= Kalman filter
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{title2=$m_t=m_t^-+K_t(y_t-H_tm_t^-)$}
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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 model>
$$
x_t=A_tx_{t-1}+B_tu_{t-1}+w_t,\qquad
y_t=H_tx_t+v_t.
$$
The <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> $m_t=\mathbb E[x_t\mid y_1,\ldots,y_t]$ 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>. https://www.cs.utexas.edu/~pstone/Courses/393Rfall15/readings/Welch%2BBishop-TR-95.pdf[Welch and Bishop, An Introduction to the Kalman Filter, section 1, UNC technical report TR 95-041, 2004 revision]{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> is
$$
x\mid y=14\sim N(13.2,0.8).
$$
With 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.
\Image[/kalman-measurement-uncertainty.png]
{title=The same observed value produces different <Kalman measurement updates> when its uncertainty changes}
{description=Both panels start from $N(10,4)$ and observe $14$. The green curve is the <likelihood function> normalized over the scalar state for this unit-coefficient observation model. The orange curve is the <posterior density>; its narrower width reflects reduced <variance>. The vertical dotted line marks the observed value, rather than a known true state.}
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. https://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/k/Kalman1960.pdf[Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering 82 (1960), pp. 35–45]{ref}.
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