Recursive Bayes Filter

The Recursive Bayes Filter provides a framework for recursive state estimation. It estimates the state $x$ of a system given observations $z$ and controls $u$:

$$p(x\mid z,u)$$

Definition of the belief

The belief is the posterior distribution of the current state $x_t$ given all observations $z_{1:t}$ and control inputs $u_{1:t}$:

$$\mathrm{bel}(x_t)=p(x_t\mid z_{1:t},u_{1:t})(1)$$

To update the belief based on new measurements and controls we apply Bayes' rule.

Mathematical derivation

Using Bayes' rule, we expand the belief in Eq. (1) as:

$$\begin{array}{rl}\mathrm{bel}(x_t)& =p(x_t\mid z_{1:t},u_{1:t})\\ & =\eta \cdot \underset{\text{likelihood}}{\underbrace{p(z_t\mid x_t,z_{1:t-1},u_{1:t})}}\cdot \underset{\text{prior}}{\underbrace{p(x_t\mid z_{1:t-1},u_{1:t})}}(2)\end{array}$$

where:

• $\eta =\frac{1}{p(z_t\mid z_{1:t-1},u_{1:t})}$ is a constant independent of the current state $x_t$. It is the normalizer (also called the evidence) for the new observation $z_t$, ensuring the posterior integrates to 1.
• $p(x_t\mid z_{1:t-1},u_{1:t})$ is the prior — our predicted belief about the current state before incorporating the latest measurement. It comes from the motion model.
• $p(z_t\mid x_t,z_{1:t-1},u_{1:t})$ is the likelihood, given by the observation model.

Continuing from Eq. (2), we introduce the Markov assumption:

The current measurement $z_t$ is conditionally independent of past measurements and control inputs given the current state $x_t$.

This simplifies to:

$$p(z_t\mid x_t,z_{1:t-1},u_{1:t})=p(z_t\mid x_t)$$

So:

$$\mathrm{bel}(x_t)=\eta \cdot p(z_t\mid x_t)\cdot p(x_t\mid z_{1:t-1},u_{1:t})(3)$$

Applying the law of total probability to Eq. (3):

$$\mathrm{bel}(x_t)=\eta \cdot p(z_t\mid x_t)\cdot {\int }_{x_{t-1}}p(x_t\mid x_{t-1},z_{1:t-1},u_{1:t})p(x_{t-1}\mid z_{1:t-1},u_{1:t})d{x}_{t-1}(4)$$

We now apply two additional Markov assumptions:

1. State transition assumption: $$p(x_t\mid x_{t-1},z_{1:t-1},u_{1:t})=p(x_t\mid x_{t-1},u_t)$$ The current state depends only on the previous state and the latest control input.
2. Control independence assumption: $$p(x_{t-1}\mid z_{1:t-1},u_{1:t})=p(x_{t-1}\mid z_{1:t-1},u_{1:t-1})$$ The state at time $t-1$ does not depend on the future control $u_t$.

With these, the belief simplifies to:

$$\mathrm{bel}(x_t)=\eta \cdot p(z_t\mid x_t)\cdot {\int }_{x_{t-1}}p(x_t\mid x_{t-1},u_t)p(x_{t-1}\mid z_{1:t-1},u_{1:t-1})d{x}_{t-1}(5)$$

The last factor $p(x_{t-1}\mid z_{1:t-1},u_{1:t-1})$ is exactly the previous belief $\mathrm{bel}(x_{t-1})$ — and that's the recursion:

$$\boxed{\mathrm{bel}(x_t)=\eta \cdot p(z_t\mid x_t)\cdot {\int }_{x_{t-1}}p(x_t\mid x_{t-1},u_t)\mathrm{bel}(x_{t-1})d{x}_{t-1}}(6)$$

Prediction and correction

The Recursive Bayes Filter can be viewed as a two-step recursive process.

Prediction step (motion model)

Estimate the new belief $\mathrm{bel}({\bar{x}}_t)$ from the previous belief and the control input:

$$\mathrm{bel}({\bar{x}}_t)=\int p(x_t\mid x_{t-1},u_t)\mathrm{bel}(x_{t-1})d{x}_{t-1}$$

• Prediction based on control commands (motion).
• $\mathrm{bel}({\bar{x}}_t)$ is the predicted belief before incorporating sensor measurements.

Correction step (observation model)

Incorporate the latest sensor measurement $z_t$ to refine the prediction:

$$\mathrm{bel}(x_t)=\eta \cdot p(z_t\mid x_t)\cdot \mathrm{bel}({\bar{x}}_t)$$

• Adjusts the prediction using actual sensor data.

Visualization1-D Bayes / Kalman cycle

Prior — bel(x_{t-1})

Prior μ0.00Prior σ²1.00Control u5.00Motion noise R2.00Measurement z8.00Meas. noise Q4.00

⏸ Pausepriorpredictionmeasurementcorrection

The animation shows the full cycle on a 1-D state: a Gaussian prior is pushed forward by the motion model (covariance grows, mean shifts), the measurement likelihood arrives, and the correction step fuses them into a sharper posterior. The Kalman gain $K$ controls how strongly the measurement pulls the estimate — large $K$ when the prediction is uncertain, small $K$ when the measurement is noisy.

Summary

• The Bayes filter is a general probabilistic framework for recursive state estimation. Two essential steps:
  • Prediction — estimate the current state from a motion model and previous state.
  • Correction — refine the prediction using the latest sensor observations.
• The Bayes filter itself does not specify how to compute the integrals or which distributions to use.
• Practical realizations of the Bayes filter include:
  • Kalman Filter (KF) — linear systems with Gaussian noise.
  • Extended Kalman Filter (EKF) — nonlinear systems using local linear approximations.
  • Particle Filter (PF) — nonlinear systems with non-Gaussian / multimodal beliefs.

Each filter is a specific solution tailored to the system dynamics, sensor models, and noise properties.

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Reading material

On the Bayes filter

• Thrun, Burgard, Fox. Probabilistic Robotics, Chapter 2.