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Recall that the model gives the joint probabilities Pr H Xfor all sequence, it also gives the posterior probability Pr H X = Pr H X / Pr X, for every possible state path H through the model, conditioned on the sequence Xwith maximum posterior probability. The measure of best is to find the path that has the maximum probability in the HMM, given the sequence X. We will adopt a Bayesian perspective, so that we treat θ tas a random variable. The various distribution in which we are interested are p ϕ y 1 …. A crucial component of this model is that the y tare independent given the set of θ tand θonly depends directly on its neighbors θ t − 1and θ t + 1. Our model is then described by the sets of probability distributions p ( y t ∣ θ t, ϕ )and p θ t θ t − 1 ϕ. In our applications, y twill either be an increase or decrease and the hidden process will determine the probability distribution of observing different letters. …, T, θ tthe value of the hidden process at location tand let ϕ represents parameters necessary to determine the probability distribution for y tgiven θ tand θ tgiven θ t − 1. Let y trepresents the observed value of the process at location tfor t = 1. Finally we may want the probability distribution for the hidden states at every location. And also determined the most likely sequence for the hidden process.
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Given such a model, we want to estimate any parameters that occur in the model. Thus a hidden Markov model is specified by the transition density of the Markov chain and the probability laws that govern what we observe given the state of the Markov chain. The fundamental idea behind a hidden Markov model is that there is a Markov process we cannot observe that determines the probability distribution for what we do observe.