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Causal state filtering estimates the current state of a dynamical system from observations up to the current time. Under quadratic loss, the conditional expectation is an optimal estimate. A Kalman filter obtains this estimate recursively for a linear Gaussian state-space model, without using future observations.

The Kalman filter alternates a Kalman prediction step with a Kalman measurement update. It carries a state estimate forward using a state-space model, then adjusts it by the part of the new observation that was not predicted: the innovation. The Kalman gain determines the adjustment from the uncertainty of the prediction and the measurement noise. A noisy measurement should have less influence than an equally surprising precise one.

For a precise discrete-time statement, let the state and observation be vectors in real vector spaces obeying the linear state-space modelThe matrices and control inputs are known. Assume a Gaussian random vector $x_0$ with mean $m_0$ and covariance matrix $P_0$, and mutually independent centered Gaussian noises $w_t,v_t$, independent of $x_0$, with covariance matrices $Q_t,R_t$. Suppose $R_t$ is positive-definite, so the observation innovation covariance is invertible. Then the recursions below give the exact posterior mean $m_t=\mathbb{E}[x_t\mid y_1,\dots ,y_t]$ and posterior covariance matrix $P_t$, hence the full conditional multivariate normal distribution. They also minimize mean squared error. Without the Gaussian assumptions, suitable second-order assumptions still give best affine state estimates, but do not generally give an exact posterior distribution. ^[ref].

For an exact scalar example, suppose the predicted state has normal distribution $N(10,4)$ and a measurement reports $y=14$ in the same units. With measurement noise variance $R=1$, the Kalman gain is $4/(4+1)=4/5$ and the posterior distribution isWith the same prediction and observed value but measurement noise variance $R=16$, the Kalman gain becomes $1/5$ and the posterior distribution is instead $N(10.8,3.2)$. Both updates move toward the observation; the more precise measurement moves the estimate further and reduces uncertainty more.

The useful compression is exact under the stated model: the two objects $m_t,P_t$ contain everything from the past observations needed for the next update. Their size depends on the state dimension, not on the length of the observation history. This makes causal state filtering practical when observations arrive continually. It also gives Linear-quadratic-Gaussian control an estimate of the state on which to act.

Rudolf E. Kalman's 1960 paper organized linear filtering around state transitions, error covariance and recursive computation, connecting estimation with optimal regulation. Its introduction acknowledges earlier work by Wiener, Bode, Shannon and others; the useful historical claim is this particular formulation, rather than that estimation began in 1960. ^[ref].

State estimation infers an unobserved state of a dynamical system from its observations and a model of its evolution. An observed position can help estimate an unobserved velocity; repeated noisy observations can help distinguish changes in the state from measurement noise. The result should describe both an estimate and its uncertainty. A Kalman filter does this exactly for a linear Gaussian state-space model, maintaining a conditional expectation and conditional covariance.

The time at which information is available matters. Causal state filtering estimates the present state from observations available now. State prediction estimates a future state from those same observations. Later observations can improve an estimate of a past state, but that is a different information set from the one a live controller had. Kalman explicitly distinguishes these estimation tasks in A New Approach to Linear Filtering and Prediction Problems (1960), section Optimal Estimates.

State prediction estimates a future state of a dynamical system using observations already available. A state-space model supplies the transition law. Unlike causal state filtering at the next time, the prediction has not yet incorporated that next observation; a Kalman prediction step therefore includes uncertainty from new process noise.