Answers.Body: <p>The input audio is splitted into overlapping frames, for instance having size 40ms at a frame-rate of 20ms. For every frame <span class="math-container">$t$</span>, some feature vector <span class="math-container">$O_t$</span> is observed. An input utterance with three frames would be represented as the sequence <span class="math-container">$O_1O_2O_3$</span>.</p>
  <p>Consider a model with no self-loops, a left-right model with three states and no skips either, with states <span class="math-container">$S=\{S_1,S_2,S_3\}$</span>. Having no skips means that the transition from <span class="math-container">$S_1$</span> to <span class="math-container">$S_3$</span> is not allowed, it has probability <span class="math-container">$a_{13}=0$</span>. This rigid model would be able to generate or recognize only observations of length <span class="math-container">$T=3$</span>, and would necessarily assign the state <span class="math-container">$S_i$</span> to the observation <span class="math-container">$O_i$</span>. There is no other possible alignment.</p>
  <p>By allowing a self-transition in the central state we give temporal flexibility to the model, so different length utterances can be properly aligned and recognized. An observation of length <span class="math-container">$T=5$</span>, for example, would be written <span class="math-container">$O_1O_2O_3O_4O_5$</span>, and the model could generate <span class="math-container">$O_1$</span> at state <span class="math-container">$S_1$</span>; <span class="math-container">$O_2$</span>, <span class="math-container">$O_3$</span> and <span class="math-container">$O_4$</span> at <span class="math-container">$S_2$</span>; and <span class="math-container">$O_5$</span> at <span class="math-container">$S_3$</span>.</p>
  <p>This self-transition models the number of observations in the second state (the duration <span class="math-container">$d_2$</span>) with an exponential distribution of average <span class="math-container">$$\overline{d_2} = \frac{1}{1-a_{22}},$$</span> where <span class="math-container">$a_{22}$</span> is the probability of remaining in <span class="math-container">$S_2$</span>. Note the extreme values <span class="math-container">$a_{22}=0$</span>, giving expected duration <span class="math-container">$1$</span>; and <span class="math-container">$\overline{d_2}$</span> growing unbounded as <span class="math-container">$a_{22}$</span> approaches <span class="math-container">$1$</span>.</p>
  <p>The symbols and formula used here come from the <a href="https://www.cs.cmu.edu/%7Ecga/behavior/rabiner1.pdf" rel="nofollow noreferrer">tutorial by Rabiner (1989)</a>.</p>
  
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Answers.CreationDate: 2022-09-28T09:11:22.670Z
Answers.Text: Thanks for the proper explanations, @JaumeOliverLafont, but could you explain the scenario where a22 tends to infinity? Would the audio input having the same phonemes be one of the scenarios?
Answers.UserDisplayName: Pam Cesar
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-09-28T21:11:55.687Z
Answers.Text: If you are modeling one phoneme with three states, a large $a_{22}$ means a long version of the phoneme, such as... in a song, for instance.
  Multiple repetitions of the same phoneme can be modeled with a transition from $S_3$ to $S_1$ with $a_{31}>0$.
Answers.UserDisplayName: Jaume Oliver Lafont
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-09-26T21:00:26.197Z
Answers.LastActivityDate: 2022-09-26T21:00:26.197Z
Answers.UserDisplayName: Jaume Oliver Lafont
Body: <p>What actually does the self-loop (within the single hidden state ) in the Hidden Markov model helpful for?</p>
  <p>I learn that one of the use cases concerning Natural language Understanding is that it helps a model to stay within the current state in case of (time variable: long sound/ short sound) pronunciation of the same word. But I can't understand what role the self-loop plays here.</p>
  <p>Any explanations would be much appreciated.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-09-23T12:54:10.583Z
Comments.Text: Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
Comments.UserDisplayName: Community
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-09-23T14:09:33.373Z
Comments.Text: Are you using three-state models for phonemes?
Comments.UserDisplayName: Jaume Oliver Lafont
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-09-24T13:43:01.133Z
Comments.Text: @JaumeOliverLafont Yes, that's right.
Comments.UserDisplayName: Pam Cesar
Tags: <machine-learning><natural-language-processing><pattern-recognition><natural-language-understanding><hidden-markov-model>
Title: What is the role of self loop in Hidden Markov Models(HMM)?
UserDisplayName: Pam Cesar
ThreadText: # What is the role of self loop in Hidden Markov Models(HMM)?
  
  (Asked by: username_0 on 2022-09-23)
  
  What actually does the self-loop (within the single hidden state ) in the Hidden Markov model helpful for?
  
  I learn that one of the use cases concerning Natural language Understanding is that it helps a model to stay within the current state in case of (time variable: long sound/ short sound) pronunciation of the same word. But I can't understand what role the self-loop plays here.
  
  Any explanations would be much appreciated.
  
  **Tags:** machine-learning, natural-language-processing, pattern-recognition, natural-language-understanding, hidden-markov-model
  > **Comments:**
  > **username_1**: Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.  
  > **username_2**: Are you using three-state models for phonemes?  
  > **username_0**: @JaumeOliverLafont Yes, that's right.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2022-09-26. Score: 0 | Accepted answer
  
  The input audio is splitted into overlapping frames, for instance having size 40ms at a frame-rate of 20ms. For every frame $t$, some feature vector $O\_t$ is observed. An input utterance with three frames would be represented as the sequence $O\_1O\_2O\_3$.
  
  Consider a model with no self-loops, a left-right model with three states and no skips either, with states $S=\{S\_1,S\_2,S\_3\}$. Having no skips means that the transition from $S\_1$ to $S\_3$ is not allowed, it has probability $a\_{13}=0$. This rigid model would be able to generate or recognize only observations of length $T=3$, and would necessarily assign the state $S\_i$ to the observation $O\_i$. There is no other possible alignment.
  
  By allowing a self-transition in the central state we give temporal flexibility to the model, so different length utterances can be properly aligned and recognized. An observation of length $T=5$, for example, would be written $O\_1O\_2O\_3O\_4O\_5$, and the model could generate $O\_1$ at state $S\_1$; $O\_2$, $O\_3$ and $O\_4$ at $S\_2$; and $O\_5$ at $S\_3$.
  
  This self-transition models the number of observations in the second state (the duration $d\_2$) with an exponential distribution of average $$\overline{d\_2} = \frac{1}{1-a\_{22}},$$ where $a\_{22}$ is the probability of remaining in $S\_2$. Note the extreme values $a\_{22}=0$, giving expected duration $1$; and $\overline{d\_2}$ growing unbounded as $a\_{22}$ approaches $1$.
  
  The symbols and formula used here come from the [tutorial by Rabiner (1989)](https://www.cs.cmu.edu/%7Ecga/behavior/rabiner1.pdf).
  
  > **Comments:**
  > **username_0**: Thanks for the proper explanations, @JaumeOliverLafont, but could you explain the scenario where a22 tends to infinity? Would the audio input having the same phonemes be one of the scenarios?  
  > **username_2**: If you are modeling one phoneme with three states, a large $a_{22}$ means a long version of the phoneme, such as... in a song, for instance. Multiple repetitions of the same phoneme can be modeled with a transition from $S_3$ to $S_1$ with $a_{31}>0$.  
  
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