<P> Using these assumptions the probability distribution over all states of the HMM can be written simply as: </P> <Dl> <Dd> p (x 0,..., x k, z 1,..., z k) = p (x 0) ∏ i = 1 k p (z i x i) p (x i x i − 1). (\ displaystyle p ((\ textbf (x)) _ (0), \ dots, (\ textbf (x)) _ (k), (\ textbf (z)) _ (1), \ dots, (\ textbf (z)) _ (k)) = p ((\ textbf (x)) _ (0)) \ prod _ (i = 1) ^ (k) p ((\ textbf (z)) _ (i) (\ textbf (x)) _ (i)) p ((\ textbf (x)) _ (i) (\ textbf (x)) _ (i - 1)).) </Dd> </Dl> <Dd> p (x 0,..., x k, z 1,..., z k) = p (x 0) ∏ i = 1 k p (z i x i) p (x i x i − 1). (\ displaystyle p ((\ textbf (x)) _ (0), \ dots, (\ textbf (x)) _ (k), (\ textbf (z)) _ (1), \ dots, (\ textbf (z)) _ (k)) = p ((\ textbf (x)) _ (0)) \ prod _ (i = 1) ^ (k) p ((\ textbf (z)) _ (i) (\ textbf (x)) _ (i)) p ((\ textbf (x)) _ (i) (\ textbf (x)) _ (i - 1)).) </Dd> <P> 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 .) </P>

Tracking by detection vs tracking by recursive bayesian filtering