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docs: minor fixes of LM rescoring texts (#1498)

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docs/source/decoding-with-langugage-models/LODR.rst
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@ -30,7 +30,7 @@ of langugae model integration.
First, let's have a look at some background information. As the predecessor of LODR, Density Ratio (DR) is first proposed `here <https://arxiv.org/abs/2002.11268>`_ First, let's have a look at some background information. As the predecessor of LODR, Density Ratio (DR) is first proposed `here <https://arxiv.org/abs/2002.11268>`_
to address the language information mismatch between the training to address the language information mismatch between the training
corpus (source domain) and the testing corpus (target domain). Assuming that the source domain and the test domain corpus (source domain) and the testing corpus (target domain). Assuming that the source domain and the test domain
are acoustically similar, DR derives the following formular for decoding with Bayes' theorem: are acoustically similar, DR derives the following formula for decoding with Bayes' theorem:
.. math:: .. math::
@ -41,7 +41,7 @@ are acoustically similar, DR derives the following formular for decoding with Ba
where :math:`\lambda_1` and :math:`\lambda_2` are the weights of LM scores for target domain and source domain respectively. where :math:`\lambda_1` and :math:`\lambda_2` are the weights of LM scores for target domain and source domain respectively.
Here, the source domain LM is trained on the training corpus. The only difference in the above formular compared to Here, the source domain LM is trained on the training corpus. The only difference in the above formula compared to
shallow fusion is the subtraction of the source domain LM. shallow fusion is the subtraction of the source domain LM.
Some works treat the predictor and the joiner of the neural transducer as its internal LM. However, the LM is Some works treat the predictor and the joiner of the neural transducer as its internal LM. However, the LM is
@ -58,7 +58,7 @@ during decoding for transducer model:
In LODR, an additional bi-gram LM estimated on the source domain (e.g training corpus) is required. Compared to DR, In LODR, an additional bi-gram LM estimated on the source domain (e.g training corpus) is required. Compared to DR,
the only difference lies in the choice of source domain LM. According to the original `paper <https://arxiv.org/abs/2203.16776>`_, the only difference lies in the choice of source domain LM. According to the original `paper <https://arxiv.org/abs/2203.16776>`_,
LODR achieves similar performance compared DR in both intra-domain and cross-domain settings. LODR achieves similar performance compared to DR in both intra-domain and cross-domain settings.
As a bi-gram is much faster to evaluate, LODR is usually much faster. As a bi-gram is much faster to evaluate, LODR is usually much faster.
Now, we will show you how to use LODR in ``icefall``. Now, we will show you how to use LODR in ``icefall``.

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docs/source/decoding-with-langugage-models/shallow-fusion.rst
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@ -139,7 +139,7 @@ A few parameters can be tuned to further boost the performance of shallow fusion
- ``--lm-scale`` - ``--lm-scale``
Controls the scale of the LM. If too small, the external language model may not be fully utilized; if too large, Controls the scale of the LM. If too small, the external language model may not be fully utilized; if too large,
the LM score may dominant during decoding, leading to bad WER. A typical value of this is around 0.3. the LM score might be dominant during decoding, leading to bad WER. A typical value of this is around 0.3.
- ``--beam-size`` - ``--beam-size``