Search results
Results from the WOW.Com Content Network
In mathematical statistics, the Kullback–Leibler (KL) divergence (also called relative entropy and I-divergence [1]), denoted (), is a type of statistical distance: a measure of how much a model probability distribution Q is different from a true probability distribution P.
where is the Kullback–Leibler divergence, and is the outer product distribution which assigns probability () to each (,).. Notice, as per property of the Kullback–Leibler divergence, that (;) is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent (and hence observing tells you nothing about ).
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.
Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection is the "closest" distribution to q of all the distributions in P. The I-projection is useful in setting up information geometry , notably because of the following inequality, valid when P is convex: [ 1 ]
The Rényi divergence is indeed a divergence, meaning simply that (‖) is greater than or equal to zero, and zero only when P = Q. For any fixed distributions P and Q , the Rényi divergence is nondecreasing as a function of its order α , and it is continuous on the set of α for which it is finite, [ 13 ] or for the sake of brevity, the ...
Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables. Let M {\displaystyle M} be a measurable subset of Ω , {\displaystyle \Omega ,} (i.e. M ∈ F , {\displaystyle M\in {\mathcal {F}},} ) and let x ∈ s u p p X . {\displaystyle x\in \mathrm {supp} \,X.}
In probability theory, an -divergence is a certain type of function (‖) that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence , Hellinger distance , and total variation distance , are special cases of f {\displaystyle f} -divergence.
The actual log-likelihood may be higher (indicating an even better fit to the distribution) because the ELBO includes a Kullback-Leibler divergence (KL divergence) term which decreases the ELBO due to an internal part of the model being inaccurate despite good fit of the model overall.