KullbackLeiblerova

Kullback-Leibler divergence, often abbreviated as KL divergence, is a measure of how one probability distribution diverges from a second, expected probability distribution. It is a fundamental concept in information theory and statistics, named after Solomon Kullback and Richard Leibler who introduced it in 1951. The KL divergence is not a true distance metric because it is not symmetric; that is, the divergence from P to Q is not necessarily the same as the divergence from Q to P. However, it is widely used in various fields such as machine learning, data compression, and pattern recognition.

The KL divergence between two discrete probability distributions P and Q is defined as:

D_KL(P || Q) = Σ P(x) log(P(x) / Q(x))

where the summation is over all possible values of x. For continuous distributions, the summation is replaced by an integral. The KL divergence is always non-negative, and it is zero if and only if P and Q are the same distribution.

One of the key properties of the KL divergence is that it measures the expected number of extra bits required to encode samples from P when using a code based on Q, rather than using a code based on P. This makes it a useful tool for comparing the similarity between two distributions.

In machine learning, the KL divergence is often used as a regularization term in variational autoencoders (VAEs) to ensure that the learned latent space distribution matches a prior distribution, typically a standard normal distribution. It is also used in other contexts such as topic modeling, where it helps to measure the difference between the true topic distribution and the inferred topic distribution.