Bayesian Filtering and Belief States › Exact Filtering: The Kalman Filter

The Bayes filter is exact but involves integrals over arbitrary distributions, which is why it is rarely computable as written. Rudolf Kálmán's 1960 insight was that an entire class of problems admits a closed-form solution: if the transition and observation models are linear, the process and observation noise are Gaussian, and the initial belief is Gaussian, the belief stays Gaussian forever. The recursion then reduces to updating a mean vector and a covariance matrix. This is a remarkable compression of the general problem. Instead of tracking a function over an infinite-dimensional space of distributions, the filter tracks finitely many numbers. Those numbers update through matrix operations that were feasible on 1960s hardware, although numerical stability required careful implementations. The filter was used in Apollo-era navigation within a decade of its publication.

Take a linear transition with additive Gaussian noise,

$$x_t = A x_{t-1} + B u_{t-1} + w_t, \qquad w_t \sim \mathcal{N}(0, Q),$$

and a linear observation with Gaussian noise,

$$o_t = H x_t + v_t, \qquad v_t \sim \mathcal{N}(0, R).$$

Assume that the initial belief, the full process-noise sequence, and the full observation-noise sequence are mutually independent, with fresh noise draws independent across time. If $b_{t-1}$ is Gaussian, then the prediction $\bar{b}_t$ is a linear transformation of a Gaussian plus independent Gaussian noise, hence Gaussian, with mean and covariance propagated by matrix algebra. The correction step multiplies the predicted Gaussian density by the likelihood $p(o_t \mid x_t)$ and normalizes the result. The product is proportional to a Gaussian. For a direct scalar observation its precision is the sum of the two scalar precisions; in the general linear case the likelihood contributes $H^\top R^{-1}H$ to state-space precision. This closure property does all the work. No integrals remain, only arithmetic on means and covariances.

It is worth spelling out the closure argument once, slowly, because the same pattern recurs whenever someone claims a filter is "exact." A distribution family is closed under an operation if applying the operation to members of the family yields another member. Linear maps take Gaussians to Gaussians, sums of independent Gaussians are Gaussian, and products of Gaussian densities are proportional to Gaussians, exactly Gaussian once normalized, which the correction step does. Since prediction uses linear maps plus sums, and correction uses normalized products, both steps stay inside the family. Nonlinearity breaks the first link, and non-Gaussianity breaks the second and third. The classical approximations below repair one of those broken links; the learned update later replaces the explicit recursion with a trained recurrent computation.

Start from an initial Gaussian belief $\mathcal{N}(\hat{x}_0, P_0)$. The mean $\hat{x}_0$ and covariance $P_0$ are part of the filter specification: they encode what is known before the first measurement.

Predict:

$$\hat{x}_{t|t-1} = A \hat{x}_{t-1} + B u_{t-1},$$ $$P_{t|t-1} = A P_{t-1} A^\top + Q.$$

Correct:

$$K_t = P_{t|t-1} H^\top \left(H P_{t|t-1} H^\top + R\right)^{-1},$$ $$\hat{x}_t = \hat{x}_{t|t-1} + K_t \left(o_t - H \hat{x}_{t|t-1}\right),$$ $$P_t = \left(I - K_t H\right) P_{t|t-1}.$$

Here $\hat{x}_t$ and $P_t$ are the posterior mean and covariance after absorbing the measurement; the $t\mid t-1$ subscript marks predicted values computed before it arrives. And $o_t - H\hat{x}_{t|t-1}$ is the innovation: the part of the measurement the model did not anticipate. Read the equations as a conversation between model and data. The predict step speaks first, asserting where the state should be and how confident the assertion is. The measurement replies, and the innovation records the disagreement. The gain decides how much the disagreement matters, and the final update moves the mean partway toward resolving it while shrinking the covariance to reflect the information gained. Every symbol plays a role in this dialogue, and none of them is decoration.

Everything interesting hides in the Kalman gain $K_t$. Look at its structure. The denominator contains $H P_{t|t-1} H^\top$, the predicted uncertainty of the measurement, plus $R$, the sensor noise. The numerator $P_{t|t-1}H^\top$ is the cross-covariance between the state and measurement. So the gain is essentially a ratio:

$$K_t \;\sim\; \frac{\text{uncertainty in the prediction}}{\text{uncertainty in the prediction} + \text{uncertainty in the sensor}}.$$

When the model is confident ($P$ small) or the sensor is poor ($R$ large), the gain shrinks and corrections barely move the estimate. When the model is uncertain or the sensor precise, the gain gives more weight to the measurement. The filter continuously re-weights trust between its own model and incoming evidence, and it does so optimally in the mean-squared-error sense for this problem class. This trust-balancing interpretation is the single most transferable idea in the chapter: every estimator, learned or analytic, faces the same question of how much to believe the internal model versus the latest evidence, and the Kalman gain is the optimal answer in the linear-Gaussian setting treated here.

For the scalar observation case plotted below, with $H=1$, the gain reduces to $K=p/(p+R)$, where $p$ is the predicted variance. In multiple dimensions $K=P H^\top(H P H^\top+R)^{-1}$ is a matrix, so neither its entries nor its eigenvalues generally follow the scalar curve.