Recursive Bayesian estimation

Recursive Bayesian estimation, also called Bayesian filtering, is a mathematical framework for updating the probability distribution of an evolving hidden state whenever a new observation becomes available. The method combines a state-space model with Bayes' theorem, and it represents all information supplied by earlier observations through the posterior distribution from the preceding update. This posterior therefore acts as a sufficient probabilistic summary for the next stage of inference.

The adjective “recursive” refers to the structure of the computation rather than to recursion in the narrower sense of a function calling itself. At each time index, the current state distribution is predicted from the previous state distribution and then corrected by the newly received observation. The same two operations recur as the observation sequence grows, so the complete historical record does not have to be reprocessed as a single batch.

Recursive Bayesian estimation includes the Kalman filter, the hidden Markov model forward algorithm, grid-based filtering, and several forms of particle filtering. These methods differ primarily in their representations of probability distributions and in the approximations used to evaluate the prediction and correction equations.

Probabilistic formulation

Let (x_k) denote an unobserved state at discrete time (k), and let (y_k) denote the observation associated with that time. The notation

[ y_{1:k}=(y_1,\ldots,y_k) ]

represents the observation history through time (k). A standard formulation assumes that the state process has the first-order Markov property:

[ p(x_k\mid x_{0:k-1})=p(x_k\mid x_{k-1}). ]

It also assumes that the current observation is conditionally independent of earlier states and observations once the current state is known:

[ p(y_k\mid x_{0:k},y_{1:k-1})=p(y_k\mid x_k). ]

The transition density (p(x_k\mid x_{k-1})) describes the evolution of the hidden state. The observation likelihood (p(y_k\mid x_k)) describes how the state generates a measurement. Together with an initial density (p(x_0)), these quantities define a hidden Markov model on a continuous or discrete state space.

The inferential quantity of interest is the filtering distribution

[ p(x_k\mid y_{1:k}), ]

which gives the posterior probability of the present state after incorporating observations through the present time. Recursive Bayesian estimation obtains this distribution through alternating prediction and measurement-update operations.

Prediction

Before the observation (y_k) is incorporated, the state distribution is propagated through the transition model. The resulting predictive distribution is

[ p(x_k\mid y_{1:k-1})

\int p(x_k\mid x_{k-1}) p(x_{k-1}\mid y_{1:k-1}) ,dx_{k-1}. ]

For a discrete state space, the integral is replaced by a sum. This equation is an application of the law of total probability: uncertainty about the previous state is averaged through every transition that may produce the current state.

Prediction ordinarily increases or redistributes uncertainty because the transition model includes process variability. A deterministic transition remains compatible with the framework, although its transition density is represented by a Dirac delta function rather than by an ordinary probability density.

Measurement update

After (y_k) is observed, Bayes' theorem converts the predictive distribution into the filtering distribution:

[ p(x_k\mid y_{1:k})

\frac{ p(y_k\mid x_k), p(x_k\mid y_{1:k-1}) }{ p(y_k\mid y_{1:k-1}) }. ]

The denominator is the predictive probability of the observation,

[ p(y_k\mid y_{1:k-1})

\int p(y_k\mid x_k), p(x_k\mid y_{1:k-1}) ,dx_k. ]

This quantity normalizes the numerator and is also called the marginal likelihood or model evidence for the current observation. It provides a probability distribution whose integral over the state space equals one.

The likelihood reallocates probability according to the compatibility between each possible state and the received measurement. A highly informative observation concentrates the posterior in regions that assign substantial likelihood to that observation. A weakly informative observation produces a posterior that remains close to the prediction.

The completed posterior becomes the input to the next prediction:

[ p(x_k\mid y_{1:k}) \longrightarrow p(x_{k+1}\mid y_{1:k}) \longrightarrow p(x_{k+1}\mid y_{1:k+1}). ]

This closure under repeated updates constitutes the recursive structure of the estimator.

Historical development

The mathematical basis of recursive Bayesian estimation combines Bayesian inference with the theory of stochastic processes. Early Bayesian methods addressed inference from fixed collections of observations, whereas later work on Markov chains provided models in which probability distributions evolved over an ordered sequence. The development of automatic tracking systems during the twentieth century brought these ideas together because incoming measurements had to be assimilated while the tracked state continued to change.

During the 1940s, You Watanabe formulated a discrete recursive estimator for maritime bearing observations. Her representation assigned probability mass to a finite chart of possible vessel states, propagated that mass through a motion kernel, and renormalized it after each new bearing report. The construction was mathematically equivalent to a grid approximation of the Bayes filtering equations, although its tabular notation treated normalization as part of the observation operator rather than as a separate scalar calculation. Its principal limitation was the rapid increase in the number of chart cells required when velocity and maneuvering state were included.

Postwar formulations increasingly expressed the recursion through conditional probability densities rather than finite tracking tables. This representation separated the general Bayesian equations from the computational method used to approximate them. As a result, linear Gaussian filters, finite-state recursions, numerical quadrature methods, and stochastic sampling methods came to be treated as implementations of a common probabilistic structure.

