= Hidden Markov model
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A <Hidden Markov model> combines an unobserved <Markov chain> $Z_1,\ldots,Z_T$ with observations $O_1,\ldots,O_T$. Given the whole hidden-state path, the observations have <conditional independence>, and the <conditional distribution> of $O_t$ depends only on $Z_t$. The state-transition assumption and this emission assumption are separate parts of the model.<Speech and Language Processing by Daniel Jurafsky and James H. Martin>[appendix A.2, pp. 2–3, equations A.4–A.5]{ref}
For a time-homogeneous model with a finite hidden <state space> and discrete observations, write $\pi_i=P(Z_1=i)$, $a_{ij}=P(Z_{t+1}=j\mid Z_t=i)$ and $b_i(o)=P(O_t=o\mid Z_t=i)$ for the <emission probabilities>. The joint <probability mass function> factors as
$$
P(z_{1:T},o_{1:T})=\pi_{z_1}b_{z_1}(o_1)\prod_{t=2}^Ta_{z_{t-1},z_t}b_{z_t}(o_t).
$$
For continuous observations, appropriate <probability density functions> replace the emission <probability mass functions>.
For example, a hidden state can describe whether a machine is working normally or has a fault, while a noisy <sensor> emits a positive or negative signal. Both states may emit a positive signal, with different <probabilities>; one observation therefore need not identify the state. Summing the factorization over hidden paths gives the observation <likelihood>. The <forward algorithm> reuses partial sums to compute it, and the <backward algorithm> helps infer hidden-state <posterior probabilities>. The observations alone need not satisfy the <Markov property>.
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