This chapter considers an alternate method of calculating likelihoods using state space representations and the Kalman filter. The power of the Kalman filter approach to calculating likelihoods is that once a problem can be formulated in state space form, it is possible to calculate the likelihood recursively without using large matrices. The innovation is the important concept in the derivation of the Kalman filter. It is this concept that allows us to calculate likelihoods. The innovation vector contains the new information in the current observations that was unpredictable from the past, and the innovations are uncorrelated with, or orthogonal to, the past observations. If observations are equally spaced with some of the observations missing, there is a simple modification to the recursion. The recursion can be started by specifying s(t1|0) and P(t1|0), where t1 is the first observation time, and entering the recursion at the third step.