KL Divergence

For discrete distributions p and q over the same outcomes, KL divergence is D_KL(p||q) = sum over outcomes p(x) log(p(x)/q(x)). It compares q with a reference p, weighting each log ratio by how often the outcome occurs under p. In information theory, it can be interpreted as expected extra coding cost when a code is optimized for q but the true distribution is p. The log base sets the unit: natural logarithms yield nats and base two yields bits. KL divergence is nonnegative and equals zero when the distributions match on outcomes with positive probability under p, subject to support considerations. It is asymmetric: D_KL(p||q) is generally not D_KL(q||p). This makes it unsuitable as a conventional geometric distance. The direction matters in applications such as variational inference, where one distribution is treated as a target and another as an approximation. In the fair-versus-biased coin example, p=(0.5,0.5) and q=(0.9,0.1). The bit-valued contributions are 0.5 log2(0.5/0.9), about -0.424, and 0.5 log2(0.5/0.1), about 1.161. Their sum is about 0.737 bits. Negative individual terms are allowed even though the total divergence is nonnegative. Reversing p and q produces a different value because the weights also change. Support mismatches matter. If p assigns positive probability to an outcome where q assigns zero, the corresponding forward divergence is infinite. If both assign zero, that outcome contributes zero by convention in the discrete sum. For continuous distributions, KL is written as an integral of the log density ratio under P; finiteness requires P to be absolutely continuous with respect to Q. SciPy's entropy function computes Shannon entropy with one distribution and relative entropy when q is provided; check input normalization and direction. KL is useful for comparing distributions, but it does not by itself say whether a model is useful for a downstream decision.