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  Where $G_{i}$ is the {\em observation matrix} which in our case will simply represent whether or not the value of the light curve was observed, and $W_{i}$ is a random zero mean vector following a set of known covariance matricies, $R_{i}$, representing the observation error.  The operation of the Kalman filter can be broken down into a prediction step and an update step. The prediction step involves finding the best linear predictor of our system, $\^{X}$, which is where we think our system should be according to our CARMA model (how is this done?). The update step compares our prediction to our true value and attempts to correct for the differences in order to find the best linear estimator, $X^{\dagger}$ for the state of the system (how is this done). (talk about how the likelihood is then calculated) Estimation of these vectors, and their corresponding covariance matricies is accomplished through the following recursive algorithm:  AGN light curves only give us a picture of the effects that the underlying physical processes have on the AGN itself, but they don't directly tell us what those physical processes are or how they evolve. Futher more, observation error makes it even more difficult to determine their structure. To understand these processes, we need a method of inspecting the state of the system based on our external observations that can also take into account our observation error. The Kalman filter is a digital filter that attempts to reduce observation error and intrinsic system noise by making predictions about future values of the time-series based on an assumed model using Markov chains. Using a Kalman filter together with our CARMA model, we can accurately model theses underlying physical processes and begin analyzing the true structure of the driving system. In order to effectively use a Kalman filter, we must transform our observations, $y(t)$, into state space, $x(t)$.  $$ y(t) = b x(t) + \epsilon(t)$$ %AGN light curves only give us a picture of the effects that the underlying physical processes have on the AGN itself, but they don't directly tell us what those physical processes are or how they evolve. Futher more, observation error makes it even more difficult to determine their structure. To understand these processes, we need a method of inspecting the state of the system based on our external observations that can also take into account our observation error. The Kalman filter is a digital filter that attempts to reduce observation error and intrinsic system noise by making predictions about future values of the time-series based on an assumed model using Markov chains. Using a Kalman filter together with our CARMA model, we can accurately model theses underlying physical processes and begin analyzing the true structure of the driving system. In order to effectively use a Kalman filter, we must transform our observations, $y(t)$, into state space, $x(t)$.  Where %$$ y(t) = b x(t) + \epsilon(t)$$  %Where  $\epsilon(t)$ is a zero mean normally distributed random variable with variance $\sigma^{2}$. %prediction  %correction  %kalman gain  The %The  Kalman filter can be used to estimate the likelihood of a proposed CARMA model by calculating the residuals of the CARMA model with the input time-series. %need more research on kfilter  Two %Two  factors contribute to the need for the Kalman filter. The intrinsic system error in the next step of the process (since Kfilter is markov) and the observation noise. \section{Fitting CARMA Models}