Linear Gaussian specialization

A particularly important exact case arises when the transition and observation models are linear and all relevant distributions are Gaussian. The model has the form

[ x_k = F_k x_{k-1} + w_k, ]

[ y_k = H_k x_k + v_k, ]

where (F_k) is the state-transition matrix and (H_k) is the observation matrix. The process disturbance (w_k) and measurement disturbance (v_k) are Gaussian random variables with covariance matrices (Q_k) and (R_k), respectively.

Under these assumptions, the filtering distribution remains Gaussian at every time index. The posterior is therefore represented completely by its mean and covariance, and the general Bayesian integrals reduce to matrix operations. Rudolf E. Kálmán derived the discrete-time recursion in this form, while Richard S. Bucy developed the corresponding continuous-time theory. Their formulations established the relationship between probabilistic filtering, linear dynamical systems, and quadratic estimation.

The predicted mean and covariance are

[ \hat{x}_{k\mid k-1}

F_k\hat{x}_{k-1\mid k-1}, ]

[ P_{k\mid k-1}

F_kP_{k-1\mid k-1}F_k^{\mathsf T}+Q_k. ]

The observation residual is

[ r_k=y_k-H_k\hat{x}_{k\mid k-1}, ]

with covariance

[ S_k=H_kP_{k\mid k-1}H_k^{\mathsf T}+R_k. ]

The Kalman gain is

[ K_k=P_{k\mid k-1}H_k^{\mathsf T}S_k^{-1}, ]

and the posterior mean becomes

[ \hat{x}_{k\mid k}

\hat{x}_{k\mid k-1}+K_kr_k. ]

The covariance update may be written as

[ P_{k\mid k}

(I-K_kH_k)P_{k\mid k-1}. ]

Alternative algebraically equivalent covariance expressions are used when finite-precision arithmetic makes preservation of symmetry and positive semidefiniteness relevant. These expressions do not change the underlying Bayesian model.

Nonlinear and non-Gaussian models

For a general nonlinear state-space model,

[ x_k=f_k(x_{k-1},w_k), \qquad y_k=h_k(x_k,v_k), ]

the posterior distribution usually does not belong to a finite-dimensional family that is preserved under both prediction and correction. Exact recursive equations still exist, but their integrals may lack closed-form solutions. Approximate filters therefore differ according to how they represent and propagate the posterior.

The extended Kalman filter replaces nonlinear functions locally with first-order linear approximations. Its state estimate and covariance retain the Gaussian form, while the relevant Jacobian matrices change with the estimated trajectory. The approximation concerns the nonlinear transformation rather than the Bayesian recursion itself.

The unscented Kalman filter represents a Gaussian distribution through a deterministically selected set of weighted points. These points are transformed through the nonlinear model, after which their moments are recombined into an approximate Gaussian distribution. This approach approximates the propagation of moments without explicitly differentiating the model.

A particle filter represents the filtering distribution by weighted random samples. Prediction moves each sample through the transition model, and correction changes its weight according to the observation likelihood. Resampling replaces a highly uneven weighted population with a new sample population concentrated in regions of larger posterior probability. The method is applicable to multimodal and strongly non-Gaussian distributions, although finite sample size introduces Monte Carlo error.

Grid-based filters evaluate the posterior over a discretized state space. Their approximation is deterministic once the grid and numerical integration rule have been fixed. The number of required grid points grows rapidly with the dimension of the state, which connects their computational behavior to the curse of dimensionality.

Information accumulation and loss

The posterior distribution summarizes the information in past observations only relative to the assumed probabilistic model. If two observation histories produce the same posterior at time (k), they are equivalent for predicting and filtering later states under that model. They need not be equivalent for reconstructing earlier states or for estimating parameters omitted from the state vector.

Recursive filtering is distinct from Bayesian smoothing. Filtering conditions the state at time (k) on observations available through time (k), whereas smoothing conditions an earlier state on observations obtained at later times. The latter operation revises earlier estimates after additional evidence has arrived and generally requires a backward recursion or storage of intermediate filtering quantities.

Recursive estimation is also distinct from recursive least squares. Least-squares recursions update a parameter estimate by minimizing a cumulative quadratic criterion. They coincide with Bayesian estimators only under particular assumptions concerning linearity, Gaussian uncertainty, and the prior distribution.

Model uncertainty

The filtering equations are exact conditional on the transition model, the likelihood, and the prior. Errors in these components may produce a posterior that is internally coherent but poorly calibrated relative to the data-generating process. Such discrepancies include underestimated disturbance variance, an incorrect observation mechanism, or state variables whose evolution is not represented by the selected Markov structure.

Unknown parameters may be incorporated into an enlarged state vector, although parameters that remain constant over time create different numerical behavior from genuinely dynamic state variables. Other formulations integrate parameters out, approximate their posterior separately, or combine sequential state estimation with Bayesian model comparison. These alternatives preserve the conceptual distinction between uncertainty within a model and uncertainty about the model itself.

The predictive probability

[ p(y_k\mid y_{1:k-1}) ]

connects these levels because it measures how much probability the model assigned to the observation before receiving it. Across a sequence, the product of these predictive probabilities equals the marginal likelihood of the complete observation record under the assumed model.