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* The future is independent of the past given the present
* if know state of the world right now, then knowing state of the world in the past is not going to help you predict the future
* There's clearly some dependency on the points that are nearby in time, can't call them iid's.
* If everything is dependent, totally intractable problem
* For the most accurate prediction of what's gonna happen in the near future is what's happening right now.
* If looking xn + 1, just look at more recent data, don't look at data from distant past
* Recent past tells you more than distant past.
* "A Markov chain makes a very strong assumption that if we want to predict the future in the sequence,all that matters is the current state." - Jurafsky
* uses: temporal data, or some sequence of data. weather, economic, language, speech recognition, automatically generated music
* fill in the blank language: what is the word at the end of this ___________?
==Definitions==
* sequential Sequential data <math>D=(x_1, ..., x_n)</math>* take random Random variables X1, .. ,Xn * Xt <math>X_t</math> depends on Xt<math>X_{t-1}, XtX_{t-2}, ... , XtX{t-m }</math> (fixed m)** Simplest case: m = 1* simplifying Simplifying assumptions: discrete time (duh) and discrete space, i.e., xi is discrete variable that happens at discrete times* discrete r.v.s Discrete random variables X1, ..., xn form a discrete time Markov Chain * i.e., joint Joint distribution <math>p(xtx_t|x1x_1, ..., xtx_{t-1}) = p(xtx_t|xtx_{t-1})</math>* AND Ergo <math>p(x1x_1,...,xnx_n) = p(x1x_1)*p(x2x_2|x1x_1)*p(x3x_3|x2x_2)*...*p(xnx_n|xnx_n-1)</math>
===different types (generalizations)===
* can also have second order markov chain where m = 2
