Bayesian Filtering Overview
Bayesian filtering is the common framework behind Kalman, ensemble, and particle methods: recursively updating a belief with each observation.
One framework, many algorithms
The Kalman, ensemble, and particle filters look different but are all instances of one idea: recursive Bayesian estimation. The filter maintains a probability distribution over the hidden state, its belief, and updates that belief every time an observation arrives, using Bayes' rule. The algorithms differ only in how they represent and propagate the distribution.
Predict and correct in Bayesian terms
The predict step uses the system dynamics to spread the belief forward in time, a convolution with the transition model. The correct step multiplies the predicted belief by the likelihood of the new observation and renormalizes. This is exactly Bayes' rule with the forecast as the prior and the measurement likelihood as the evidence.
How the methods specialize it
- Kalman filter: belief is Gaussian, dynamics linear, so the update is a closed-form formula
- Extended and unscented filters: keep the Gaussian belief but handle nonlinear dynamics approximately
- Ensemble filter: represent the Gaussian belief with samples for high dimensions
- Particle filter: represent an arbitrary belief with weighted samples
Why the framework helps
Seeing all filters as approximations of the same Bayesian recursion makes the trade-offs clear. Choosing a filter is choosing which approximation the problem tolerates: Gaussian or not, differentiable or not, low or high dimension. It also clarifies what every filter delivers, a full distribution rather than a single number, which is what allows a twin to report calibrated uncertainty.
In a fusion twin
A Kronos twin uses different specializations for different subsystems, but all share this Bayesian backbone, so their uncertainty estimates combine coherently. The framework also connects to offline calibration, where the same Bayesian logic estimates fixed model parameters rather than time-varying state. See calibration and parameter estimation and data assimilation.