TLDR: NLP metric ranging from 1 to infinity. Lower is better.

In natural language processing, perplexity is the most common metric used to measure the performance of a language model. To calculate perplexity, we use the following formula:

$perplexity = e^z$

where

$z = -{1 \over N} \sum_{i=0}^N ln(P_{n})$

Typically we use base e when calculating perplexity, but this is not required. Any base will do, so sometimes the formula will use base 2 or base 10, along with logarithms to the corresponding base.

## Example

Imagine that we have a language model which generates the following sequence of tokens:

<start> jack and jill went up the hill

And suppose that the conditional probabilities for each of the tokens are as follows:

token probability
<start> 15%
jack 5%
and 12%
jill 18%
went 25%
up 40%
the 33%
hill 50%

For the purposes of calculating perplexity it doesn’t matter how the sequence was generated. It may be using an n-gram model or an LSTM or a transformer. All that matters is the probabilities the model assigns to each of the tokens. To calculate perplexity, we calculate the logarithm of each of the values above:

token P ln(P)
<start> 15% -1.897
jack 5% -2.996
and 12% -2.120
jill 18% -1.715
went 25% -1.386
up 40% -0.916
the 33% -1.109
hill 50% -0.693

Summing the logs, we get -12.832. Since there are 8 tokens, we divide -12.832 by 8 to get -1.604. Negating that allows us to calculate the final perplexity:

$perplexity = e^{1.604} = 4.973$

Therefore the perplexity of this sequence is about 4.973.