The Kalman filter estimates the hidden state of a linear dynamical system by alternating between prediction from the state model and correction from a noisy measurement.

Assume the model

$$\mathbf x_k=A\mathbf x_{k-1}+B\mathbf u_k+\mathbf w_k,$$

$$\mathbf z_k=H\mathbf x_k+\mathbf v_k,$$

with Gaussian process and measurement noise.

Predict

From the previous estimate,

$$\hat{\mathbf x}{k|k-1}=A\hat{\mathbf x}{k-1|k-1}+B\mathbf u_k,$$

and the covariance is propagated as

$$P_{k|k-1}=AP_{k-1|k-1}A^T+Q.$$

The prediction generally becomes more uncertain because process noise $Q$ is added.

Compare with the measurement

The innovation is

$$\mathbf y_k=\mathbf z_k-H\hat{\mathbf x}_{k|k-1}.$$

It measures the disagreement between what the model predicted and what the sensor observed.

Correct

The Kalman gain is

$$K_k=P_{k|k-1}H^T(HP_{k|k-1}H^T+R)^{-1}.$$

The state estimate becomes

$$\hat{\mathbf x}{k|k}=\hat{\mathbf x}{k|k-1}+K_k\mathbf y_k.$$

When measurement uncertainty $R$ is small relative to prediction uncertainty, the update trusts the measurement more strongly; when measurements are noisy, it relies more heavily on the model.

The filter is not merely smoothing data. It propagates both a state estimate and a quantified uncertainty, then combines model and measurement according to their relative uncertainty.