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@@ -165,6 +165,7 @@ conversion utilities for the following models:
     :caption: Research
 
     bertology
+    perplexity
     benchmarks
 
 .. toctree::
diff --git a/docs/source/perplexity.rst b/docs/source/perplexity.rst
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+Perplexity of fixed-length models
+=================================
+
+Perplexity (PPL) is one of the most common metrics for evaluating language
+models. Before diving in, we should note that the metric applies specifically
+to classical language models (sometimes called autoregressive or causal
+language models) and is not well defined for masked language models like BERT
+(see :doc:`summary of the models <model_summary>`).
+
+Perplexity is defined as the exponentiated average log-likelihood of a
+sequence. If we have a tokenized sequence :math:`X = (x_0, x_1, \dots, x_t)`,
+then the perplexity of :math:`X` is,
+
+.. math::
+
+    \text{PPL}(X)
+    = \exp \left\{ {-\frac{1}{t}\sum_i^t \log p_\theta (x_i|x_{<i}) } \right\}
+
+where :math:`\log p_\theta (x_i|x_{<i})` is the log-likelihood of the ith
+token conditioned on the preceding tokens :math:`x_{<i}` according to our
+model. Intuitively, it can be thought of as an evaluation of the model's
+ability to predict uniformly among the set of specified tokens in a corpus.
+Importantly, this means that the tokenization procedure has a direct impact
+on a model's perplexity which should always be taken into consideration when
+comparing different models.
+
+This is also equivalent to the exponentiation of the cross-entropy between
+the data and model predictions. For more intuition about perplexity and its
+relationship to Bits Per Character (BPC) and data compression, check out this
+`fantastic blog post on The Gradient
+<https://thegradient.pub/understanding-evaluation-metrics-for-language-models/>`_.
+
+Calculating PPL with fixed-length models
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+If we weren't limited by a model's context size, we would evaluate the
+model's perplexity by autoregressively factorizing a sequence and
+conditioning on the entire preceding subsequence at each step, as shown
+below.
+
+.. image:: imgs/ppl_full.gif
+    :width: 600
+    :alt: Full decomposition of a sequence with unlimited context length
+
+When working with approximate models, however, we typically have a constraint
+on the number of tokens the model can process. The largest version
+of :doc:`GPT-2 <model_doc/gpt2>`, for example, has a fixed length of 1024
+tokens, so we cannot calculate :math:`p_\theta(x_t|x_{<t})` directly when
+:math:`t` is greater than 1024.
+
+Instead, the sequence is typically broken into subsequences equal to the
+model's maximum input size. If a model's max input size is :math:`k`, we
+then approximate the likelihood of a token :math:`x_t` by conditioning only
+on the :math:`k-1` tokens that precede it rather than the entire context.
+When evaluating the model's perplexity of a sequence, a tempting but
+suboptimal approach is to break the sequence into disjoint chunks and
+add up the decomposed log-likelihoods of each segment independently.
+
+.. image:: imgs/ppl_chunked.gif
+    :width: 600
+    :alt: Suboptimal PPL not taking advantage of full available context
+
+This is quick to compute since the perplexity of each segment can be computed
+in one forward pass, but serves as a poor approximation of the
+fully-factorized perplexity and will typically yield a higher (worse) PPL
+because the model will have less context at most of the prediction steps.
+
+Instead, the PPL of fixed-length models should be evaluated with a
+sliding-window strategy. This involves repeatedly sliding the
+context window so that the model has more context when making each
+prediction.
+
+.. image:: imgs/ppl_sliding.gif
+    :width: 600
+    :alt: Sliding window PPL taking advantage of all available context
+
+This is a closer approximation to the true decomposition of the
+sequence probability and will typically yield a more favorable score.
+The downside is that it requires a separate forward pass for each token in
+the corpus. A good practical compromise is to employ a strided sliding
+window, moving the context by larger strides rather than sliding by 1 token a
+time. This allows computation to procede much faster while still giving the
+model a large context to make predictions at each step.
+
+Example: Calculating perplexity with GPT-2 in 🤗 Transformers
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Let's demonstrate this process with GPT-2.
+
+.. code-block:: python
+
+    from transformers import GPT2LMHeadModel, GPT2TokenizerFast
+    device = 'cuda'
+    model_id = 'gpt2-large'
+    model = GPT2LMHeadModel.from_pretrained(model_id).to(device)
+    tokenizer = GPT2TokenizerFast.from_pretrained(model_id)
+
+We'll load in the WikiText-2 dataset and evaluate the perplexity using a few
+different sliding-window strategies. Since this dataset is small and we're
+just doing one forward pass over the set, we can just load and encode the
+entire dataset in memory.
+
+.. code-block:: python
+
+    from nlp import load_dataset
+    test = load_dataset('wikitext', 'wikitext-2-raw-v1', split='test')
+    encodings = tokenizer('\n\n'.join(test['text']), return_tensors='pt')
+
+With 🤗 Transformers, we can simply pass the ``input_ids`` as the ``labels``
+to our model, and the average log-likelihood for each token is returned as
+the loss. With our sliding window approach, however, there is overlap in the
+tokens we pass to the model at each iteration. We don't want the
+log-likelihood for the tokens we're just treating as context to be included
+in our loss, so we can set these targets to ``-100`` so that they are
+ignored. The following is an example of how we could do this with a stride of
+``512``. This means that the model will have at least 512 tokens for context
+when calculating the conditional likelihood of any one token (provided there
+are 512 preceding tokens available to condition on).
+
+.. code-block:: python
+
+    max_length = model.config.n_positions
+    stride = 512
+
+    lls = []
+    for i in tqdm(range(1, encodings.input_ids.size(1), stride)):
+        begin_loc = max(i + stride - max_length, 0)
+        end_loc = i + stride
+        input_ids = encodings.input_ids[:,begin_loc:end_loc].to(device)
+        target_ids = input_ids.clone()
+        target_ids[:,:-stride] = -100
+
+        with torch.no_grad():
+            outputs = model(input_ids, labels=target_ids)
+            log_likelihood = outputs[0] * stride
+
+        lls.append(log_likelihood)
+    
+    ppl = torch.exp(torch.stack(lls).sum() / i)
+
+Running this with the stride length equal to the max input length is
+equivalent to the suboptimal, non-sliding-window strategy we discussed above.
+The smaller the stride, the more context the model will have in making each
+prediction, and the better the reported perplexity will typically be.
+
+When we run the above with ``stride = 1024``, i.e. no overlap, the resulting
+PPL is ``19.64``, which is about the same as the ``19.93`` reported in the
+GPT-2 paper. By using ``stride = 512`` and thereby employing our striding
+window strategy, this jumps down to ``16.53``. This is not only a more
+favorable score, but is calculated in a way that is closer to the true
+autoregressive decomposition of a sequence likelihood.