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However, when using the Kalman filter to estimate the state '''x''', the probability distribution of interest is associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalising out the previous states and dividing by the probability of the measurement set.)
This leads to the ''predict'' and ''update'' steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (''k'' - 1)-th timestep to the ''k''-th and the probability distribution associated with the previous state, over all possible <math>x_{
:<math> p(\textbf{x}_k|\textbf{z}_{1:k-1}) = \int p(\textbf{x}_k | \textbf{x}_{k-1}) p(\textbf{x}_{k-1} | \textbf{z}_{1:k-1} ) \, d\textbf{x}_{k-1} </math>
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