So, what have we learned from this investigation? Our goal here was to give you a sense about how coarse-graining of data leads to transformations in the model and in fact we saw a couple things. One is is that when you simplify the data, when you coarse-grain you don't necessarily get a simpler model. Sometimes the relationship between the model and the transformed model once we do the coarse-graining can be somewhat hard to see. If you remember when we took that three-state Markov chain that we began with, when we coarse-grain the data and ask what the best fit model would be it actually had more non-zero transitions, for example. One of the big things that you learned or encountered for the first time was the idea of fixed points. A fixed point is a model such that when you coarse-grain the data and you ask how the model transformed you get back the original model. If you simplify the data, the model doesn't change. It kind of has a fractal feel to it. And in fact in the case of the Markov chains we found a continuum of fixed points. All of the Markov chains where they all go to the same states with the same probabilities, such that the input probability to state Q for example is equal for the transition from all the other states. Markov chains of that particular form that lie on that lower dimensional manifold in the space of all possible Markov Chain models with that number of states, those models act as fixed points and not only do they act as fixed points, they actually act as attractive fixed points, meaning as continue to coarse-grain the data and continue to transform the model, yeah, it kind of ??? for a while, maybe, but at the end it's going to end up somewhere on this lower-dimensional plane in the case of the two-state model that lower-dimensional plane is actually just a line. The final thing we got actually was a little fun point, which is that it's hard to take a square root of a stochastic matrix. But what that means for us is that sometimes you have a model and you say what fine-grained theory could this have coarse grained from. Where does that model come from what's the more fine-grained, the more detailed story that I could use to describe or to explain where the coarse-grained model came from. And if we go all the way back to the beginning of this module we talked about the relationship between microeconomics and macroeconomics In the macroeconomic case, in this case here when we ask what the square root of the T is what we're saying is what's the micro economic story that corresponds to the macroeconomic pattern? In that case the micro story is not just a different model, but an entirely different model class. You have to give that model greater sophistication then you had in the coarse-grained version. That's a wrinkle that we'll see over and over again the fact that when you do this sometimes you stay within the same model space. But other times things can go, well, not necessarily wrong but things can get interesting. And in fact in the next modules what we're going to see is the opposite case. what you're going to do is you're going to take some data, coarse-grain it, and the new model you're going to get out the other side is going to be a different class, a more complicated, richer class, than the one you began with. Here when we went from A to T, we simplified the world. It turned out by coarse graining we could forget things, we could forget details. The story we had to tell about the system got simpler. In other cases when we coarse-grain what we're going to find is that we have to enlarge our model space, that we have to make our models more sophisticated. In those cases, as we see, simplifying the data makes the model harder to do.