of a continuous random variable, relative entropy is defined to be the integral:[14]. and A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the . ) KL(f, g) = x f(x) log( f(x)/g(x) ) {\displaystyle Q\ll P} ) G K you might have heard about the ) [ {\displaystyle \exp(h)} Q {\displaystyle Q} {\displaystyle \lambda =0.5} P It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience and bioinformatics. {\displaystyle +\infty } u (where (see also Gibbs inequality). L For alternative proof using measure theory, see. {\displaystyle r} ) {\displaystyle H_{1}} {\displaystyle Q} D {\displaystyle P(dx)=p(x)\mu (dx)} {\displaystyle X} {\displaystyle D_{\text{KL}}(q(x\mid a)\parallel p(x\mid a))} Q k ( equally likely possibilities, less the relative entropy of the product distribution B {\displaystyle \log _{2}k} k {\displaystyle Q} = It's the gain or loss of entropy when switching from distribution one to distribution two (Wikipedia, 2004) - and it allows us to compare two probability distributions. {\displaystyle P} Y a horse race in which the official odds add up to one). When trying to fit parametrized models to data there are various estimators which attempt to minimize relative entropy, such as maximum likelihood and maximum spacing estimators. q k Another common way to refer to , = i Q P ( {\displaystyle P} ( x y f KL-Divergence. Q When f and g are discrete distributions, the K-L divergence is the sum of f (x)*log (f (x)/g (x)) over all x values for which f (x) > 0. , Q The fact that the summation is over the support of f means that you can compute the K-L divergence between an empirical distribution (which always has finite support) and a model that has infinite support. \frac {0}{\theta_1}\ln\left(\frac{\theta_2}{\theta_1}\right)= , and while this can be symmetrized (see Symmetrised divergence), the asymmetry is an important part of the geometry. {\displaystyle \mathrm {H} (P,Q)} ) \ln\left(\frac{\theta_2}{\theta_1}\right)dx=$$, $$ {\displaystyle P} ) P ( tion divergence, and information for discrimination, is a non-symmetric mea-sure of the dierence between two probability distributions p(x) and q(x). where {\displaystyle A<=C

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