add LODR

docs/source/decoding-with-langugage-models/LODR.rst

@@ -2,3 +2,99 @@
 
 LODR for RNN Transducer
 =======================
+
+
+As a type of E2E model, neural transducers are usually considered as having an internal 
+language model, which learns the language level information on the training corpus. 
+In real-life scenario, there is often a mismatch between the training corpus and the target corpus space. 
+This mismatch can be a problem when decoding for neural transducer models with language models as its internal
+language can act "against" the external LM. In this tutorial, we show how to use
+`Low-order Density Ratio <>`_ to alleviate this effect to further improve the performance
+of langugae model integration. 
+
+.. note::
+
+    This tutorial is based on the recipe 
+    `pruned_transducer_stateless7_streaming <https://github.com/k2-fsa/icefall/tree/master/egs/librispeech/ASR/pruned_transducer_stateless7_streaming>`_,
+    which is a streaming transducer model trained on `LibriSpeech`_. 
+    However, you can easily apply shallow fusion to other recipes.
+    If you encounter any problems, please open an issue here `icefall <https://github.com/k2-fsa/icefall/issues>`_.
+
+
+.. note::
+
+    For simplicity, the training and testing corpus in this tutorial is the same (`LibriSpeech`_). However, 
+    you can change the testing set to any other domains (e.g GigaSpeech) and prepare the language models 
+    using that corpus.
+
+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
+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:
+
+.. math::
+
+    \text{score}\left(y_u|\mathit{x},y\right) = 
+    \log p\left(y_u|\mathit{x},y_{1:u-1}\right) + 
+    \lambda_1 \log p_{\text{source LM}}\left(y_u|\mathit{x},y_{1:u-1}\right) - 
+    \lambda_2 \log p_{\text{target LM}}\left(y_u|\mathit{x},y_{1:u-1}\right)
+
+
+where :math:`\lambda_1` and :math:`\lambda_2` are the LM score for source domain and target domain respectively. 
+Here, the source domain LM is trained on the training corpus. The only difference in the above formular compared to 
+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 
+considered to be weak and can only capture low-level language information. Therefore, `LODR <https://arxiv.org/abs/2203.16776>`_ propose to use
+a low-order n-gram LM as an approximation of the ILM of the neural transducer. This leads to the following formula
+during decoding for RNNT model:
+
+.. math::
+
+    \text{score}\left(y_u|\mathit{x},y\right) = 
+    \log p_{rnnt}\left(y_u|\mathit{x},y_{1:u-1}\right) + 
+    \lambda_1 \log p_{\text{LM}}\left(y_u|\mathit{x},y_{1:u-1}\right) - 
+    \lambda_2 \log p_{\text{bi-gram}}\left(y_u|\mathit{x},y_{1:u-1}\right)
+
+In LODR, an additional bi-gram LM estimated on the training corpus is required apart from the neural LM. Comared to DR, 
+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. As a bi-gram is much faster to evaluate, LODR
+is a suitable decoding method for faster inference.
+
+
+Now, we will show you how to use LODR in ``icefall``.
+For illustration purpose, we will use a pre-trained ASR model from this `link <https://huggingface.co/Zengwei/icefall-asr-librispeech-pruned-transducer-stateless7-streaming-2022-12-29>`_.
+If you want to train your model from scratch, please have a look at :ref:`non_streaming_librispeech_pruned_transducer_stateless`.
+
+As the initial step, let's download the pre-trained model.
+
+.. code-block:: bash
+
+    $ git lfs install
+    $ git clone https://huggingface.co/Zengwei/icefall-asr-librispeech-pruned-transducer-stateless7-streaming-2022-12-29
+    $ pushd icefall-asr-librispeech-pruned-transducer-stateless7-streaming-2022-12-29/exp
+    $ ln -s pretrained.pt epoch-99.pt # create a symbolic link so that the checkpoint can be loaded
+
+To test the model, let's have a look at the decoding results without using LM. This can be done via the following command:
+
+.. code-block:: bash
+
+    $ exp_dir=./icefall-asr-librispeech-pruned-transducer-stateless7-streaming-2022-12-29/exp/
+    $ ./pruned_transducer_stateless7_streaming/decode.py \
+        --epoch 30 \
+        --avg 9 \
+        --exp-dir $exp_dir \
+        --max-duration 600 \
+        --decode-chunk-len 32 \
+        --decoding-method modified_beam_search
+
+The following WERs are achieved on test-clean and test-other:
+
+.. code-block:: bash
+
+    $ For test-clean, WER of different settings are:
+    $ beam_size_4	3.11	best for test-clean
+    $ For test-other, WER of different settings are:
+    $ beam_size_4	7.93	best for test-other
+
+