Hidden Markov Models Fundamentals
نویسنده
چکیده
How can we apply machine learning to data that is represented as a sequence of observations over time? For instance, we might be interested in discovering the sequence of words that someone spoke based on an audio recording of their speech. Or we might be interested in annotating a sequence of words with their part-of-speech tags. These notes provides a thorough mathematical introduction to the concept of Markov Models a formalism for reasoning about states over time and Hidden Markov Models where we wish to recover a series of states from a series of observations. The nal section includes some pointers to resources that present this material from other perspectives. 1 Markov Models Given a set of states S = {s1, s2, ...s|S|} we can observe a series over time ~z ∈ S . For example, we might have the states from a weather system S = {sun, cloud, rain} with |S| = 3 and observe the weather over a few days {z1 = ssun, z2 = scloud, z3 = scloud, z4 = srain, z5 = scloud} with T = 5. The observed states of our weather example represent the output of a random process over time. Without some further assumptions, state sj at time t could be a function of any number of variables, including all the states from times 1 to t− 1 and possibly many others that we don't even model. However, we will make two Markov assumptions that will allow us to tractably reason about time series. The limited horizon assumption is that the probability of being in a state at time t depends only on the state at time t−1. The intuition underlying this assumption is that the state at time t represents enough summary of the past to reasonably predict the future. Formally: P (zt|zt−1, zt−2, ..., z1) = P (zt|zt−1) The stationary process assumption is that the conditional distribution over next state given current state does not change over time. Formally:
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