Editing Viterbi algorithm (section)

== Example ==
A doctor wishes to determine whether patients are healthy or have a fever. The only information the doctor can obtain is by asking patients how they feel. The patients may report that they either feel normal, dizzy, or cold.

It is believed that the health condition of the patients operates as a discrete [[Markov chain]]. There are two states, "healthy" and "fever", but the doctor cannot observe them directly; they are ''hidden'' from the doctor. On each day, the chance that a patient tells the doctor "I feel normal", "I feel cold", or "I feel dizzy", depends only on the patient's health condition on that day.

The ''observations'' (normal, cold, dizzy) along with the ''hidden'' states (healthy, fever) form a hidden Markov model (HMM). From past experience, the probabilities of this model have been estimated as:
<pre>
init = {"Healthy": 0.6, "Fever": 0.4}
trans = {
    "Healthy": {"Healthy": 0.7, "Fever": 0.3},
    "Fever": {"Healthy": 0.4, "Fever": 0.6},
}
emit = {
    "Healthy": {"normal": 0.5, "cold": 0.4, "dizzy": 0.1},
    "Fever": {"normal": 0.1, "cold": 0.3, "dizzy": 0.6},
}
</pre>
In this code, <code>init</code> represents the doctor's belief about how likely the patient is to be healthy initially. Note that the particular probability distribution used here is not the equilibrium one, which would be <code>{'Healthy': 0.57, 'Fever': 0.43}</code> according to the transition probabilities. The transition probabilities <code>trans</code> represent the change of health condition in the underlying Markov chain. In this example, a patient who is healthy today has only a 30% chance of having a fever tomorrow. The emission probabilities <code>emit</code> represent how likely each possible observation (normal, cold, or dizzy) is, given the underlying condition (healthy or fever). A patient who is healthy has a 50% chance of feeling normal; one who has a fever has a 60% chance of feeling dizzy.

[[File:An example of HMM.png|thumb|center|300px|Graphical representation of the given HMM]]

A particular patient visits three days in a row, and reports feeling normal on the first day, cold on the second day, and dizzy on the third day.

Firstly, the probabilities of being healthy or having a fever on the first day are calculated. The probability that a patient will be healthy on the first day and report feeling normal is <math>0.6 \times 0.5 = 0.3</math>. Similarly, the probability that a patient will have a fever on the first day and report feeling normal is <math>0.4 \times 0.1 = 0.04</math>.

The probabilities for each of the following days can be calculated from the previous day directly. For example, the highest chance of being healthy on the second day and reporting to be cold, following reporting being normal on the first day, is the maximum of <math>0.3 \times 0.7 \times 0.4 = 0.084</math> and <math>0.04 \times 0.4 \times 0.4 = 0.0064</math>. This suggests it is more likely that the patient was healthy for both of those days, rather than having a fever and recovering.

The rest of the probabilities are summarised in the following table:

{| class="wikitable"
|-
! Day !! 1 !! 2 !! 3
|-
! Observation
| Normal || Cold || Dizzy
|-
! Healthy
| '''0.3'''|| '''0.084'''|| 0.00588
|-
! Fever
| 0.04 || 0.027 || '''0.01512'''
|}

From the table, it can be seen that the patient most likely had a fever on the third day. Furthermore, there exists a sequence of states ending on "fever", of which the probability of producing the given observations is 0.01512. This sequence is precisely (healthy, healthy, fever), which can be found be tracing back which states were used when calculating the maxima (which happens to be the best guess from each day but will not always be). In other words, given the observed activities, the patient was most likely to have been healthy on the first day and also on the second day (despite feeling cold that day), and only to have contracted a fever on the third day.

The operation of Viterbi's algorithm can be visualized by means of a [[Trellis diagram#Trellis diagram|trellis diagram]]. The Viterbi path is essentially the shortest path through this trellis.