Editing Hidden Markov model (section)

== Examples ==
=== Drawing balls from hidden urns ===

[[File:HiddenMarkovModel.svg|right|thumb|300px|
Figure 1. Probabilistic parameters of a hidden Markov model (example)<br/>
''X'' — states<br/>
''y'' — possible observations<br/>
''a'' — state transition probabilities<br/>
''b'' — output probabilities]]
In its discrete form, a hidden Markov process can be visualized as a generalization of the [[urn problem]] with replacement (where each item from the urn is returned to the original urn before the next step).<ref>{{cite journal |author=Lawrence R. Rabiner |author-link=Lawrence Rabiner |date=February 1989 |title=A tutorial on Hidden Markov Models and selected applications in speech recognition |url=http://www.ece.ucsb.edu/Faculty/Rabiner/ece259/Reprints/tutorial%20on%20hmm%20and%20applications.pdf |journal=Proceedings of the IEEE |volume=77 |issue=2 |pages=257–286 |citeseerx=10.1.1.381.3454 |doi=10.1109/5.18626 |s2cid=13618539}} [http://www.cs.cornell.edu/courses/cs481/2004fa/rabiner.pdf]</ref> Consider this example: in a room that is not visible to an observer there is a genie. The room contains urns X1, X2, X3, ... each of which contains a known mix of balls, with each ball having a unique label y1, y2, y3, ... . The genie chooses an urn in that room and randomly draws a ball from that urn. It then puts the ball onto a conveyor belt, where the observer can observe the sequence of the balls but not the sequence of urns from which they were drawn. The genie has some procedure to choose urns; the choice of the urn for the ''n''-th ball depends only upon a random number and the choice of the urn for the {{nowrap|(''n'' − 1)-th}} ball. The choice of urn does not directly depend on the urns chosen before this single previous urn; therefore, this is called a [[Markov process]]. It can be described by the upper part of Figure 1.

The Markov process cannot be observed, only the sequence of labeled balls, thus this arrangement is called a ''hidden Markov process''. This is illustrated by the lower part of the diagram shown in Figure 1, where one can see that balls y1, y2, y3, y4 can be drawn at each state. Even if the observer knows the composition of the urns and has just observed a sequence of three balls, ''e.g.'' y1, y2 and y3 on the conveyor belt, the observer still cannot be ''sure'' which urn (''i.e.'', at which state) the genie has drawn the third ball from. However, the observer can work out other information, such as the likelihood that the third ball came from each of the urns.

=== Weather guessing game ===
Consider two friends, Alice and Bob, who live far apart from each other and who talk together daily over the telephone about what they did that day. Bob is only interested in three activities: walking in the park, shopping, and cleaning his apartment. The choice of what to do is determined exclusively by the weather on a given day. Alice has no definite information about the weather, but she knows general trends. Based on what Bob tells her he did each day, Alice tries to guess what the weather must have been like.

Alice believes that the weather operates as a discrete [[Markov chain]]. There are two states, "Rainy" and "Sunny", but she cannot observe them directly, that is, they are ''hidden'' from her. On each day, there is a certain chance that Bob will perform one of the following activities, depending on the weather: "walk", "shop", or "clean". Since Bob tells Alice about his activities, those are the ''observations''. The entire system is that of a hidden Markov model (HMM).

Alice knows the general weather trends in the area, and what Bob likes to do on average. In other words, the parameters of the HMM are known. They can be represented as follows in [[Python programming language|Python]]:
<syntaxhighlight lang="python">
states = ("Rainy", "Sunny")

observations = ("walk", "shop", "clean")

start_probability = {"Rainy": 0.6, "Sunny": 0.4}

transition_probability = {
    "Rainy": {"Rainy": 0.7, "Sunny": 0.3},
    "Sunny": {"Rainy": 0.4, "Sunny": 0.6},
}

emission_probability = {
    "Rainy": {"walk": 0.1, "shop": 0.4, "clean": 0.5},
    "Sunny": {"walk": 0.6, "shop": 0.3, "clean": 0.1},
}
</syntaxhighlight>
In this piece of code, <code>start_probability</code> represents Alice's belief about which state the HMM is in when Bob first calls her (all she knows is that it tends to be rainy on average). The particular probability distribution used here is not the equilibrium one, which is (given the transition probabilities) approximately <code>{'Rainy': 0.57, 'Sunny': 0.43}</code>. The <code>transition_probability</code> represents the change of the weather in the underlying Markov chain. In this example, there is only a 30% chance that tomorrow will be sunny if today is rainy. The <code>emission_probability</code> represents how likely Bob is to perform a certain activity on each day. If it is rainy, there is a 50% chance that he is cleaning his apartment; if it is sunny, there is a 60% chance that he is outside for a walk.

[[File:HMMGraph.svg|border|center|400px|Graphical representation of the given HMM]]

''A similar example is further elaborated in the [[Viterbi algorithm#Example|Viterbi algorithm]] page.''