In information theory, perplexity is a measurement of how well a probability distribution or probability model predicts a sample. It may be used to compare probability models. A low perplexity indicates the probability distribution is good at predicting the sample.
Perplexity of a Probability Distribution The perplexity PP of a discrete probability distribution p is a concept widely used in information theory, machine learning, and statistical modeling. It is defined as
P P ( p ) := 2 H ( p ) = 2 − ∑ x p ( x ) log 2 p ( x ) = ∏ x p ( x ) − p ( x ) {\displaystyle {\mathit {PP}}(p):=2^{H(p)}=2^{-\sum _{x}p(x)\log _{2}p(x)}=\prod _{x}p(x)^{-p(x)}}
where H(p) is the entropy (in bits) of the distribution, and x ranges over the events. The base of the logarithm need not be 2: The perplexity is independent of the base, provided that the entropy and the exponentiation use the same base. In some contexts, this measure is also referred to as the (order-1 true) diversity. Perplexity of a random variable X may be defined as the perplexity of the distribution over its possible values x. It can be thought of as a measure of uncertainty or "surprise" related to the outcomes. In the special case where p models a fair k-sided die (a uniform distribution over k discrete events), its perplexity is k. A random variable with perplexity k has the same uncertainty as a fair k-sided die. One is said to be "k-ways perplexed" about the value of the random variable. Unless it is a fair k-sided die, more than k values may be possible, but the overall uncertainty is not greater because some values may have a probability greater than 1/k. Perplexity is sometimes used as a measure of the difficulty of a prediction problem. However, it's not always an accurate representation. For example, if you have two choices, one with probability 0.9, your chances of a correct guess using the optimal strategy are 90 percent. Yet, the perplexity is 2−0.9 log2 0.9 - 0.1 log2 0.1= 1.38. The inverse of the perplexity, 1/1.38 = 0.72, does not correspond to the 0.9 probability. The perplexity is the exponentiation of the entropy, a more straightforward quantity. Entropy measures the expected or "average" number of bits required to encode the outcome of the random variable using an optimal variable-length code. It can also be regarded as the expected information gain from learning the outcome of the random variable, providing insight into the uncertainty and complexity of the underlying probability distribution.
Perplexity of a probability model A model of an unknown probability distribution p, may be proposed based on a training sample that was drawn from p. Given a proposed probability model q, one may evaluate q by asking how well it predicts a separate test sample x1, x2, ..., xN also drawn from p. The perplexity of the model q is defined as
b − 1 N ∑ i = 1 N log b q ( x i ) = ( ∏ i q ( x i ) ) − 1 / N {\displaystyle b^{-{\frac {1}{N}}\sum _{i=1}^{N}\log _{b}q(x_{i})}=\left(\prod _{i}q(x_{i})\right)^{-1/N}}
where b {\displaystyle b} is customarily 2. Better models q of the unknown distribution p will tend to assign higher probabilities q(xi) to the test events. Thus, they have lower perplexity: they are less surprised by the test sample. The exponent above may be regarded as the average number of bits needed to represent a test event xi if one uses an optimal code based on q. Low-perplexity models do a better job of compressing the test sample, requiring few bits per test element on average because q(xi) tends to be high. The exponent − 1 N ∑ i = 1 N log b q ( x i ) {\displaystyle -{\frac {1}{N}}\sum _{i=1}^{N}\log _{b}q(x_{i})} may also be interpreted as a cross-entropy:
H ( p ~ , q ) = − ∑ x p ~ ( x ) log b q ( x ) {\displaystyle H({\tilde {p}},q)=-\sum _{x}{\tilde {p}}(x)\log _{b}q(x)}
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