal weights $1/N$.
5. Summarise: Before or after a correctly performed resampling step, approximate a state expectation by $\sum_iw_t^{(i)}h(x_t^{(i)})$ using the corresponding particles and weights. Repeat as observations arrive.

Resampling removes tiny-weight particles and concentrates effort on plausible states, but duplicates ancestors and introduces additional Monte Carlo randomness. “Effective sample size” is a weight-degeneracy diagnostic, not the number of independent patients or observations. Systematic, stratified and residual resampling are alternatives to multinomial resampling; each must preserve the intended sampling distribution. If every likelihood is numerically zero, check model-data conflict and underflow instead of dividing by zero; log weights and stable normalisation can help computationally.

Worked update with three particles

Imagine a deliberately small teaching model in which a latent probability of an event is represented before a new binary observation by particles $0.10$, $0.30$ and $0.60$, each with weight $1/3$. The observation is $Y_t=1$, and the Bernoulli likelihood for that observation at latent value $x$ is $p(Y_t=1\mid X_t=x)=x$. Assume these particles have already completed the prediction step. Their likelihoods are $0.10$, $0.30$ and $0.60$, so normalising the equally weighted likelihoods gives updated weights $0.10$, $0.30$ and $0.60$.

The filtered mean estimate is $0.01+0.09+0.36=0.46$, compared with the equally weighted predicted mean $1/3\approx0.3333$. The weight diagnostic is $N_{\mathrm{eff}}=1/(0.10^2+0.30^2+0.60^2)=1/0.46\approx2.17$ out of three particles. A hypothetical multinomial resampling draw could produce $0.30,0.60,0.60$ with weights $1/3$ each; its mean would be $0.50$ in this particular random draw. That variation illustrates why a tiny particle count is unsuitable for precise inference. The exact numerical path depends on random draws, while the pre-resampling weighted mean of these three displayed particles is 0.46.

Where it helps in health research

In an epidemic or health service monitoring model, latent states might include currently infectious people, unobserved demand, or an evolving risk rate; observations might be reported cases or hospital activity. Transition dynamics describe how states evolve, and an observation model accounts for reporting, measurement or sampling. A filter can update the distribution of those states as reports arrive, and the resulting forecast can inform planning of staffing, beds or supplies. These are possible applications, not evidence that a given model is calibrated or fit for any particular jurisdiction.

In health economic decision modelling, sequential estimates can feed a downstream resource or outcome model when decisions are revised over time. Carry forward uncertainty in the latent states and in estimated parameters rather than treating one filtered mean as a known input. A particle filter estimates states conditional on the specified model and parameters; parameter learning, model comparison and policy evaluation require additional methods or an augmented state with careful attention to degeneracy.

Design choices, failure modes and validation

The bootstrap proposal uses the transition distribution and can waste particles when a highly informative observation makes most propagated states unlikely. Better proposals can incorporate the current observation; sequential Monte Carlo methods then need the correct importance-weight ratio rather than simply the observation likelihood. Particle impoverishment can also occur when repeated resampling leaves too few distinct lineages, especially with nearly deterministic transitions. Higher-dimensional latent states often require many particles or specialised methods.

Check the transition and observation models, temporal alignment, missing-observation rule, parameter uncertainty and measurement process. Track effective sample size, estimated log likelihood where applicable, Monte Carlo variation over repeated random seeds, and sensitivity to particle count and resampling thresholds. Validate on simulated data with known latent states, and compare with a tractable special case or independent implementation where possible. Filtering estimates are model-conditional; a narrow-looking particle distribution is no safeguard against a misspecified likelihood or biased reporting.

A Kalman filter computes an exact recursive solution under its linear-Gaussian assumptions, whereas particle filtering approximates more general distributions and may preserve skewness or multiple modes. This broader flexibility has computational and Monte Carlo costs; select the method for the model and required accuracy. A particle filter is not an ordinary random sample of patients, and resampling particles is not the same as bootstrap resampling of a clinical dataset.

Sources and reporting

Gordon, Salmond and Smith introduced the bootstrap filter for nonlinear and non-Gaussian Bayesian state estimation. Doucet and Johansen's particle filtering and smoothing tutorial develops the sequential Monte Carlo framework and practical variants. A transparent applied report should specify the state and observation models, likelihood, proposal distribution, particle count, resampling method and threshold, random-seed policy, diagnostics and validation evidence, so that the meaning and stability of the estimated state can be assessed.