Cross entropy

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In information theory, the cross entropy between two probability distributions and over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set, if a coding scheme is used that is optimized for an "unnatural" probability distribution , rather than the "true" distribution .

The cross entropy for the distributions and over a given set is defined as follows:

where is the entropy of , and is the Kullback–Leibler divergence of from (also known as the relative entropy of p with respect to q — note the reversal of emphasis).

For discrete and this means

The situation for continuous distributions is analogous. We have to assume that and are absolutely continuous with respect to some reference measure (usually is a Lebesgue measure on a Borel σ-algebra). Let and be probability density functions of and with respect to . Then

NB: The notation is also used for a different concept, the joint entropy of and .

Motivation[edit]

In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value out of a set of possibilities can be seen as representing an implicit probability distribution over , where is the length of the code for in bits. Therefore, cross entropy can be interpreted as the expected message-length per datum when a wrong distribution is assumed while the data actually follows a distribution . That is why the expectation is taken over the probability distribution and not .

Estimation[edit]

There are many situations where cross-entropy needs to be measured but the distribution of is unknown. An example is language modeling, where a model is created based on a training set , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, is the true distribution of words in any corpus, and is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula:

where is the size of the test set, and is the probability of event estimated from the training set. The sum is calculated over . This is a Monte Carlo estimate of the true cross entropy, where the training set is treated as samples from .

Cross-entropy minimization[edit]

Cross-entropy minimization is frequently used in optimization and rare-event probability estimation; see the cross-entropy method.

When comparing a distribution against a fixed reference distribution , cross entropy and KL divergence are identical up to an additive constant (since is fixed): both take on their minimal values when , which is for KL divergence, and for cross entropy. In the engineering literature, the principle of minimising KL Divergence (Kullback's "Principle of Minimum Discrimination Information") is often called the Principle of Minimum Cross-Entropy (MCE), or Minxent.

However, as discussed in the article Kullback–Leibler divergence, sometimes the distribution is the fixed prior reference distribution, and the distribution is optimised to be as close to as possible, subject to some constraint. In this case the two minimisations are not equivalent. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by redefining cross-entropy to be , rather than .

Cross-entropy error function and logistic regression[edit]

Cross entropy can be used to define the loss function in machine learning and optimization. The true probability is the true label, and the given distribution is the predicted value of the current model.

More specifically, let us consider logistic regression, which (in its most basic form) deals with classifying a given set of data points into two possible classes generically labelled and . The logistic regression model thus predicts an output , given an input vector . The probability is modeled using the logistic function . Namely, the probability of finding the output is given by

where the vector of weights is optimized through some appropriate algorithm such as gradient descent. Similarly, the complementary probability of finding the output is simply given by

The true (observed) probabilities can be expressed similarly as and .


Having set up our notation, and , we can use cross entropy to get a measure for similarity between and :

The typical loss function that one uses in logistic regression is computed by taking the average of all cross-entropies in the sample. For example, suppose we have samples with each sample labeled by . The loss function is then given by:

where , with the logistic function as before.


The logistic loss is sometimes called cross-entropy loss. It's also known as log loss (In this case, the binary label is often denoted by {-1,+1}).[1]

Notes[edit]

  1. ^ Murphy, Kevin (2012). Machine Learning: A Probabilistic Perspective. MIT. ISBN 978-0262018029. 

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