# What is the role of self loop in Hidden Markov Models(HMM)?

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.

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Sep 23, 2022 at 12:54
• Are you using three-state models for phonemes? Sep 23, 2022 at 14:09
• @JaumeOliverLafont Yes, that's right. Sep 24, 2022 at 13:43

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).

• 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? Sep 28, 2022 at 9:11
• 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 $a_{31}>0$. Sep 28, 2022 at 21:11
• Got your point. Thanks! Sep 30, 2022 at 6:06