 What I want to introduce in this lecture is the idea of the forecast analysis cycle. The forecast analysis cycle is something that's key to any time we want to do iterative forecasting, and it's based on the idea that when we make forecasts, when we then have new information that becomes available, we want to be able to update those forecasts iteratively. So we'll be making a forecast, we'll get some new observations, we want to update and then forecast again. To do this, we're going to set up two steps in the forecast analysis cycle. First, the forecast step where we actually make a projection in the future, and the analysis step which is where we take that forecast, combine it with new information to update our understanding of the state of the system and then use that updated state to forecast again. The key idea of the forecast analysis cycle is that we're going to leverage Bayes' Theorem to do this. Bayes' Theorem is an inherently iterative approach to inference. It starts with a prior, which is our current understanding of the system. It multiplies that by the likelihood, the probability of seeing the data that we saw in the real world, given our model, and then it gives us the posterior, our updated understanding of the state of the system given data. So in a forecast, unlike most typical applications of Bayes' Theorem, we put a lot of emphasis on that prior and specifically we're leveraging the idea that prior to observing the future state of the system, that our forecast is our best estimate of that state, that the forecast combines all of our understanding of the process, all of our previous synthesis, all of the data that we've previously assimilated are all baked into that forecast. And so because all of our understanding is in that forecast and it represents our best understanding of the system, we use that as an informative prior during the forecast analysis cycle. So a key part of this cycle is to translate that forecast to the formal prior in a Bayesian sense and then we can use the likelihood of the data to update that prior. There are many alternative ways to implement this forecast analysis cycle and it is very flexible. Most of the existing examples you'll find in the literature and the ones that we're going to cover in the next few videos essentially map onto each of the different approaches to uncertainty propagation we covered previously in the uncertainty propagation series of lectures. So we're going to talk about the classic Kalman filter which maps onto using an analytical moment approach to transforming uncertainties from the present into the future. So the forecast is done using an analytical approximation to the mean invariance. We'll then talk about the extended Kalman filter where we then use the Taylor series approach to propagate uncertainties. Then the ensemble Kalman filter where we use the ensemble approach to uncertainty propagation in the forecast step and then the particle filter where we use the Monte Carlo approach to uncertainty propagation. So you can see that we can map each of our approaches to data assimilation onto the different approaches that we use for uncertainty propagation in the forecast. On the analysis step we're going to see that a lot of the classic literature focused on analytically tractable solutions to the analysis step using conjugate likelihoods and conjugate priors that have analytical solutions but that in the general sense we're not limited to that that we can use any of our Bayesian tools such as MCMC to perform the analysis step by combining the forecast prior with the likelihood of new data. The final thing I want to emphasize about the forecast analysis cycle is that it is all just a special case of the state space model that we covered earlier. One important difference between the state space model and the forecast analysis cycle is that in the state space model our estimates of the state are informed both by information that came in the past and information that comes in the future. While in the forecast analysis cycle we've optimized that to work in a way that only looks directionally forward in time so when we're making a forecast we don't have observations of the future at the time that we're estimating the current state so we're always working just from the past forward.